COLORADO STATE UNIVERSITY March 1, 2001 WE HEREBY RECOMMEND THAT THE THESIS PREPARED UNDER OUR SUPERVISION BY MICHAEL L. ANDERSON ENTITLED MIMO ROBUST

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1 THESIS MIMO ROBUST CONTROL FOR HEATING VENTILATING AND AIR CONDITIONING (HVAC) SYSTEMS Submitted by Michael L. Anderson Department of Electrical and Computer Engineering In partial fulfillment of the requirements for the Degree of Master of Science Colorado State University Fort Collins, Colorado Spring 2001

2 COLORADO STATE UNIVERSITY March 1, 2001 WE HEREBY RECOMMEND THAT THE THESIS PREPARED UNDER OUR SUPERVISION BY MICHAEL L. ANDERSON ENTITLED MIMO ROBUST CONTROL FOR HEATING VENTILATING AND AIR CONDITION- ING (HVAC) SYSTEMS BE ACCEPTED AS FULFILLING IN PART RE- QUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE. Committee on Graduate Work Adviser Department Head ii

3 ABSTRACT OF THESIS MIMO ROBUST CONTROL FOR HEATING VENTILATING AND AIR CONDITIONING (HVAC) SYSTEMS Potential improvements in heating ventilation and air conditioning (HVAC) system performance are investigated through the application of multiple-input, multipleoutput (MIMO) robust controllers. This approach differs dramatically from today's prevalent, if not state of the art, method of building HVAC controllers using multiple single-input, single-output (SISO) controls. Issues important to the application of MIMO robust controllers to HVAC systems are identified and investigated. To aid in the design and testing of controllers, a simulation model of an experimental HVAC system is developed. While simulation can be insightful, the only way to truly verify the performance provided by different HVAC controller designs is by actually using them to control an HVAC system. To verify controller designs, an experimental HVAC system, intended for controlling the discharge air temperature and flow rate, is constructed. This discharge air system (DAS), consisting of external and return air dampers, a variable speed blower and a heating coil, is one of the most basic HVAC system components. While the experimental system is only a portion of an overall HVAC system, it was built using standard HVAC components and is representative of a typical hot water to air, heating system. Central water supplies serving multiple central air supplies (DASs) are utilized in iii

4 conventional HVAC systems. Such an HVAC interconnection restricts the controller design and impacts the performance of the resulting system. A full MIMO controller requires independent control of the temperature and flow rates of both the air and water flowing through the heating coil. Between today's HVAC systems employing SISO based controllers and a system using a full MIMO robust controller, a wide assortment of configurations are possible. To gain insight into potential performance improvements as well as the constraints associated with this breath of controllers, several diverse designs are implemented and tested, both in simulation and on an experimental system. While (simple) optimal and (SISO) robust HVAC controllers have been designed and their performance experimentally verified, this investigation is the first to do so using MIMO robust HVAC controllers. Michael L. Anderson Department of Electrical and Computer Engineering Colorado State University Fort Collins, Colorado Spring 2001 iv

5 ACKNOWLEDGEMENTS I would first like to thank the National Science Foundation for providing funding for this project under grants and I would like to thank Dr. Peter Young (Electrical and Computer Engineering), Dr. Douglas Hittle (Mechanical Engineering), and Dr. Charles Anderson (Computer Science), who conceived of and provided me the opportunity to pursue this project. Special recognition is due my advisor, Dr. Peter Young. His expertise and instruction in MIMO robust control theory was central to my graduate studies. The guidance, encouragement, assistance and support that he has provided me throughout this project and throughout my graduate career are deeply appreciated. Special credit is due to Dr. Douglas Hittle, of the Mechanical Engineering Department, for his effort in pursuing the ideas behind this project. His expertise and guidance regarding HVAC systems was crucial to the success of the project. Thanks is also due to fellow graduate research assistant Christopher Delnero of the Mechanical Engineering Department for his assistance in building the experimental HVAC system. I would especially like to thank my wife, Wendy, whose unwavering love, support and encouragement have sustained me throughout this project. v

6 TABLE OF CONTENTS Notation and Symbols List of Abbreviations List of Units xix xxi xxii 1 Project Overview Background Objectives Contribution Overview of Remaining Chapters Literature Review Modeling of HVAC System Heating Coil Models HVAC Simulation HVAC Control Proportional-plus-Integral (PI) Control Optimal Control Adaptive Control vi

7 2.3.4 Robust Control An Introduction to Robust Control Design Robustness Modeling Uncertainty Representing Model Uncertainty Robust Stability Robust Performance Complex Structured Singular Value, μ Linear Fractional Transformations DK-iteration Experimental HVAC System Components and Interconnection Air Handling Unit Hot Water Subsystem Heat Exchanger Measurements and Sensors Control Devices Control Hardware and Software Control Hardware Control Software Modeling the Experimental HVAC System Data Acquisition Blower Model Mixing Box Model Boiler Model Water Flow Control Valve Model vii

8 5.6 Heating Coil Model Overall System Model Linear Model of the Experimental System Controller Design and Simulation Types of HVAC Controls PI Based HVAC Control Structure Robust MIMO Control Structure Weight Selection Model Uncertainty Weights Sensor Noise Weights Disturbance Weights Control Weights Performance Weights System Startup PI Based Controller Design Robust MIMO Control Design MIMO Robust Controller with Ext. Water Temp. Control MIMO Robust Controller with Ref. Water Temp. Control MIMO Robust Controller with Free Water Temp. Control Summary of Controller Simulation Performance Experimental Verification of Controller Designs Experimental System Configuration Reference PI Controller Verification K R1 Controller Verification K R2 Controller Verification K R3 Controller Verification viii

9 8 Analysis and Discussion HVAC System Architecture and Its Impact on Control Architecture Analysis of Controller Performance Factors Affecting Performance Merits of HVAC Designs Conclusions and Recommendations Recommendations for Future Work A Routines and Models Used in Developing the HVAC Model 173 A.1 Data Analysis A.2 Blower A.3 Mixing Chamber A.4 Boiler A.5 Water Flow Control Valve A.6 Linear Coil Model A.7 HVAC Model Validation A.8 Linear State-Space Component Models B Controller Design: Details, Routines and Models 203 B.1 Controller K R2 Weight Selection B.2 Controller K R3 Weight Selection B.3 Bumpless Transfer C-MEX S-Function B.3.1 Ordering Controller States (for Bumpless Transfer) B.4 Routines Used for Robust Controller Designs B.4.1 Routine Forming Linear Subsystem Models B.4.2 Routine for Plotting Design Weights B.4.3 Routine for Reducing Order of Controller B.4.4 Routine for Plotting Simulation Test Data ix

10 B.5 Routines used in Designing K R B.5.1 Script Generating Weights for K R1 Design B.5.2 Script (SYSIC) Used to Build System in for K R1 Design B.5.3 Main Routine for K R1 Design B.5.4 Auxiliary Routine for K R1 Design B.6 Routines used in Designing K R B.6.1 Script Generating Weights for K R2 Design B.6.2 Script (SYSIC) Used to Build System for K R2 Design B.6.3 Main Routine for K R2 Design B.6.4 Auxiliary Routine for K R2 Design B.7 Routines used in Designing K R B.7.1 Script Generating Weights for K R3 Design B.7.2 Script (SYSIC) Used to Build System infor K R32 Design B.7.3 Main Routine for K R3 Design B.7.4 Auxiliary Routine for K R3 Design C Controller Verification Testing 249 C.1 Routines For Manipulating Experimental Data C.1.1 Routine For Joining Two Data Files C.1.2 Routine for Filtering Experimental Data C.1.3 Routine for Filtering and Decimating Data C.1.4 Routine For Plotting Experimental Results C.2 Controller Verification Raw" Data C.2.1 K PI Controller Verification C.2.2 K R1 Controller Verification C.2.3 K R2 Controller Verification C.2.4 K R3 Controller Verification x

11 REFERENCES 267 xi

12 LIST OF FIGURES 3.1 Spinning satellite Singular value plot for spinning satellite, G(s) Spinning satellite loop-at-a-time stability check Stability robustness analysis of spinning satellite Plant G p formed using additive uncertainty Plant G p formed using multiplicative uncertainty Nyquist diagram of nominal plant G(j!) Nyquist diagram of perturbed plant G(j!)+w 1 (j!) (j!) Feedback loop with multiplicative uncertainty Nyquist plot showing uncertainty's destabilizing effect Feedback system with multiplicative uncertainty for RP analysis Nyquist plot for robust performance M interconnection Feedback system in lower LFT form Feedback system in upper LFT form General feedback interconnection for μ synthesis General configuration for analysis (F L (G; K)) and synthesis (F U (G; K)) Experimental HVAC system xii

13 4.2 Diagram showing the experimental HVAC system and interface signals Blower (variable speed fan) Water heater, pump and expansion tank Water flow control valve End view of heating coil Return air RTD temperature sensor and damper Pitot array airflow sensor Coriolis water flow sensor PC based control hardware PC and MATLAB RTW/WT based interface Interface cabinet(top) and drive cabinet (bottom) Boiler electrical cabinet Simulink model used for data acquisition Diagram of experimental HVAC system Overall model of experimental HVAC system Overall blower model Airflow rate versus blower speed (with C dr fully open) Airflow rate versus (return air) damper position Measured and modeled airflow rates for various damper positions Return air, linear actuator to rotary damper translation External air, linear actuator to rotary damper translation Preshaping return air command to linearize translation Preshaping external air command to linearize translation Model of mixing box Boiler model Block modeling the temperature of water returned to boiler Command to duty cycle (power) converter model xiii

14 5.15 Heater model for temperature of water out of boiler Comparison of model and measured boiler output water temperature Blocks forming model for water flow control valve Water flow rate thru coil versus valve position Total water flow versus water flow through coil Comparison of measured and modeled water flow Main coil model with subsystem blocks Water-side coil subsystem Air-side coil subsystem Step response of individual coil transfer functions Overall model configuration used for validation testing Model (solid) and experimental system (dashed) outputs Heating coil capacity versus water flow rate System Model Configuration for Controller Synthesis Controller implementation for bumpless transfer Discrete-time PI controller with antiwindup HVAC Controller Based Upon Three SISO PI Controllers Simulation model used in verifying K PI design Controller K PI simulation test results HVAC System using MIMO Robust Controller K R Detailed model for controller K R1 synthesis Disturbance & reference weights used in the design of K R Sensor noise weights used in the design of K R Control weights used in the design of K R Performance weights used in the design of K R Simulation model used to test controller K R Controller K R1 simulation test results xiv

15 6.16 HVAC System using MIMO Robust Controller K R Detailed model for controller K R2 synthesis Simulation model used to test controller K R Controller K R2 simulation test results HVAC System using MIMO Robust Controller K R Detailed model for controller K R3 synthesis Simulation model used to test controller KR Controller K R3 simulation test results Simulink model during verification testing of controller K PI Controller K PI experimental test results Simulink model during verification testing of controller K R Controller K R1 experimental test results Simulink model during verification testing of controller K R Controller K R2 experimental test results Simulink model during verification testing of controller K R Controller K R3 experimental test results Comparison of controller settling times Interrelationship of controller design objectives/parameters A.1 Blower validation model A.2 Simulation model used in validation of boiler model A.3 Simulink model for valve A.4 Simulink model for testing T ao (F a ) A.5 Simulink model for testing T wo (F a ) A.6 Simulink model for testing T ao (T ai ) A.7 Simulink model for testing T wo (T ai ) A.8 Simulink model for testing T ao (F w ) xv

16 A.9 Simulink model for testing T wo (F w ) A.10 Simulink model for testing T ao (T wi ) A.11 Simulink model for testing T wo (T wi ) A.12 Transport delay for water flowing from boiler to coil A.13 Delay associated with discrete-time-controller B.1 Disturbance & reference weights used in the design of K R B.2 Sensor noise weights used in the design of K R B.3 Control weights used in the design of K R B.4 Performance weights used in the design of K R B.5 Uncertainty weights used in the design of K R B.6 Disturbance & reference weights used in the design of K R B.7 Sensor noise weights used in the design of K R B.8 Control weights used in the design of K R B.9 Performance weights used in the design of K R C.1 Controller K PI experimental test results C.2 Controller K R1 experimental test results C.3 Controller K R2 experimental test results C.4 Controller K R3 experimental test results xvi

17 LIST OF TABLES 4.1 RTD sensor types used for temperature measurements The experimental model's five subsystems The experimental model's inputs and outputs Operating point for linear coil model Proportional and Integral Gains for PI Controllers Controller performance measurements Summary of controller K PI simulation performance Model uncertainty weights used in the design of K R Disturbance (& reference weights) used in the design of K R Sensor noise weights used in the design of K R Control weights used in the design of K R Performance weights used in the design of K R Listing of design file dsn4et.m" Listing of design file dsn4eta.m" Summary of controller K R1 simulation performance Summary of controller K R2 simulation performance Listing of design file dsn5t.m" xvii

18 6.14 Listing of design file dsn5ta.m" Summary of controller K R3 simulation performance Comparison of controller simulation performance Summary of controller K PI performance Summary of controller K R1 performance Summary of controller K R2 performance Summary of controller K R3 performance Comparison of controller performance (exp. test) HVAC system and controller architecture Significant project outcomes B.1 Model uncertainty weights used in the design of K R B.2 Disturbance (& reference weights) used in the design of K R B.3 Sensor noise weights used in the design of K R B.4 Control weights used in the design of K R B.5 Performance weights used in the design of K R B.6 Model uncertainty weights used in the design of K R B.7 Disturbance (& reference weights) used in the design of K R B.8 Sensor noise weights used in the design of K R B.9 Control weights used in the design of K R B.10 Performance weights used in the design of K R xviii

19 NOTATION AND SYMBOLS R and C fields of real and complex numbers 2 belongs to ρ subset [ union intersection 2 end of proof 3 end of remark := defined as ff jffj Re(ff) I n a ij diag(a 1 ;::: ;a n ) A T and A Λ A 1 and A + det(a) trace(a) (A) ρ(a) ff(a) and ff(a) complex conjugate of ff 2 C absolute value of ff 2 C real part of ff 2 C n x n identity matrix a matrix with a i as its ith row and jth column element an n x n diagonal matrix with a i as its ith diagonal element transpose and complex conjugate transpose of A inverse and pseudoinverse of A determinant ofa trace of A eigenvalue of A spectral radius of A the largest and smallest singular values of A xix

20 A 1 A 1(pseudo) 8x ff i (A) jjaj sup inverse of A such that A Λ A 1 = I Moore-Penrose pseudoinverse of A (non-square) for all values of x ith singular value of A spectral norm of A : jjajj = ff(a) supremum, the least upper bound L 1 (jr) functions bounded on Re(s) = 0 including at 1 L 2 (jr) square integrable functions on C 0 including at 1 H 1 the set of L 1 (jr) functions analytic in Re(s) > 0 H 2 subspace of L 2 (jr) with functions analytic in Re(s) > 0 G K L plant model controller, in what ever configuration loop transfer function L = GK S sensitivity function, S =(I + L) 1 T complementary sensitivity function T =(I + L) 1 uncertainty xx

21 LIST OF ABBREVIATIONS A de A dr A vp C de C dr C bs C vp C wh F a F w F ws P wh T ae T ar T ai T ao T wi T wo T ws fi Actual position of External air Damper Actual position of Return air Damper Actual position of water flow control valve Commanded position of External air Damper Commanded position of Return air Damper Commanded Blower (fan) Speed Commanded position of water flow control valve Commanded electric power input to Water Heater Flow rate of Air into the Heating Coil Flow rate of Water through the Heating Coil Flow rate of Water through the Water heater/boiler actual electric Power input to Water Heater/boiler Temperature of External Air Temperature of Return Air Temperature of Air Input to coil Temperature of Air Out of coil Temperature of Water input to Coil Temperature of Water Output from Coil Temperature of Water Output from Supply/boiler Time constant (seconds) xxi

22 LIST OF UNITS ffi C Degrees Centigrade! frequency in radians per second (rad=sec) AC DC Hz kg kv kw m min mm ma mv m 3 s kpa rpm s or sec VAC VDC Alternating Current Direct Current Hertz, frequency in (cycles sec) kilogram ((10 3 grams) kilovolt (10 3 volts) kilowatts (10 3 watts) meter minutes millimeter (10 3 meter) milliamperes (10 3 ampere) millivolts flow rate in cubic meters per second kilopascals, pressure (10 3 pascals) revolution per minute (revolutions=min)) second Volts, AC Volts, DC W watts, power (1 J s or 1 Nm s ) xxii

23 Chapter 1 Project Overview This thesis investigates potential performance improvements in heating ventilation and air conditioning (HVAC) systems through the application of multiple-input, multiple-output (MIMO) robust controls. This approach differs dramatically from today's prevalent, if not state of the art, method of building HVAC controllers using multiple single-input, single-output (SISO) controls. 1.1 Background In commercial HVAC systems, a central air supply provides air at a controlled temperature and flow rate for use in heating (or cooling) a space. A heating (or cooling) coil is used in the central air supply for heating (or cooling) the discharged air. The temperature of the discharged air is controlled by regulating the rate at which hot (or chilled) water flows through its heating (or cooling) coil(s). The flow rate of the discharged air is regulated to maintain a predetermined static air pressure within the temperature controlled space. Typically, the space within a building is divided into smaller zones, allowing the temperature within each zone to be maintained independently of the others. Each zone contains a reheat (and/or cooling) coil which is used to moderate the final temperature of the air discharged into the zone. Acharacteristic of today's HVAC systems is the use of a centralized hot (or chilled) water supplies in servicing multiple central air supplies. Such anhvac interconnec- 1

24 tion (architecture) restricts the controller design and impacts the performance of the resulting system. A full MIMO controller requires control of the temperature and flow rates of both the air and water flowing through the heating coil. Consequently, independent air and water supplies would be required for each coil. Such a system represents a major shift from current HVAC design paradigms. Between today's HVAC systems employing SISO (proportional-plus-integral, PI) based controllers and a system using a full MIMO robust controller, a wide assortment of configurations are possible. To gain insight into potential performance improvements as well as the constraints associated with this breadth of controllers, several diverse designs are implemented and tested, both in simulation and on an experimental system. 1.2 Objectives Issues important to the application of MIMO robust controllers to HVAC systems must first be identified and investigated. To facilitate the design and testing of controllers, a simulation model of the experimental HVAC system must be developed. While simulation of HVAC systems and their associated controllers can be very insightful, the only way to truly verify the performance provided by different HVAC controller designs is by actually using them to control an HVAC system. This requires that a simple, experimental HVAC system, intended for controlling the temperature and flow rate of the air that it supplies, be designed, built and calibrated. This discharge air system (DAS), consisting of external and return air dampers, a variable speed blower and a heating coil, is one of the most basic HVAC system components. It is comparable to the central air supply in a commercial HVAC system. While the experimental system is only a portion of an overall HVAC system, it was built using standard HVAC components and is representative of a typical hot water to air, heating system. 2

25 MIMO robust controllers are conceived, designed and tested on the experimental system. The selected HVAC controller designs represent several distinct system configurations, in an attempt to capture the breath of possible designs. The performance of the MIMO robust controllers will be quantified relative totoday's HVAC standard control technique. This necessitates the design and testing of a PI based, reference" HVAC controller. In order to simplify the formulation and to facilitate a more intuitive understanding, the controller designs explored herein are restricted to discharge air temperature and airflow rate control for a DAS. Potential MIMO robust controller designs will be developed and tested using both the simulation model and the experimental HVAC system. 1.3 Contribution While (simple) optimal and (SISO) robust HVAC controllers have been designed and their performance experimentally verified, this project is the first to do so using MIMO robust HVAC controllers. The design and testing of these controllers provides valuable insight into potential improvements in performance, as well as constraints, associated with applying this control methodology to HVAC systems. Test results presented herein, demonstrate that performance gains (reductions in discharge air temperature settle time) in excess of 300% maybeachieved using MIMO robust HVAC controllers. Furthermore, it may be possible for such performance gains be achieved without significant impact to current HVAC system architecture (interconnection). 1.4 Overview of Remaining Chapters The content of the remaining chapters are summarized in this section. Chapter 2 reviews previous HVAC related research concerning HVAC component and system modeling, HVAC simulation and HVAC control techniques. 3

26 Chapter 3 provides a brief introduction to robust control theory. An example problem, involving a spinning satellite, is used to motive the need for MIMO robustness techniques. Such methods are necessary in dealing with issues such as (in MIMO system) cross-coupling and model uncertainty. A means for representing model uncertainty and a robust controller design method, based upon the maximum singular value (μ), are introduced. Chapter 4 provides an overview of the experimental HVAC system that was constructed and used as part of this project. This discharge air system (DAS), consisting of external and return air dampers, a variable speed blower and a hot-water to air heat exchanger, allows the discharge air temperature and flow rate to be independently controlled. Chapter 5 covers the development of a model of the experimental HVAC system. This model is later used in the design of MIMO robust controllers and their simulation testing. Chapter 6 details the design and simulation testing of three MIMO robust HVAC controllers, as well as a P-I based (reference) HVAC controller. Chapter 7 covers the verification testing of the controller designs (developed in Chapter 6) on the experimental HVAC system. Chapter 8 discusses and compares the results of the controller verification testing. The relationship between system/controller architecture and controller performance is explored. Chapter 9 discusses the significant outcomes of the project and offers recommendations for further work. The Appendices (A, B and C) contain key MATLAB cfl routines and Simulink cfl models used during the course of this investigation. 4

27 Chapter 2 Literature Review The literature is rich with heating, ventilation and air conditioning (HVAC) related articles. Since this investigation concerns the design and testing of HVAC controllers, as well as formulating the necessary design and simulation models, a brief review of the relevant literature follows. 2.1 Modeling of HVAC System Models for HVAC components, particularly heat exchangers, have been the subject of a number of articles over the past thirty years. The ASHRAE publication Reference Guide for Dynamic Models of HVAC Equipment [3] provides a concise overview of the dynamic models available for HVAC-related equipment (air and water handling equipment, heating and cooling equipment and control equipment). While providing an excellent section covering heat exchanger models, its review of other HVAC component models, which are less represented in the literature, make it an even more valuable reference Heating Coil Models Several dynamic heating coil models [9; 32, among others] are described in the ASHRAE publication Reference Guide for Dynamic Models of HVAC Equipment [3]. Compact Heat Exchangers by London and Kays [19] is an excellent reference for 5

28 general heat exchangers. Gartner and Harrison developed an early dynamic model for a serpentine cross flow heat exchanger [9]. This model deals primarily with water temperature disturbances and was extended by Garner and Dane [8] to handle disturbances in water velocity. Tamm wrote a companion paper [32] which which extended the the Gartner and Dane model to multi-row cross flow heating coils. Tamm and Green validated this model experimentally [33]. By solving the partial differential equations (PDEs) corresponding to the individual finned elements of a cross flow heat exchanger, Kabelac and others developed overall models [14; 26]. Using the basic element approach, a variety of configurations and sizes of cross flow heat exchangers could be modeled. However, as the number of basic elements increase, so do the number of PDE to be solved, contributing to lengthy computations. Continuing Gartner and Harrison's work, McCutchan developed a first-principles model for a finned serpentine cross flow heat exchanger [24]. At the time, a solution to the resulting PDEs was not known and an approximate solution was substituted. Rather than assuming an approximate solution to the PDEs, Delnero found an exact solution (for changes in air flow rate) [6]. When complete, this model would deal with simultaneously changing inputs. While showing great promise, the current state of the model (as well as its complexity), greatly limits its usefulness for the purposes of simulation and controller design. 2.2 HVAC Simulation Several HVAC simulation products, incorporating HVAC component models, such as HVACSIM + [29] and BLAST [22] have been developed. Such packages were not intended for the purpose of controller design and are not readily compatible with 6

29 applications (e.g., MATLAB cfl 1 and Simulink cfl) commonly used in the design of controllers. However, upon completion of this project, an HVAC simulation package based upon MATLAB became available. The SIMBAD Building and HVAC Toolbox [28] provides models of of HVAC system components suited for controller design and overall HVAC system simulation. 2.3 HVAC Control Early publications dealing with HVAC control focused on proportional-plus-integral (PI) controllers. More recently, a number of advanced HVAC control methodologies have been proposed. The following sections provide an overview of HVAC control methodologies from PI to the latest proposed techniques Proportional-plus-Integral (PI) Control It has long been recognized that the necessarily low gains and tedious and inaccurate tuning of PI-based HVAC controllers contributed to poor performance. While this dilemma has prompted much research and produced numerous papers, PI controllers prevail as the industries preferred control methodology. Shavit and Brant [30] performed research at Honeywell regarding the dynamic performance of discharge air systems using PI control. Their work confirmed that unless the coil and valve are matched to provide a system which is linear overall, system response is altered. Later work attests to the difficulties encountered in properly determining the PI controller gains [20; 13]. Sub-optimal tuning of the PI controller further worsened the already slow HVAC response. Later references to PI-based HVAC controllers appear in articles proposing alternate control solutions. In these articles, the performance provided by PI controllers was used as a reference for con- 1 An application integrating computation, visualization and programming for use in science, engineering, mathematics and related fields. 7

30 trasting the increases in performance afforded by advanced HVAC controllers Optimal Control Zaheer-Uddin [37] used linear HVAC models in developing controllers using optimal regulator theory and adaptive control. In simulation tests, the performance of the controllers in response to disturbances emphasized the benefits offered by advanced HVAC design techniques. Simulation testing of the advanced controllers demonstrates advantages over PI controllers in decoupling system interrelationships and in rejecting external disturbances. Kasahara, et al. [15] developed a multivariable autoregressive (AR) HVAC model for use in implementing optimal HVAC controllers. Optimal linear quadratic Gaussian (LQG) and optimal preview controllers (3 2), regulating room temperature and humidity, were developed and verified (both in simulation and experimentally). Using published data 2, Zheng [39] developed a state-space model of a discharge air temperature system. An optimal tracking controller was designed for the model and used to augment the output of a PI controller. The tracking performance of the combined system was evaluated in simulation. An optimal controller's inability to tolerate discrepancies between the model used in designing the controller and that of the physical plant(model uncertainty), poses a major problem in HVAC systems, where accurate models are not readily available. Furthermore, the characteristics of an HVAC plant change (deteriorate) over time. Even an optimal controller designed, using an accurate model, performs less-than optimally over time. 2 Zheng, 1997, Dynamic Modeling and Global Optimal Operation of Multizone Variable Air Volume HVAC Systems, Ph.D. thesis, Center for Building Studies, Concordia University, Montreal, Canada 8

31 2.3.3 Adaptive Control Adaptive HVAC controllers have the potential to eliminate the need for manual tuning of the controller and have the ability to adapt over time to changes in the system. For these reasons, they have been proposed as alternatives to current fixed gain controllers [23; 25; 21]. More recently, neural networks (NNs) have been proposed [12; 27; 1] as a means of implementing adaptive controllers for HVAC applications. As with all controllers adapting within a control loop, ensuring the stability of the system is paramount [4]. A technique based upon the synthesis of robust control theory and reinforcement learning has been proposed [17] for resolving this issue Robust Control The application of robust control theory in controlling HVAC systems has only recently been considered. Chen and Lee [5] proposed an adaptive robust controller to account for the uncertainty under which all HVAC systems operate. They proposed a zone temperature control scheme that would account for and compensate for both uncertainty associated with the model used in controller design (an advantage over Optimal controllers) and the nonlinear nature of HVAC systems. Zaheer-uddin [38] proposed a load-preview decentralized robust controller for zone temperature control. By assuming that the variations in outdoor air temperature are previewable," a preview controller was formed using a parameter optimization method. The controller was considered robust in that it achieved regulation in the presence of non-destabilizing changes in parameters and external disturbances. Kasahara, et al. [16] developed and experimentally verified a robust PID controller for zone temperature control. The controller design used a geometric method for robustness" (solving a two-disk type of mixed sensitivity problem) by generating a series of gains for each of the PID terms (K p, K i, K d ). The gain scheduling of the PID terms used a method based upon approximating the ratio of the system's lag to 9

32 response times. Underwood [34; 35] also developed a SISO robust controller for a minimal HVAC system. The discharge air temperature of this system was regulated by varying the valve position to adjust the flow rate of water through the heating coil. An H 1 controller was designed, using a linear system model (1 st order lag, plus delay) and parametric uncertainty (associated with the heating coil gain, time constant and delay). The performance of the robust controller (verified in simulation), was found to compare favorably with a locally optimized PID controller. Robust controllers have been proposed for HVAC applications. A (SISO) robust PID controller has been implemented and its performance verified on an experimental system. However, this investigation is the first in which a MIMO robust controller was designed and its performance verified using an experimental HVAC system. Ultimately, it has provided valuable insight into potential improvements in performance, as well as constraints associated with applying the control methodology to HVAC systems. 10

33 Chapter 3 An Introduction to Robust Control This chapter provides a brief introduction to robust control theory. Inevitability, the models used in controller design only approximate the physical systems they are intended to represent. Robust control theory addresses the effects that discrepancies between the model and the physical system (model uncertainty) may have on the design and performance of linear feedback systems. These uncertainties may arise from model approximation, neglected or unmodelled dynamics, unknown parameters, or even sensor and/or actuator imperfections. The techniques introduced in this chapter are based upon the structured singular value (μ), which is frequently used in dealing with model uncertainty. Robust control provides a unified design approach under which the concepts of gain margin, phase margin, tracking, disturbance rejection and noise rejection are generalized into a single framework. Typically, the uncertainties considered in robust control theory are bounded using norms. The H 1 and H 2 norms are frequently applied in the robust controller design process, as they may be used to bound, respectively, the magnitudes or the energy content of signals. In this chapter, the H 1 robust control design methodology is introduced. 3.1 Design Robustness Most controller designs are based upon a mathematical model of the physical system to be controlled. Clearly, the model fails to fully capture the fundamental nature 11

34 of the physical system. These differences are commonly referred to as model uncertainties. Since even small uncertainties can have an adverse impact on performance and stability, an effective design technique must account for uncertainty. A control system that is insensitive to the uncertainties inherent in the system model used in the design of the controller is said to be robust. Notions regarding robustness for single-input, single-output (SISO) systems indicate that if a single-loop system exhibits good (robust) stability and nominal performance characteristics, it should provide good robust performance as well. In general, these notions do not extend to multiple-input, multiple-output (MIMO) systems. In addition, analyzing the robustness of MIMO systems by individual feedback loops does not account for interactions between the loops and can yield misleading results. To demonstrate the shortcomings of applying classical control analysis techniques to a MIMO system, consider an example based upon the spinning satellite, as shown in Figure 3.1. The original form and mathematics of this example are credited to John Doyle and appear in [31, pp. 91]. The satellite of Figure 3.1 is spinning about its z axis at a constant angular velocity (Ω). The inertia of the satellite in the direction of the x, y and z axes are I x, I y and I z, respectively. The angular velocities of the satellite with respect to the x and y axes are! x and! y. The satellite is controlled using two torque inputs, T x and T y along the x and y axes. The relationships governing the torques and the rates of rotation about the x and y axes are given by Equations 3.1 and 3.2. I x _! x! y Ω(I x I z )=T x (3.1) I x _! y! x Ω(I z I x )=T y (3.2) The system input vector (U) is defined by Equation

35 z Ω! x x y! y Figure 3.1: Spinning satellite U :=» ux := u y " Tx I x T y I y # (3.3) From Equations 3.1, 3.2 and 3.3, and defining a := (1 I z =I x )Ω, the dynamics of the spinning satellite is expressed in Equation 3.4.» _!x = _! y» 0 a a 0»!x +! y» ux u y (3.4) By setting a = tan, the angular velocities! x and! y are translated to the coordinates of Equation 3.5.» yx y y = 1» cos sin cos sin cos»!x =! y» 1 a a 1»!x! y (3.5) 13

36 From the preceding equations, the transfer matrix of Equation 3.6 is obtained. Y (s) =G(s)U(s) where :» G(s) = 1 s 2 +a 2 s a 2 (3.6) a(s +1) a(s +1) s a 2 Choosing a = 10, and using G(s) = C(sI A) 1 B + D, a minimal state-space representation is given by Equation 3.7.» A B C D = a a 0 1 a a (3.7) 0 0 As is evident from G(s), the plant has a pair of poles on the imaginary axis at s = ±ja and thus is not open-loop stable. In order to stabilize it, negative feedback is applied using the constant diagonal controller, K = I. The resulting closed-loop system's complementary sensitivity function, T (s), is given by Equation 3.8. T (s) =GK(I + GK) 1 = 1» 1 1 s +1 a 1 (3.8) From Equation 3.8 we see that the closed-loop system has two poles at s = 1 and thus stability is assured. This is verified by checking the closed-loop state matrix (A cl ) of Equation 3.9. _x = Ax Bu; y = Cx and u = Ky A cl = _x = Ax BKy _x = Ax BKCx =) A cl = A BKC» 0 a a 0» 1 a a 1 =» (3.9) 14

37 Magnitude ff(l) ff(l) Frequency (rad/sec) Figure 3.2: Singular value plot for spinning satellite, G(s) The magnitude of the singular values of the loop transfer function (L(s) = G(s)K(s)) provide an indication of performance. In Figure 3.2, the minimum singular value of the loop transfer function (ff(l)) equals unity at low frequencies and rolls off at around! = 10. Since ff(l) never exceeds unity, the system is only loosely controlled and poor performance can be expected. This is further confirmed by evaluating the maximum singular values of the closed-loop sensitivity and complementary sensitivity functions, ff(s) and ff(t ), respectively. At steady-state, ff(s) and ff(t ) are both found to be approximately 10. Furthermore, the large off-diagonal elements of T (s) indicate that strong interactions exist between the two inputs. By considering how sensitive the system is to perturbations on the input channels, it can be determined how robustly stable the system is. To do this, consider the system of Figure 3.3, where the first feedback loop is broken just prior to the input to the plant. The loop transfer function from w 1 to z 1 is L 1 (s) = T 11 (s)k(s) = 1=s. This transfer function provides an infinite gain margin and 90 ffi of phase margin. While 15

38 z 1 w 1 r 1 ± y 1 - G r 2 ± y 2 - K Figure 3.3: Spinning satellite loop-at-a-time stability check ffi 1 u 1 ^u 1 r 1 ± ± y 1 - G r 2 ± ± y 2 - u 2 ^u 2 ffi 2 K Figure 3.4: Stability robustness analysis of spinning satellite this would indicate that the system is robust independent of the value of K, this is misleading, as will now be demonstrated. Consider the system of Figure 3.4 with input gain uncertainty. Where the uncertainty for the respective channels is ffi 1 and ffi 2, the plant inputs are given in Equation Using the relationships given by Equation 3.10, the matrix B of the statespace model is replaced with ^B, asgiven in Equation ^u 1 =(1+ffi 1 )u 1 ; ^u 2 =(1+ffi 2 )u 2 (3.10) 16

39 » 1+ffi1 0 ^B = 0 1+ffi 2 (3.11) The closed-loop state space matrix for the system of Figure 3.4, ^ A cl, is given in Equation The characteristic polynomial for ^ A cl is given in Equation 3.13.» A^ cl = A ^BKC 0 a = a 0»» 1+ffi1 0 1 a 0 1+ffi 2 a 1 (3.12) det(s IA ^ cl )=s 2 +(2+ffi 1 + ffi 2 ) s +1+ffi z } 1 + ffi 2 +(a 2 +1)ffi 1 ffi z } 2 a1 a0 (3.13) The system, having uncertain input gain, is stable if and only if both of the coefficients, a 0 and a 1 are positive. It can be shown that the system is stable if only one input-channel at a time is subject to gain uncertainty. In this scenario, the system is stable for 1 <ffi 1 < 1 (ffi 2 =0)and 1 <ffi 2 < 1 (ffi 1 = 0), confirming the infinite gain margin obtained previously. For the system to be truly robust, it must tolerate simultaneous gain uncertainties on the input channels. By setting ffi 1 = ffi 2, the resulting inequality given in Equation 3.14 shows that the system is unstable for relatively small values of gain uncertainty. Hence, applying one loop at a time, SISO stability criterion to a MIMO system may yield misleading results. Also note that large maximum singular values of the sensitivity and inverse-sensitivity functions (ff(s) and ff(t )) are indicative of a lack of robustness. jffi 1 j > 1 p a2 +1 ß 0:1 (3.14) In the next section, some sources of model uncertainty are introduced. Also, a methodology for representing uncertainty is presented and some of the stability and performance implications of this uncertainty are investigated. 17

40 3.2 Modeling Uncertainty In the spinning satellite example, the destabilizing effect of gain uncertainty was demonstrated. In this section, other forms of model uncertainty are examined. Model uncertainty may involve variation in a model's real parameters (real uncertainty) or uncertainty regarding a model's dynamic characteristics (complex uncertainty). The manner in which uncertainty is modeled reflects both the nature of the uncertainty considered and the structure imposed by the analysis and synthesis techniques to be employed. If the structure of a system's model is known, and the uncertainty involves only the values of a few real parameters, then the uncertainty may be modeled as real parametric uncertainty. If, however, the uncertainty pertains to a system's dynamic characteristics, the likely choice would be to model it as complex dynamic uncertainty. The analysis and synthesis techniques for real parametric uncertainty [36] are much more complicated and require significantly more computational resources than those for complex uncertainty. Since the HVAC control problem does not require the use of real parametric uncertainty, complex model uncertainty is used exclusively in the discussion herein. Model uncertainty is accounted for by assuming that the dynamic characteristics of a plant are described by a set of possible linear time invariant (LTI) models, rather than a single model. This set of possible perturbed models, Π, is sometimes referred to as the uncertainty set. The following notation is adopted for specifying the models making up this set: G(s) 2 Π : the nominal plant model G p (s) 2 Π : a perturbed plant model A norm-bounded uncertainty method is used, where Π is generated by allowing H 1 norm-bounded stable perturbations to the nominal plant G(s). This method yields 18

41 an infinite set of possible plants (Π), collectively providing a continuous description of the model uncertainty Representing Model Uncertainty Model uncertainty arises from such things as model approximation/errors, nonlinearities and sensor imperfections. Model uncertainty may be categorized as either parametric uncertainty or neglected/unmodeled dynamics uncertainty. In parametric uncertainty, the structure of the model is known, but some of the parameters are uncertain. In dynamic uncertainty, the uncertainty involves the dynamics of the system, usually at higher frequencies. Regardless of the source, uncertainty may be lumped into a single perturbation of a chosen structure. The perturbed plant, G p (s), in Figure 3.5 is formed using an additive uncertainty structure. The uncertainty block,, is defined as any stable transfer function such that jj jj 1» 1 8!. At each frequency, (j!) corresponds to a transfer function, whose Nyquist plot is entirely contained within a circle having a radius of one, centered at the origin. Considering all possible 's generates" a disc-shaped region of radius one, centered at the origin. The weighting function, w 1 (s), is used to scale the magnitude of the uncertainty as a function of frequency. Adding this weighted uncertainty to a nominal model yields a model with additive uncertainty. Using additive uncertainty, the perturbed plant model G p (s) shown in Figure 3.5 and represented by Equation 3.15 is formed. G p (s) =G(s)+w 1 (s) (s); j (j!)j»1 8! (3.15) Similarly, the multiplicative uncertainty block shown in Figure 3.6, is defined by Equation G p (s) =G(s)(1+w 2 (s) (s)) ; j (j!)j»1 8! (3.16) 19

42 w 1 G p G ± Figure 3.5: Plant G p formed using additive uncertainty w 2 G p ± G Figure 3.6: Plant G p formed using multiplicative uncertainty It should be noted that the form of these uncertainties need not be limited to the additive and multiplicative constructs presented in the preceding discussion, but may take many other forms (e.g., inverse additive, inverse multiplicative, output, feedback) Robust Stability Model uncertainty affects the stability of a system. Consider a system whose nominal dynamic characteristics are represented by the Nyquist diagram shown in Figure 3.7. If the nominal plant isg(s), the set of perturbed plants formed by additive uncertainty are given by Equation The set of perturbed plants, G p, arising from Equation 3.15 clearly has two components, that of the nominal plant G(s) and the uncertainty w 1 (s) (s). The weighting function, w 1, acts to scale the uncertainty,. At each individual frequency, the Nyquist plots corresponding to the set of all possible perturbed plants, G p, are contained within a disc of radius kw 1 (j!)k 1. In Figure 3.8, the Nyquist path of the 20

43 Im 1 Re G(j!) Figure 3.7: Nyquist diagram of nominal plant G(j!) nominal plant is plotted, as well as disks representing the uncertainty bound at a number of individual frequencies. In the Nyquist diagram, as uncertainty is increased, the set of perturbed plants will eventually encircle the critical point, thus revealing the potentially destabilizing effect of uncertainty. In further investigating the stability of feedback control loops in the presence of uncertainty, we begin with a few definitions. Given the set of perturbed plant models Π, a controller, K is designed based upon a nominal plant model G n 2 Π to meet specific performance specifications. The following is said of the resulting closed-loop system: Definition 1 Nominal Stability (NS): if K internally stabilizes the nominal closed-loop system (having plant G n ) Definition 2 Robust Stability (RS): if K internally stabilizes the perturbed closed loop system (having any perturbed plant G p 2 Π) In Figure 3.9, the stability of the SISO system in the presence of multiplicative 21

44 Im 1 jw(j!j) Re G(j!) Figure 3.8: Nyquist diagram of perturbed plant G(j!)+w 1 (j!) (j!) w 2 G p ± ± - G K Figure 3.9: Feedback loop with multiplicative uncertainty uncertainty is to be determined. Including uncertainty, the transfer function (T p ) and the loop transfer function (L p ) are given by Equations 3.17 and 3.18, respectively. T p = G p 1+G p K = G(1 + w 2 ) 1+G(1 + w 2 )K (3.17) L p = G p K = GK(1 + w 2 ) = L + w 2 L ; j (j!)j < 1; 8! (3.18) Assume that the nominal closed-loop system is stable and that its Nyquist plot does not encircle the critical point. While this restriction may be dropped without 22

45 Im 1 j1+l(j!)j jw 2 (j!)lj Re L(j!) Figure 3.10: Nyquist plot showing uncertainty's destabilizing effect loss of generality, it is imposed here for reasons of simplicity. To test the closed-loop system for robust stability, the Nyquist criterion is applied. According to the Nyquist criterion, stability is ensured if the loop transfer function does not encircle the critical point ( 1). Consider the Nyquist plot of L p shown in Figure Begin by noting that the distance between the center of the uncertainty disc (w 2 ) and the critical point is j 1 Lj = j1+lj. Also note that the radius of the uncertainty disc is jw 2 (j!)lj. Encirclements of the critical point are avoided and robust stability ensured if none of the discs cover the critical point. As Equation 3.19 shows, this requires that the magnitude of the weighted transfer function be less than unity across all frequencies. For SISO systems, meeting this condition ensures robust stability. jw 2 Lj < j1+lj; 8! () fi fi fi w2lfi < 1; 8! 1+L () jjw 2 T jj 1 < 1; 8! (3.19) Robust Performance Robust performance is a measure of how well a feedback system achieves its performance specification in the presence of model uncertainties. A few definitions regarding robust performance follow. Given the set of perturbed plant models, Π, a controller 23

46 r w p w p e w 2 e ± K ± - G G p y Figure 3.11: Feedback system with multiplicative uncertainty for RP analysis K is designed based upon a nominal plant model G n 2 Π to meet specific performance specifications. The following is said of the resulting closed-loop system: Definition 3 Nominal Performance (NP): if nominally stable and the performance specifications are met for the nominal closed loop system (having plant G n ) Definition 4 Robust Perfomance (RP): if robustly stable and the performance specifications are met for the perturbed closed loop system (having any perturbed plant G p 2 Π) Extending the effects of model uncertainty to feedback control loops, consider the SISO system of Figure Here, the performance of the system in the presence of multiplicative uncertainty is to be determined. Including uncertainty, the loop transfer function is given in Equation L p = G p K = L(1 + w 2 ) = L + w 2 L (3.20) The use of frequency weighting functions allow a higher level of performance to be specified at (lower) frequencies where it is critical, while imposing a less stringent requirement at (higher) frequencies where it is not. Typically, performance weights 24

47 Im jw p (j!)j 1 j1+l(j!)j jw 2 (j!)lj Re L(j!) Figure 3.12: Nyquist plot for robust performance are constructed as low-pass filters, having non-zero low-frequency and finite highfrequency gains. In Figure 3.11 tracking performance is specified using the frequency weighted performance function w p. For this system, a high-pass performance weight (w p ) specifies a small error signal (e) at low frequencies, while permitting a larger error at higher frequencies. The performance specification requires that the weighted error signal w p e, be less than unity for all frequencies (jw p ej < 1; 8!, or alternately, e<1=w p ; 8!). This corresponds to the nominal performance requirement given by Equation 3.21 (for derivation see [7, pp ]). For the (SISO) case of Figure 3.12, the performance specification may be expressed as in Equation ff(sj!) < 1=jw p (j!)j; 8!, ff(w p S) < 1; 8!, jjw p Sjj 1 < 1; 8! (3.21) jjw p Sjj 1 < 1; 8!, j wp(j!) j < 1; 8! 1+L(j!), jw p (j!)j < j1+l(j!)j; 8! (3.22) A graphical interpretation of the SISO robust performance condition is obtained using Figure Begin by considering the nominal performance requirement of 25

48 Equation 3.22 and define a disc of radius jw p (j!)j, centered on the critical point ( 1). The nominal performance condition is satisfied if the nominal locus (without uncertainty) does not enter this disc. Hence, robust performance requires that the locus of uncertain loop transfer functions L(j!) also avoid entering the disc. Since at each distinct frequency, L(j!) is bounded by a disc of radius jw 2 (j!)j, centered at the point L(j!), this requires that the two discs not meet. Using the preceding criteria, and the fact that the centers of the two discs are j1+l(j!)j apart, the condition for SISO robust performance is given by Equation jw p j + jw 2 Lj < j1+lj; 8! () jw p 1+L j + jw L 2 j < 1; 8! (3.23) 1+L The general robust performance (RP) condition, expressed in terms of the sensitivity function (S) and the complementary sensitivity function (T ), is given by Equation (For the derivation and proof of Equation 3.24 see [7, pp ].) k jw p Sj + jw 2 T jk 1 < 1 (3.24) Note the robust performance condition (Equation 3.24) considers both nominal performance (jw p Sj) and robust stability (jw 2 T j). This ensures that both the performance specification and robust stability are achieved, over the entire uncertainty set. In this section several methods of including complex dynamic uncertainties in feedback models have been presented. Nyquist plots were used to gain and intuitive understanding of robust stability and robust performance criteria in SISO systems. In the next section the mathematical foundation for robust MIMO controller synthesis/design will be investigated. 26

49 3.3 Complex Structured Singular Value, μ In this section, μ( ) the structured singular value is defined. The standard notation from linear algebra and control theory is used. R denotes the set of real numbers, whereas C denotes the set of complex numbers, j j is the absolute value of the elements in R or C, R n is the set of real n vectors, C n is the set of complex n vectors, kvk is the Euclidean norm for v 2 C n, R n m is the set of n m real matrices, C n m is the set of n m complex matrices and I n is the n n identity matrix. For M 2 C n m, M T is the transpose of M, M Λ is the complex-conjugate transpose of M and ff(m) is the maximum singular value of M. For M 2 C n m, i (M) is the i th eigenvalue of M, ρ(m) := max i j i (M)j is the spectral radius of M. Consider matrices M 2 C n n. The definition of μ depends upon the underlying block structure of the uncertainties ( ). This structure depends upon the uncertainty and performance goals unique to the problem. The structure of is defined by specifying the type of each block, the total number of blocks and their dimensions. Two types of blocks, repeated scalar and full blocks are considered. The non-negative integers S and F specify the number of repeated scalar blocks and the number of full blocks. To track the block dimensions, the positiveintegers r 1 ;::: ;r S and m 1 ;::: ;m F are introduced. The ith repeated scalar block is r i r i, while the jth full block is m j m j. Using these integers, ρ C n n is defined as = Φ =block diag[ffi 1 I r 1;::: ;ffi s I rs ; 1;::: ; F ]:ffi i 2 C ; j 2 C m j m j Ψ (3.25) For consistency among all dimensions, P S i=1 r i + P F j=1 m j must equal n. Often, the norm bounded subsets of will be needed, and the notation B := f 2 :ff( )» 1g is introduced. In Equation 3.25, all of the repeated scalar blocks appear first, followed by the full blocks. This is done to simplify the notation and can easily be relaxed. Similarly, the full blocks are assumed to be square, but 27

50 u v M Figure 3.13: M interconnection again, this is only to simplify notation. Definition 5 For M 2 C n n, μ (M) is defined μ (M) = min : det(i M )=0g 1 2 fff( ) (3.26) with μ (M) :=0if no 2 solves det(i M ) = 0. The computation of the exact value of μ is not numerically tractable and thus an estimate of μ is required. An estimate is obtained by calculating lower and upper bounds for μ. The standard lower and upper bounds are given by Equations 3.27 and 3.28, respectively. μ LB : max ρ(qm)» μ (M) (3.27) Q2Q μ UB : μ» inf D2D ff(dmd 1 ) (3.28) Using the feedback loop shown in Figure 3.13, a feedback interpretation of μ (M) is considered. The equations for the feedback system shown in Figure 3.13 are: u = Mv and v = u. As long as 1 M is nonsingular, the only solutions of u, v to the loop equations are u = v = 0. However, if I M is nonsingular, there are an infinite number of solutions to the equations, and the norms kuk, kvk of the solutions can be arbitrarily large. From a systems point of view, the feedback system would be considered 28

51 unstable. Conversely, stability is ensured only when the only solutions are zero. In this context, μ (M) provides a measure of the smallest structured that causes instability. Thus, an alternative definition of μ (M) is: μ (M) = max ρ( M) (3.29) 2B For the proof of Equation 3.29, see [2, pp ]. As Figure 3.13 is nearly synonymous with the small gain theorem, it is introduced here. The small gain theorem states that if M(s) (s) is stable, the closed loop system will be stable. km(s)kk (s)k < 1 (3.30) Equation 3.30 requires that the locus of M(j!) (j!) be within the unit circle in the complex plane. This is a considerably more conservative than the Nyquist stability criterion, which only requires that the 1 point not be encircled by the locus (an odd number of times). The utility of the small gain theorem is that it provides a bound on (s) that guarantees closed-loop stability. For the proof of the small gain theorem see [18, pp ]. When is one of two extreme sets, μ (M) can be related to common algebra quantities. ffl If =fffii : ffi 2 C g (S =1;F =0;r 1 = n), then μ (M) =ρ(m), the spectral radius of M. ffl If = C n n (S =0;F =1;m 1 = n), then μ = ff(m). 3.4 Linear Fractional Transformations The application of μ in robust control theory relies heavily upon a class of linear feedback loops called linear fractional transformations (LFTs). In this section two 29

52 M L z y G u w k Figure 3.14: Feedback system in lower LFT form key linear fractional transformations are introduced as well as associated theorems pertinent to μ and its application to controller analysis and synthesis. Using the definition of μ and a few basic matrix transformations, several key theorems governing LFTs can be obtained. The block partitioned complex matrix, G, given in Equation 3.31, is introduced and a matrix K having dimensions compatible with matrix G 22 is identified. The loop equations given in Equation 3.32 and the feedback system of Figure 3.14 are then considered. G =» G11 G 12 G 21 G 22 (3.31) z = G 11 w + G 12 u y = G 21 w + G 22 u u = ky (3.32) This system of equations is said to be well posed, if, for any vector w, there exists unique vectors u, y and z solving the loop equations given in Equation The solutions to the loop equations are well posed if and only if the inverse of I G 22 2 exists. In Equation 3.33 the loop equations given in Equation 3.32 are solved to find an expression for z in terms of w. 30

53 M U z y G u w Figure 3.15: Feedback system in upper LFT form y = G 21 w + G 22 u y = G 21 w + G 22 ky (I G 22 k)y = G 21 w y = (I G 22 k) 1 G 21 w z = G 11 w + G 12 ky z = G 11 w + G 12 k(i G 22 k) 1 G 21 w z = (G 11 + G 12 k(i G 22 k) 1 G 21 )w (3.33) The equation z =(G 11 +G 12 k(i G 22 k) 1 G 21 )w is a linear fractional transformation on G by K, and corresponds to the lower loop of the feedback system shown in Figure As designated by the subscript L, Equation 3.34 is the lower fractional transformation of the feedback system. F L (G; K) =M L = w 1 z = G 11 + G 12 k(i G 22 k) 1 G 21 (3.34) The upper LFT given by Equation 3.35 is defined in a similar manner. A feedback system in upper LFT form is shown in Figure F U (G; ) = M U = w 1 z = G 22 + G 21 (I G 11 ) 1 G 12 (3.35) In the control systems context, the matrix G 22 represents the nominal model and is a norm bounded perturbation from an allowable perturbation set ( ). The 31

54 matrices G 12,G 21 and G 11 and the upper LFT represent how the perturbation affects the nominal map, G 22. Solving the constant matrix problem involves two issues. The first is determining if the LFT is well posed for the set of norm bounded perturbations 2 B := f 2 :ff( ) < 1g. Secondly, if the LFT is well posed, how large can F U (G; ) become for the norm bounded set of perturbations. Three theorems which help answer this problem are now examined. Assume that for the complex, block partitioned matrix M, there is block diagonal matrix having block structures 1 compatible with M 11 and 2 compatible with M 22. This yields three s that may be used in calculating μ: μ 1(M 11 ), using 1, μ 2(M 22 ), using 2 and μ (M) using. If I M 11 1 is invertible, then the upper LFT: F U (M; 1) =M 22 + M 21 1(I M 11 1) 1 M 12 is well posed. Theorem 1 (Well Posedness): The linear fractional transformation F U (M; 1) is well posed for all 1 2 B 1 if and only if μ 1(M 11 )» 1. Theorem 1 is just another variation of the definition of μ. As the perturbation increases, F U (M; ) diverges from M 22. How μ 2 (F U (M; 1)) varies with the perturbation is connected to μ (M), in the following Theorem: Theorem 2 (Main Loop Theorem): The following are equivalent: 1: μ (M)» 1 2: μ 1 (M 11 )» 1; and max 12B μ 2 (F U (M; 1)) < 1 3: μ 2 (M 22 )» 1; and max 22B μ 1 (F L (M; 2)) < 1 For further development and application of LFTs see [40, pp ]. To facilitate the task of controller synthesis, the general configuration shown in Figure 3.16 is adopted. Although some feedback interconnections may be quite complex, they can still be rearranged to fit this general configuration. The choice of this structure recasts the control problem into a form that permits the application of linear fractional transformation (LFT) techniques in the task of controller synthesis. 32

55 y u z G w y K u Figure 3.16: General feedback interconnection for μ synthesis In the generalized configuration, the block labeled G is the open-loop model and contains all of the components of the nominal plant model, along with the performance and uncertainty weighting functions. The block labeled is block-diagonal complex uncertainty set. The block labeled K represents the controller. G has three input vectors: perturbations (u ), disturbances (w) and control (u) and three output vectors: perturbations y, errors (z) and measurements (y). The set of perturbed plants (Π) to be controlled is defined by the LFT of Equation F U (G; ) : 2 ; sup ff[ (j!)] < 1 (3.36)! The design objective is to find a controller (K), that under all uncertainties (jj jj 1 < 1), stabilizes the closed-loop system and satisfies Equation jjf L [F U (G; );K]jj z } 1» 1 (3.37) Π The LFT equality given in Equation 3.38 is easily confirmed by considering Figure

56 y u F U (G; ) y u z G w z G w y u K F L (G; K) y K u Figure 3.17: General configuration for analysis (F L (G; K)) and synthesis (F U (G; K)) F U [F L (G; K); ] = F L [F U (G; );K] (3.38) From the preceding, the design objective can be restated as finding a controller, K, that for all normalized perturbations,, stabilizes the closed loop system and satisfies Equation jj F U [F L (G; K); ] jj 1» 1 (3.39) Theorem 3 is applied to F L (G; K) in determining whether the controller meets its robust performance objectives. Theorem 3 Suppose F L (G; K) and are stable systems, define the augmented block diagonal uncertainty structure:» 0 ^ = 0 p then the perturbed closed-loop system is stable and satisfies jj F U [F L (G; K); ^ ] jj 1 < 1 for all ^ :jj jj 1 < 1 if and only if: sup μ (F L (G; K)(j!))» 1! 34

57 The standard H 1 optimal control problem is to find all stabilizing controllers K that minimize Equation DK-iteration jjf L (G; K)jj 1 = sup ff(f L (G; K)(j!)) (3.40)! There currently is no method to directly synthesis an H 1 optimal controller. However, for complex perturbations, a method known as DK-iteration provides a good approximation. DK-Interation is a mix of H 1 -synthesis and μ 1 -analysis. Previously, the upper bound on μ was found (Equation 3.28), which as a matter of convenience is repeated here. μ UB : μ (M)» inf D2D ff(dmd 1 ) The optimal controller minimizes the peak value of μ UB over frequency. This is expressed more precisely by Equation min K stabilizing ( min jjdf L (G; K)D 1 jj 1 ) (3.41) D2D stable;min phase The process is to alternate between minimizing jjdf L (G; K)D 1 jj 1 with respect to either K or D, while holding the other constant. A simplified version of the DK-iteration procedure is summarized in the following: Prior to the first iteration, the transfer matrix D is initialized (often it initialized to an identity matrix). The iteration then proceeds as follows: 1. K-step Holding D fixed, find the stabilizing H 1 controller, K, that solves min jjdf L (G; K)D 1 jj 1 K stabilizing 35

58 2. D-step Find D(j!) to minimize at each frequency ff(df L (G; K)D 1 (j!)) holding F L (G; K) fixed. 3. Fit the magnitude of each elementofd(j!) to a stable, minimum phase transfer function D(s) and return to step 1. Iteration continues until jjdf L (G; K)D 1 jj 1 < 1 or until the H 1 norm ceases to decrease. While preceding D-K iteration process would need to be modified to handle full" D-blocks, it succinctly demonstrates the basic procedure. This chapter has dealt with developing the concepts and tools required for robust controller design. While the actual process of H 1 robust controller design/synthesis is covered in Chapter 6, the process is summarized by the following: ffl developing a system model, including model uncertainties ffl choosing a controller architecture and methodology ffl specifying design performance objectives using dynamic weights ffl setting-up the problem for solution using a robust controller design tool such as the μ-analysis and Synthesis Toolbox cfl ffl iterating between controller synthesis and testing, making performance tradeoffs (weight functions) as required to obtain a suitable robust controller In this Chapter, some of the origins of model uncertainty were introduced and the effects of this uncertainty on the system's closed-loop stability and performance were explored. Some of the deficiencies of applying classical SISO analysis and design techniques to MIMO systems were demonstrated. Several techniques were shown for including model uncertainty into our design models. Concepts regarding and the conditions for robust stability and robust performance were introduced and defined. 36

59 Relative to controller design, the complex singular value, μ, linear fractional transformations and D-K iteration were introduced. The next chapter provides an overview of the experimental HVAC system. 37

60 Chapter 4 Experimental HVAC System The role of experiments in confirming the validity of theoretical work is fundamental to the scientific method. Thus, the availability ofa suitable physical HVAC system was central to accomplishing the objectives of this project. Although an experimental HVAC system already existed and was available for use on this project, it was not suited for the required project tasks. It was decided to upgrade the existing HVAC system to meet the needs of this project. While the core HVAC components were reused, the system was extensively redesigned and rebuilt. The original experimental HVAC system was a simple heating system consisting of a boiler, a three-way water flow control valve, hot-water-to-air heat exchanger, constant speed centrifugal fan (blower) and two parallel blade dampers. The discharge air temperature control system consisted of multiple, single-input, single-output (SISO) proportional-integral (PI) control loops and electric actuators. In order to achieve the data acquisition (DAQ) and multiple-input, multiple-output (MIMO) control capabilities required by the project, the instrumentation and control hardware was entirely replaced. This involved designing a personal computer (PC) based DAQ and a control system consisting of the sensor and control interface (and associated enclosures), power control/distribution, safety interlock circuitry, variable frequency drive (variable air volume (VAV) subsystem), electric water heater subsystem, cabling and graphic user interface. In order to decrease the actuation times of the dampers and flow control valve, the electric actuators were replaced with pneumatic 38

61 Figure 4.1: Experimental HVAC system actuators (and the requisite electric-to-pneumatic transducers). After the interface and drive cabinets had been assembled and wired, the cabinets interfaced to the PC, sensors and actuators mounted and interfaced, the interface hardware and software were debugged and tested. The sensors were then calibrated and the actuators adjusted. Prior to commissioning, an energy balance analysis was performed to verify that the system was properly adjusted and calibrated. The completed experimental HVAC system is shown in Figure 4.1. This discharge air system (DAS) corresponds closely to the central air supply in a commercial HVAC system. The main difference being that in a central air supply, a cooling coil would be used in place of (or in conjunction with) the heating coil used in the experimental system. The experimental system, while only a portion of an overall HVAC system, was built using standard HVAC components and is representative ofatypical industrial hot water to air heating system. While there was nothing special regarding the 39

62 Cde Cdr Cwh Cvp Cbs Tae Ade Adr Tar Tai Twi Fw Tws Fws Pwh Avp Two Tao Fa experimental HVAC system itself, virtually every system parameter was monitored and available at the PC for the purpose of control and data logging. The PC based controller permits the implementation of advanced controls such as multiple-input, multiple-output (MIMO) robust and/or adaptive/learning controllers. Of course, it is also possible to implement the HVAC industry's standard" controls consisting of multiple single-input, single-output (SISO) controllers. The experimental HVAC system resulting from the redesign and rebuild effort, proved to be an excellent platform for the purposes of this project. The remainder of this chapter, provides an overview of the experimental HVAC system. 4.1 Components and Interconnection Heating Coil External Air T Mixing Box (Filter) T T ) Blower Discharge Air P E P E T T T Flow Control Valve T P E Interface Return Air Boiler Variable Freq. Drive Inputs Outputs Figure 4.2: Diagram showing the experimental HVAC system and interface signals A diagram representing the experimental system is shown in Figure 4.2. The main components comprising the HVAC experimental system are the heat exchanger (coil), water flow control valve, blower (fan) and the mixing chamber, which consists of two dampers to control the mix of external and return air. Pneumatic actuators are 40

63 used to position the dampers and control valve. Sensors are used to measure the air and water flow rates, external air, return air, input air and output air temperatures, electrical power input to the hot water heater, actuator positions and differential duct air pressure. The experimental system consists of the air and water subsystems, which converge at the heating coil. Here energy is transferred from the hot water flowing transversely through the coil to a cool air mass flowing over the coil fins, consequently warming the discharged air Air Handling Unit Airflow in the HVAC experiment is confined within a duct system constructed of oneinch thick fiberglass duct board". While the exterior surface of the duct is coated with a thin aluminum foil, the interior surface of the duct is not. This exposed fiberglass surface is rough and the resistance of the duct wall slows the air stream considerably. This causes a large differential in the airflow velocity from the center of the duct to the walls. To reduce this effect, a thin layer of polypropylene sheeting was bonded to the inside surfaces of the ducting, starting at the output of the coil to the entrance of the vertical duct, which houses the blower unit. A variable air volume (VAV) unit is used to move the air through the experimental system. A variable frequency drive (VFD) is used to vary the speed of a centrifugal blower (fan) and consequently change the volume of air transported through the system and into the heated space. The blower shown in Figure 4.3 is located in the discharge air duct. Cold, external (outside) air is combined at the mixing box, with warmer air returned from the heated space. The mix of external and return air is dependent upon the positions of the external and return air dampers. The positions of the two dampers are controlled so as to maintain a constant" combined damper opening and to mini- 41

64 Figure 4.3: Blower (variable speed fan) mize their effect on the airflow rate. Thus, a mix of 25% external and 75% return air is achieved by opening the external air damper 25% and the return air damper 75%. This effectively maintains a combined 100% damper opening. By varying the damper openings in this manner, it is possible to regulate the temperature of the air input to the heating coil to any level between the external and return air temperatures, with minimal effect on the airflow rate. In fact, the primary reason that the air flow rate is affected at all, is that the external and return air ducts differ in length and in the number of bends Hot Water Subsystem The closed loop hot water circuit was formed using both one inch and three-quarters inch copper tubing. To minimize heat losses, the tubing was insulated with cylinders of extruded foam, split to slip over the copper tubing. The main components comprising the hot water subsystem are shown in Figure 4.4. Water is propelled through the circuit by a single stage, in-line pump. The pump is placed between the coil return and the input to the water heater. A four element, 42

65 Figure 4.4: Water heater, pump and expansion tank electric hydronic block hot water heater 1 is used to heat the water to operating temperature. The water from the hot water heater flows to an expansion tank which maintains a constant 138 kpa working pressure over the operating water temperature range. The water then flows to the inlet of the coil and the inlet of the three-way flow control valve shown in Figure 4.5. Depending upon the position of the valve spindle, hot water can flow entirely through the coil or bypass the coil altogether, or travel through both the coil and the bypass. The water flow control valve is an equal percentage type. Each increment of valve opening, increases the flow by an equal percentage over the previous value. Such valves have an opening versus flow relationship such that the water flow increases exponentially as the valve opens Heat Exchanger The heat exchanger (coil) is the key energy transfer device. An end view of the experimental system's heating coil is shown in Figure 4.6. It is here that the air handling unit and hot water subsystems connect. A 23 row, four-pass, 24 inch x 24 inch cross section, counter flow heat exchanger was used in the experimental system. Attached to each 3=8 inch tube coil are inch-thick aluminum fins. 1 ARGO Industries, model fitted with four 5 kw heating elements 43

66 Figure 4.5: Water flow control valve Measurements and Sensors Various sensors were utilized to measure the system on the experimental HVAC system. Air and water temperatures were measured using industry standard 2 resistance temperature detectors (RTDs). Both point and averaging type RTD sensors were utilized in measuring air temperatures. The averaging Return Air RTD temperature sensor is shown in Figure 4.7. The air temperatures were measured using the type and model of RTD indicated in Table 4.1. Table 4.1: RTD sensor types used for temperature measurements Temp. Measurement Sensor Type Model No Air, External 1000 Ω Duct Point RTD Sensor Air, Return 1000 Ω Duct Averaging RTD Sensor Air, In to Coil 1000 Ω Duct Averaging RTD Sensor Air, Out of Coil 1000 Ω Duct Averaging RTD Sensor Water, Supply 1000 Ω Immersion RTD Sensor Water, In to Coil 1000 Ω Immersion RTD Sensor Water, Out of Coil 1000 Ω Immersion RTD Sensor The airflow rate through the experimental system was measured using a pitot 2 Landis and Staefa 1000 ohm RTDs having thin film platinum elements. 44

67 Figure 4.6: End view of heating coil array 3 and pressure to voltage transducer 4 to obtain the average airflow rate through the coil/system. The sensor and transducer (mounted below the sensor) are shown in Figure 4.8. Initially, the accuracy of the airflow rate measurement, provided by this instrument, was very poor. As previously mentioned, the rough interior surfaces of the duct caused a large difference in airflow velocity throughout the duct. The application of a thin layer of polypropylene to the inside surfaces of the ducting greatly reduced this effect. Even with this smooth coating, the airflow sensor measurements did not agree with the calibration measurements obtained by averaging individual pitot tube measurements 5 across the duct cross section. A linear correction of the measured value was required to obtain a more accurate airflow measurement. The linear correction was implemented into the interface application routine discussed later in this chapter. 3 Brandt model DSK x12 4 Setra 264 series low pressure transducer 5 using a pitot tube and manometer to measure the pressure increase over atmospheric 45

68 Figure 4.7: Return air RTD temperature sensor and damper The flow rate of water through the heating coil was measured using the coriolis flow sensor 6 shown in Figure 4.9. The water flow rate through the pump was also measured using an in-line turbine type sensor 7. The air pressure drops across various components of the experimental system were measured using pressure-to-voltage transducers 8. The pressure drops across the mixing box, first input filters, coil, duct and blower were measured. These sensors, while fully integrated and calibrated, were not used during the course of this project; they are however, available for future measurements. The actuator pressure sensors associated with each pneumatic control device (external damper, return air damper and water flow valve) were also fully integrated and calibrated, and while they were not used as part of this project, they are available for use in future projects. 6 Micro Motion model DS065S239SU sensor and model RFT9712 1PNU transmitter 7 ONICON model F-1310 flow sensor (voltage output) 8 Setra 264 Series (low) pressure to voltage transducer 46

69 Figure 4.8: Pitot array airflow sensor While not used by the closed loop controller, the actual actuator positions are measured using linear or angular potentiometers. These measurements were used in developing models for the HVAC system components. The linear potentiometer used to measure valve position is shown in Figure 4.5. The final value measured in the experimental system was the electrical power supplied to the hot water heater. The power is measured using an AC watt transducer 9 and a simple toroidal core transformer Control Devices The operation of the experimental HVAC system is governed by the following five control parameters External and return air damper positions determined the mix, thus the temperature of the air output from the mixing box to the input side of the heating 9 Ohio Semitronics Inc, model PC

70 Figure 4.9: Coriolis water flow sensor coil. The dual louvered dampers are positioned using linear pneumatic actuators 10. The return air damper is shown in Figure 4.7. Voltage-to-pressure transducers 11 control the pressure applied to extend the pneumatic actuators. Water flow valve position determined the flow rate of hot water through the heating coil. The equal percentage, three-way valve 12 was positioned using a linear pneumatic actuator 13. Voltage-to-pressure transducers controlled the pressure applied to extend the pneumatic actuators. Electrical power input to the hot water heater determined the heat energy available to raise the water temperature. The input power was controlled using a time-proportioning controller in conjunction with a solid state contactor to vary the on-off duty cycle of the 208 VAC power supplied to the heater elements. 10 Landis and Staefa, No. 4 Damper Actuator 11 Landis and Staefa, AO-P transducer model Landis and Staefa, No Landis and Staefa, 8-inch Valve Actuator No

71 Blower speed (with to a much lesser degree, the position of the return and external air dampers) determined the airflow rate through the system. The speed at which the three-phase electric motor, used to drive the blower 14 rotates was varied by changing the frequency of the AC voltage used to drive the motor. A variable frequency drive (VFD) was used to vary the drive frequency, consequently controlling the speed of the blower. 4.2 Control Hardware and Software As previously stated, the control hardware used in the experimental HVAC system was designed and built specifically to meet the control and measurement requirements of this (and future) HVAC research projects. While building this control system was a considerable undertaking, it was essential in providing the flexibility and capabilities required for developing advanced HVAC controls. Figure 4.10: PC based control hardware The personal computer 15 shown in Figure 4.10 and represented in the diagram of 14 Dayton model 4C770A with a 3-phase /460VAC 1725rpm, motor 15 Dell OptiPlex GX1p, 500 MHz Pentium with 256 MB RAM, having 4 ISA expansion card slots 49

72 Experimental HVAC System AT- MIO- 64E Input Card Computer Application SW MATLAB Simulink Real-time Workshop Windows Target PCL- 726 Output Card Figure 4.11: PC and MATLAB RTW/WT based interface Figure 4.11 (together with the control interface) was used for both data acquisition (DAQ) and control functions. MATLAB cfl 16, MATLAB toolboxes and MATLAB supported PC interface cards were used to provide DAQ and control capability in one application environment. In addition, system modeling and simulation were also carried out in the MATLAB environment Control Hardware Two interface cards were used in interfacing the computer and the experimental system. A 12-bit, 32 channel, differential analog input card 17, also having two analog outputs, and a user configurable, digital input or output port (8-bits) was used to interface the analog sensor signals. A 12-bit, six-channel, analog output card 18, also having 16 digital inputs and 16 digital outputs provides the control outputs. The external hardware was connected with the interface cards in the PC us- 16 MATLAB is published by the MATH WORKS Inc.; Natick, Mass. 17 National Instruments, AT-MIO-64E-1, Multifunction I/O interface (ISA) card 18 Advantec, PCL-726, 6 Channel D/A Output (ISA) card 50

73 ing additional hardware for signal conditioning, signal attenuation/amplification or switching. These operations were carried out using hardware contained within the interface (I/F) and drive cabinets shown in Figure Figure 4.12: Interface cabinet(top) and drive cabinet (bottom) As the name implies, the interface cabinet contained most of the hardware used to connect the computer to the experimental systems sensor inputs and control outputs. The differential analog input card, having 100 I/O connections, required the use of an interconnection adapter 19 to break-out the individual signals for wiring. Another specialized interface card 20, having a 1.0 ma current source and input filtering, was used in interfacing the RTD temperature sensors. This card was connected to the 19 National Instruments SC-2056 adapter and SH shielded cable assembly 20 National Instruments, SC-2042-RTD, 8-Channel RTD Signal Conditioning Accessory 51

74 interconnection adapter using a 26-pin ribbon cable. The remaining 24 differential analog input pairs were broken-out from the the interconnection adapter to terminal block connections using three, 26-pin ribbon cables and three ribbon cable to terminal block adaptor modules 21. The differential analog input card also provided two analog outputs and either eight digital inputs or eight outputs. The analog outputs were broken out from the interconnection adapter to terminal block connections while the digital inputs/outputs were not. The analog output card, having six analog outputs, 16 digital inputs and 16 digital outputs, was connected to four terminal block connectors 22 in the interface enclosure via four 20-pin ribbon cables. The drive cabinet contains the variable frequency drive and associated hardware used to power the blower motor. It also housed the logic and power devices used in controlling the power distributed to the interface cabinet and major system components. The bulk of the control signals used within the drive cabinet originated, or passed through the interface cabinet. The three-phase, 3=4 horsepower blower motor 23 is driven at speeds ranging from zero revolutions per minute (rpm) to 3450 rpm using a variable frequency drive 24 (VFD). The VFD was configured to turn the motor in only one direction. In addition to the analog speed reference supplied to the drive, two digital signals were required for initialization and cycling of the drives forward and run inputs. The control and timing logic required for the drives forward and run inputs was implemented in the Simulink I/F model. The VFD forms the three variable frequency alternating current (AC) outputs by switching at four kilohertz (KHz) current through transistors in the 21 Phoenix Contact, No , 26-pin ribbon cable to terminal block assembly 22 Advantec, ADAM-3920, 20-pin ribbon cable to terminal block assembly 23 Dayton 3-phase motor, P/N 3N042L 24 Allen-Bradley, 1-ffi 240v input VFD P/N 160S-AA04NSF1P1 52

75 drives power section. Due to this switching, quite a bit of noise is produced. In order to minimize the amount of switching noise transferred to the control inputs, a line-filter 25 was used at the 208 VAC supply to the VFD. To ensure the safe operation of the experimental HVAC system, a watch-dog-timer (WDT) circuit was implemented to prevent the distribution of power to devices when the control outputs were not under the control of an interface routine running on the PC. Since the analog (and digital) outputs retained their past values when a program is terminated, the WDT was required to prevent, among other things, overheating the boiler when a control program was not running on the PC. Figure 4.13: Boiler electrical cabinet The single-phase 208 VAC power supplied power applied the to the boilers four resistive heating elements was modulated in accordance to an analog reference signal. The heater power and control circuitry was housed in the boiler cabinet shown in Figure An analog voltage to duty cycle converter circuit 26 was used to drive 25 Allen Bradley, Line Filter module P/N 160S-RFA-9-A 53