Interaction Analysis and Control of Bioreactors for Nitrogen Removal

Transcription

1 IT Licentiate theses 27- Interaction Analysis and Control of Bioreactors for Nitrogen Removal BJÖRN HALVARSSON UPPSALA UNIVERSITY Department of Information Technology

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3 Interaction Analysis and Control of Bioreactors for Nitrogen Removal BY BJÖRN HALVARSSON December 27 DIVISION OF SYSTEMS AND CONTROL DEPARTMENT OF INFORMATION TECHNOLOGY UPPSALA UNIVERSITY UPPSALA SWEDEN Dissertation for the degree of Licentiate of Philosophy in Electrical Engineering with Specialization in Automatic Control at Uppsala University 27

4 Interaction Analysis and Control of Bioreactors for Nitrogen Removal Björn Halvarsson Division of Systems and Control Department of Information Technology Uppsala University Box 337 SE Uppsala Sweden c Björn Halvarsson 27 ISSN Printed by the Department of Information Technology, Uppsala University, Sweden

5 Abstract Efficient control of wastewater treatment processes are of great importance. The requirements on the treated water (effluent standards) have to be met at a feasible cost. This motivates the use of advanced control strategies. In this thesis the activated sludge process, commonly found in the biological wastewater treatment step for nitrogen removal, was considered. Multivariable interactions present in this process were analysed. Furthermore, control strategies were suggested and tested in simulation studies. The relative gain array (RGA), Gramian based interaction measures and an interaction measure based on the H 2 norm were considered and compared. Properties of the H 2 norm based measure were derived. It was found that the Gramian based measures, and particularly the H 2 norm based measure, in most of the considered cases were able to properly indicate the interactions. The information was used in the design of multivariable controllers. These were found to be less sensitive to disturbances compared to controllers designed on the basis of information from the RGA. The conditions for cost-efficient operation of the activated sludge process were investigated. Different fee functions for the effluent discharges were considered. It was found that the economic difference between operation in optimal and non-optimal set points may be significant even though the treatment performance was the same. This was illustrated graphically in operational maps. Strategies for efficient control were also discussed. Finally, the importance of proper aeration in the activated sludge process was illustrated. Strategies for control of a variable aeration volume were compared. These performed overall well in terms of treatment efficiency, disturbance rejection and process economy. Keywords: activated sludge process; biological nitrogen removal; bioreactor models; cost-efficient operation; interaction measures; multivariable control; wastewater treatment.

6 Acknowledgements First of all, I would like to express my sincere gratitude to my supervisor, Professor Bengt Carlsson, for all his help and encouragement during my research so far. Special thanks also go to my co-author and mentor Dr. Pär Samuelsson at Dalarna University (formerly with us here at the Division of Systems and Control) for all fruitful discussions and good advice. He read previous versions of this thesis and his suggestions certainly improved the quality. Furthermore, I wish to thank all colleagues at the Division of Systems and Control and the Division of Scientific Computing for providing such a pleasant working atmosphere. Part of this work has been financially supported by the EC th Framework programme as a Specific Targeted Research or Innovation Project (HIPCON, Contract number NMP2-CT ). Furthermore, I would like to thank Stiftelsen J. Gust. Richerts Minne for financial support. I am also grateful to Dr. Ulf Jeppsson for letting me use his Simulink implementation of BSM1 and to Assistant Professor Torsten Wik, Chalmers University of Technology, Göteborg, for taking the time of being my licentiate opponent. Finally, very special thanks go to my friends and to my family.

7 Contents 1 Introduction Interaction measures Motivational example Wastewater treatment systems The activated sludge process (ASP) The benchmark model BSM Control of WWTPs Thesis outline Controllability and Interaction Measures Introduction Systems description Controllability State controllability for continuous-time systems State controllability for discrete-time systems Output controllability The Relative Gain Array (RGA) Gramian based interaction measures The Hankel norm Energy interpretations of the controllability and observability Gramians for discrete-time systems The Hankel Interaction Index Array (HIIA) The Participation Matrix (PM) The selection of proper scaling An interaction measure based on the H 2 norm The Σ 2 interaction measure The H 2 norm Calculation of the H 2 norm Energy interpretation for discrete-time systems Properties of the H 2 norm based interaction measure Σ The H 2 norm and induced norms Examples

8 2.7.1 Example Example Example Example Example Conclusions Interaction Analysis in a Bioreactor Model Introduction The bioreactor model RGA analysis HIIA analysis Discussion Conclusions Interaction Analysis and Control of the Denitrification Process Introduction The bioreactor model Analysis of the model Linearization and scaling of the model RGA analysis of the model HIIA analysis of the model Control simulations Decentralized control Multivariable control Discussion Conclusions Economic Efficient Operation of a Pre-denitrifying Activated Sludge Process Introduction The model and the operational cost functions The nitrate fee The ammonium fee Simulation results Simulation results for the denitrification process Simulation results for the combined denitrification and nitrification process Discussion Conclusions

9 Aeration Volume Control in an Activated Sludge Process Discussion of Some Strategies Involving On-Line Ammonium Measurements 95.1 Introduction The simulation setup Description of the proposed control strategies The reference aeration control strategies Feedforward aeration volume control I and II Supervisory feedback aeration volume control Simulation results The reference aeration control strategies Feedforward aeration volume control I Feedforward aeration volume control II Supervisory feedback aeration volume control Conclusions A The minimized condition number 19 5

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11 Chapter 1 Introduction This thesis concerns the interaction analysis and the control of bioreactors for nitrogen removal. More precisely, models of the activated sludge process commonly found in the biological treatment of wastewater are considered. The interactions present in these processes will be analysed and different controller structures will be compared in simulation studies. The influence of various input signals on the treatment efficiency, both in terms of the treatment performance and in terms of the process economy, will also be investigated. 1.1 Interaction measures Many control systems of today are multivariable. This means that they have multiple inputs and multiple outputs. Such systems are called multiple-input multiple-output (MIMO) systems. Compared to single-input single-output (SISO) systems, the control design for MIMO systems is more elaborate. One reason for this is that different parts of a multivariable system may interact and cause couplings in the system. As an example, consider a shower with separate flow control for hot and cold water. This is a MIMO system since the two inputs, the flow of hot water and the flow of cold water, are utilized to control the two outputs which are the flow from the tap and the temperature of the effluent water. Evidently, when changing one of the inputs, both of the outputs will be affected. This means that there are significant couplings in the system. In other words, interaction occurs if a change in one input affects several outputs. Often, an easy way to control a fairly decoupled MIMO system is to use a multi-loop strategy, i.e. to separate the control problem into several single-loop SISO systems and then use conventional SISO control on each of the loops, see Kinnaert (1995) and Wittenmark et al. (1995). This gives rise to the pairing problem: Which input signal should be selected to control which output signal to get 7

12 the most efficient control with a low degree of interaction? In real-life applications the considered MIMO system could be rather complex: In the chemical process industry a complexity of several hundred control loops is not unusual, see Wittenmark et al. (1995). The proper pairing selection is thus often not at all obvious. Also, the choice of pairing is crucial since a bad choice may give unstable systems even though each loop separately is stable. This problem could arise due to interaction between the different loops. Generally, the stronger the interactions, the harder it is to obtain satisfactory control performance using a multi-loop strategy. Evidently, there is a need for a measure that can both give some advise when solving the pairing problem and that also quantifies the level of interaction occurring in the system. One such measure is the Relative Gain Array (RGA) developed by Bristol (19). The RGA considers steady-state properties of the plant and gives a suggestion on how to solve the pairing problem in the case of a decoupled (decentralized) control structure. Such structure will be diagonal. It also indicates which pairings that should be avoided due to possible stability and performance problems. Later, a dynamic extension of the RGA was proposed in the literature, see e.g. Kinnaert (1995) for a survey. With the extension, the RGA could be used to analyse the considered plant at any frequency but still only at one single frequency at a time. A recent approach to define a dynamic relative gain array was made by Mc Avoy et al. (23). Moreover, the RGA can be generalized for non-square plants and be employed as a screening tool to get a suggestion on what inputs or outputs that should be removed in the case of excess signals, see Skogestad and Postlethwaite (199). Over the years, several resembling tools have been developed. One such example is the Partial Relative Gain (PRG) suggested by Häggblom (1997) that is intended to handle the pairing problem for larger systems in a more reliable way than the conventional RGA. Other examples are the µ interaction index (Grosdidier and Morari, 1987) and the Performance Relative Gain Array (PRGA) (Hovd and Skogestad, 1992). An interesting novel approach is found in (He and Cai, 24) where pairings are found by minimizing the loop interaction energy characterized by the General Interaction (GI) measure. This measure is used in combination with the pairing rules of the RGA and of the Niederlinski Index (NI) (Niederlinski, 1971). The NI can be used as an indicator of possible instability issues when solving the pairing problem. In the Effective RGA (ERGA) proposed by Xiong et al. (25) the steady state gain and the bandwidth of the process are utilized to form a dynamic interaction measure. He et al. (2) suggest an algorithm for control structure selection where the ideas by He and Cai (24) are further developed. Other examples are given by Kinnaert (1995) where a survey of interaction measures for MIMO systems can be found. 8

13 The RGA provides only limited knowledge about when to use multivariable controllers and gives no indication of how to choose multivariable controller structures. A somewhat different approach for investigating channel interaction was therefore employed by Conley and Salgado (2) and Salgado and Conley (24) when considering observability and controllability Gramians in so called Participation Matrices (PM). In a similar approach Wittenmark and Salgado (22) introduced the Hankel Interaction Index Array (HIIA). These Gramian based interaction measures seem to overcome most of the disadvantages of the RGA. One key property of these is that the whole frequency range is taken into account in one single measure. Furthermore, these measures seem to give appropriate suggestions for controller structures both when a decentralized structure is desired as well as when a full multivariable structure is needed. The use of the system H 2 norm as a base for an interaction measure has been proposed by Birk and Medvedev (23) as an alternative to the HIIA Motivational example As a first motivational example, consider a system previously analysed by Goodwin et al. (25). The system has a transfer function G(s) = 1(s+.4) (s+4)(s+1) 2 s s+3.5 s+1 2(s.4) (s+4)(s+2) 3 s+3 1 s+1 1 s+2 3(s+.4) (s+4)(s+3) and a steady-state gain of G() = The interaction measures are: Λ(G()) = Σ H = , Φ = , Σ 2 = , 9

14 where Λ is the RGA, Σ H is the HIIA, Φ is the PM and finally, Σ 2 is an H 2 norm based interaction measure. All of these will be defined in Chapter 2. The aim in this example is to find the decentralized pairing recommendation so that each input signal is paired uniquely with one output signal. In the case of the RGA input-output pairings corresponding to elements close to one should be selected and negative elements should be avoided. The other of the considered interaction measures recommend the input-output pairings that result in the largest sum when adding the corresponding elements in the measure. Evidently, all interaction measures suggest the diagonal pairing: input 1 output 1, input 2 output 2 and input 3 output 3. However, no useful pairing information can be found by inspecting G() or G(). This demonstrates the need of dedicated interaction measures even for pairing suggestions relevant for operation in steady state. Even though, the considered interaction measures are rather similar in this particular example, this is not generally the case. Further examples and theoretical differences between these will be examined in Chapter Wastewater treatment systems Until some time during the 19:th century, the activity of man had not affected the environment to any appreciable extent. When the industrial revolution came, a rapidly increased standard of living as well as a substantially population growth followed. The society became more and more urbanized and the problem of taking care of the human waste products and waste disposal became a serious (hygienic) problem. The introduction of the water closet solved the problem locally, but only locally, since the problem was instead moved to the surrounding environment with an increased load on the recipients (e.g. lakes and rivers). This could not be handled by the recipients without heavily disturbed local ecosystems. The degradation of organic material present in the wastewater, consumes oxygen and the recipient will thus suffer from lack of oxygen after some while. Even if most of the organic matter is removed before the wastewater reaches the recipient, chemical compounds such as phosphorous and nitrogen are still present, and may cause eutrophication (i.e. over-fertilization). Eventually, this will also result in a lack of oxygen. Therefore, the aim of wastewater treatment should be to remove both the content of organic matter and suspended solids as well as the content of nitrogen and phosphorous to a reasonable extent. In the beginning of the 2:th century, the first wastewater treatment plants were introduced in Sweden. They were simple plants using only a mechanical treatment step. This step could consist of a grid and a sand filter to remove larger objects and particles. In the late 195 s the biological treatment step, was introduced. Hereby, microorganisms (e.g. bacteria) are used to remove organic matter present in the incoming wastewater. Later, in 1

15 the 197 s, the chemical treatment step, was employed to reduce the content of phosphorous. Nowadays, the biological step is also utilized to reduce the content of nitrogen and phosphorous. A general wastewater treatment plant (WWTP), consisting of the above mentioned steps, is given in Figure 1.1. The sludge also needs to be treated. The main procedures are depicted in Figure 1.1. In the thickening procedure, the sludge is concentrated. Then, the sludge is stabilized in order to reduce odor and pathogenic content. Finally, the moisture content of sludge can be reduced by the use of dewatering. For a description of how to practically realize these steps, see e.g. Hammer and Hammer Jr. (28). 1 Grid Mechanical treatment 2 Biological treatment 3 Sand filter Primary Sedimentation Activated sludge Chemical treatment Chemicals Preciptation Effluent water 4 Sludge treatment Dewatering Dewatered sludge Sludge thickening Stabilization Supernatants + Backwashing Figure 1.1: A general WWTP (Kommunförbundet, 1988). In the complex process of wastewater treatment, many different causeeffect relationships exist, and therefore, there are many possible choices of input and output signals, see Olsson and Jeppsson (1994). This makes the WWTP models particularly interesting to study with respect to the interactions present and the selection of proper control structures. When treating wastewater, the aim is to reduce as much as possible of the undesired constituents such as organic matter, nitrogen and phosphorous. This is commonly done using wastewater treatment plants. In a WWTP several biological processes occur simultaneously. These processes need to be properly controlled in order to maintain the concentrations of undesired constituents in the outlet water within the legislated limits. As the public awareness of environmental issues increases, the environmental legislation becomes stricter, and thus, the requirements on WWTPs become even harder to fulfill. The used control strategies need then to be as efficient as possible, see e.g. Olsson and Newell (1999). Therefore, models of the WWTP processes are interesting to study with respect to the choice of e.g. control structure. An example of such models are the bioreactor models. From a theoretical point of view, the bioreactor models are non-linear 11

16 multivariable systems that may contain a significant degree of coupling. Hence, this also gives an interesting opportunity to test the performance of the methods for input-output pairing selection mentioned in the previous section. The aim of Section 1.2 is to give a brief description of the bioreactor models that will be analysed in the forthcoming chapters The activated sludge process (ASP) The biological treatment step can be realized in several different ways. One of the most common is the activated sludge process where activated sludge, i.e. microorganisms (mainly bacteria), is employed to degrade (i.e. oxidize) organic material. The basic set-up consists of an aerated basin where oxygen is added by blowing air into the water, and a settler tank, see Figure 1.2. In the aerated basin, the bacteria degrade the incoming organic material while consuming oxygen. In this way the microorganisms fulfill their need of energy and as a result bacterial growth will occur. Together with decayed microorganisms and other particulate material, the living microorganisms form sludge. To separate the sludge from the purified water a settler, where the sludge settles, can be used directly after the aerated tank. Since the amount of microorganisms needs to be kept at a high level, some sludge is recirculated as shown in Figure 1.2, while the rest is removed as excess sludge. With the excess sludge, some nitrogen (and phosphorus) is removed, but still far too much remains. Influent Effluent Aerobic Settler Sludge recirculation Excess sludge Figure 1.2: A basic activated sludge process with an aerated basin and a settler. However, if the activated sludge process is extended to consist of both aerated and non-aerated (anoxic) basins, then bacteria may be employed for efficient nitrogen removal. In the aerated basins, bacteria oxidize ammonium to nitrate in a two-step process called nitrification: NH O 2 NO 2 + H 2O + 2H +, NO 2 +.5O 2 NO 3. For these processes to occur, the concentration of dissolved oxygen (DO) 12

17 must be sufficiently high and a long sludge age (the average time each particle stays in the system) is required due to slow bacteria growth. In the anoxic tanks, another type of bacteria is employed in the denitrification process, described by 2NO 3 + 2H+ N 2 (g) + H 2 O + 2.5O 2 i.e., the bacteria convert nitrate into nitrogen gas using the oxygen in the nitrate ions. However, no dissolved oxygen should be present for this process to take place, instead, a sufficient amount of readily biodegradable substrate is needed. Hence, together, nitrification and denitrification convert ammonium into nitrogen gas which is harmless to the environment. For further descriptions of these processes, see Henze et al. (1995). Nitrogen removal can be performed in several different types of WWTPs. One of the most popular is the pre-denitrification system (ibid.). In this design, the anoxic tanks are placed before the aerated basins, and thus, denitrification is performed before the nitrification process, see Figure 1.3. Influent Effluent Anoxic Anoxic Aerobic Aerobic Aerobic Internal recirculation Sludge recirculation Settler Excess sludge Figure 1.3: An activated sludge process configured for nitrogen removal (predenitrification). To supply the denitrification process with nitrate, there is a feedback flow from the last tank as shown in Figure 1.3. In some cases, when the influent water has a low content of carbon, the bacteria in the anoxic tank need to be fed with an external carbon source. For this purpose, methanol or ethanol is often used. For a further discussion about the ASP, see e.g. Olsson and Newell (1999) and Hammer and Hammer Jr. (28) The benchmark model BSM1 The comparison between different control strategies for a WWTP is often difficult due to the variable influent conditions and the high complexity of a WWTP. Therefore, to enable objective comparisons between different control strategies, a simulation benchmark activated sludge process, Benchmark Simulation Model No.1 (BSM1), has been developed by the COST 82 Working Group No.2, see Copp (22) and IWA (November 19, 27). 13

18 In the BSM1 a typical activated sludge process with pre-denitrification is implemented. It consists of five biological reactor tanks configured in-series. The first two tanks have a volume of 1 m 3 each, and are anoxic and assumed to be fully mixed. The remaining three tanks are aerated and have a volume of 1333 m 3 each. All biological reactors are modelled according to the ASM1 model. Finally, there is a secondary settler modelled using the double-exponential settling velocity function of Takács et al. (1991). To get an objective view of the performance of the applied control strategy, it is important to run the BSM1 simulation with different influent disturbances. Therefore, influent input files for three different weather conditions dry, stormy and rainy weather are available together with the benchmark implementation. A number of different performance criteria are defined, such as various quality indices and formulas for calculating different operational costs Control of WWTPs As previously stated, WWTPs may be seen as complex multivariable systems. Therefore, to obtain satisfactory control performance, it is often necessary to use more advanced control strategies. However, since wastewater treatment traditionally has been seen as non-productive compared to the industry, the extra investments needed to employ such advanced control strategies have been hard to justify economically. Nowadays, as the effluent demands get tighter, the interest for more advanced control strategies is again awakening, see Olsson and Newell (1999). The plant has to be run economically and at the same time the discharges to the recipient should be kept at a low level. The control problem is hence twofold. The economical aspect involves minimizing operational costs such as pumping energy, aeration energy and dosage of different chemicals. Consequently, the main problem is how to keep the effluent discharges below a certain pre-specified limit to the lowest possible cost (ibid.). One way of solving this conflict of interest is to design the control algorithms in such a way that the overall operational costs are minimized. To make sure that also the wastewater treatment performance demands are fulfilled, the effluent discharges can be economically penalized. The corresponding cost can then be included together with the actual costs (energy and chemicals) in the calculation of the overall cost. Control handles for nitrogen removal In the nitrogen removal process, there are several variables that can be used as actuators, or control handles, to control the outputs. In a predenitrification system, there are five main control handles, as stated by Ingildsen (22): 14

19 1. The airflow rate (in the aerated compartments); 2. The internal recirculation flow rate; 3. The external carbon dosage; 4. The sludge outtake flow rate (excess sludge); 5. The sludge recirculation flow rate. In this thesis, only the three first of these are considered. The last two control handles are described by for example Yuan et al. (21) and Yuan et al. (22). The first control handle, the airflow rate, is employed to affect the DO concentration in the aerated compartments. Hereby, the performance of the autotrophic nitrification bacteria will be influenced. Most common today is to control the airflow rate to maintain a specific DO level. Another way is to make use of online-measurements of the ammonium concentration in the last aerated compartment, and let these control the time-varying DO set point, see e.g. Lindberg (1997). The internal recirculation flow rate affects the supply of nitrate for the denitrification process but also the DO concentration in the anoxic compartments since some DO may be transported from the last aerated compartment. The DO transportation between the processes, can however, be reduced by introducing an anoxic tank after the last aerated basin. External carbon dosage can be applied when the influent water does not have enough readily biodegradable substrate to feed the denitrification bacteria. Controlled output signals for nitrogen removal The primary outputs from a WWTP are the effluent ammonium concentration, the organic matter, the nitrate concentration and the suspended solids, see Ingildsen (22). For a more thorough discussion on cause-effect relationships in activated sludge plants, see Olsson and Jeppsson (1994). 1.3 Thesis outline Chapter 2 In Chapter 2 different interaction measures are reviewed and compared for some MIMO plants. In addition a simulation study is performed where the influence of a time delay on the coupling is examined. State controllability and output controllability are also discussed and further motivations for incorporating the concept of output controllability in an interaction measure are given. A H 2 norm based interaction measure is investigated. 15

20 Chapter 3 In Chapter 3 the interactions in a multivariable ASP model configured for nitrogen removal are studied. The RGA and the HIIA are utilized to quantify the degree of coupling present in the system. Both the nitrification and the denitrification process are studied since the output signals (the controlled signals) are the effluent concentration of ammonium and the effluent concentration of nitrate. The input signals (control handles) are the dissolved oxygen concentration set point in the aerobic compartment and the internal recirculation flow rate. The material is based on: Halvarsson, B., P. Samuelsson and B. Carlsson (25). Applications of Coupling Analysis on Bioreactor Models. In: Proceedings of the 1th IFAC World Congress, Prague, Czech Republic, July 4-8. Chapter 4 Chapter 4 once again considers the interactions present in an ASP. Here, the focus is on controlling the denitrification process when an external carbon source is added. Thus, one of the two considered control handles (input signals) is the readily biodegradable organic substrate in the influent water (which has the same influence as an external carbon source would have). The other input signal is the internal recirculation flow rate. The output signals (controlled signals) are the nitrate concentration in the anoxic compartment and the nitrate concentration in the effluent. The model is analysed using the RGA and the HIIA. The results are discussed from a process knowledge point of view, and are also illustrated with some control experiments. The chapter is based on: Samuelsson, P., B. Halvarsson and B. Carlsson (25). Interaction Analysis and Control Structure Selection in a Wastewater Treatment Plant Model. IEEE Transactions on Control Systems Technology 13(). Samuelsson, P., B. Halvarsson and B. Carlsson (24). Analysis of the Input-Output Couplings in a Wastewater Treatment Plant Model. Technical Report Division of Systems and Control, Department of Information Technology, Uppsala University, Uppsala, Sweden. Chapter 5 In this chapter, the focus is on finding optimal set-points and cost minimizing control strategies for the activated sludge process. Both the denitrification and the nitrification process are considered. In order to compare different 1

21 criterion functions, simulations utilizing the COST/IWA simulation benchmark (BSM1) are considered. By means of operational maps the results are visualized. It is also discussed how efficient control strategies may be accomplished. The material is based on: Halvarsson, B. and B. Carlsson (2). Economic Efficient Operation of a Predenitrifying Activated Sludge Process. HIPCON Report number HIP-8-v1-R Deliverable D.5. Uppsala University, Uppsala, Sweden. 1 which is an extended version of: Samuelsson, P., B. Halvarsson and B. Carlsson (27). Cost-Efficient Operation of a Denitrifying Activated Sludge Process. Water Research 41(27) Samuelsson, P., B. Halvarsson and B. Carlsson (25). Cost Efficient Operation of a Denitrifying Activated Sludge Plant An Initial Study. Technical report Division of Systems and Control, Department of Information Technology, Uppsala University, Uppsala, Sweden. In these two references only the denitrification process is studied. Chapter In the concluding chapter the influence of the aeration on the efficiency of the nitrogen removal in an ASP is studied. Different strategies for controlling the DO set point as well as the aerated volume are compared in terms of efficiency in a simulation study. Chapter is based on: Halvarsson, B. and B. Carlsson (2). Aeration Volume control in an activated sludge process Discussion of some strategies involving on-line ammonia measurements. HIPCON Report number HIP-8- v1-r Deliverable D.5. Uppsala University, Uppsala, Sweden. 1 1 This paper is an internal EU project report which is available from the author. 17

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23 Chapter 2 Controllability and Interaction Measures In this chapter the concept of controllability is discussed and different interaction measures are reviewed and compared. In particular, the two Gramian based interaction measures the Hankel Interaction Index Array (HIIA) and the Participation Mtarix (PM) are considered. Moreover, motivations for incorporating the concept of output controllability in an interaction measure are given and a H 2 norm based interaction measure is investigated. 2.1 Introduction There are today several different measures for quantifying the level of inputoutput interactions in multivariable systems. The perhaps most commonly used is the Relative Gain Array (RGA) introduced by Bristol (19). The RGA is a measure that can be employed in order to decide a suitable inputoutput pairing when applying a decentralized control structure. It can also be used to decide whether a certain pairing should be avoided. This measure, however, suffers from some major disadvantages. For instance it only considers the plant in one frequency at the time and it often provides limited knowledge about when to use multivariable controllers. Neither is the RGA able to give advice on how to select an appropriate multivariable controller structure. The RGA is also unable to suggest a proper pairing in the case of plants with triangular structure or large off-diagonal elements (this particular situation is further investigated in Chapter 3). A different approach for investigating channel interaction was employed by Conley and Salgado (2) when considering observability and controllability Gramians in so called Participation Matrices (PM). In a similar approach Wittenmark and Salgado (22) introduced the Hankel Interaction Index Array (HIIA). These Gramian based interaction measures seem to overcome most of the disadvantages of the RGA. One key property of 19

24 these is that the whole frequency range is taken into account in one single measure. Furthermore, these measures seem to give appropriate suggestions for controller structures both when a decentralized structure is desired and when a full multivariable structure is needed. For applications and comparisons between the RGA and various types of Gramian based interaction measures, see for instance Salgado and Conley (24), Birk and Medvedev (23), Samuelsson et al. (25c) and Halvarsson et al. (25). The use of the system H 2 norm as a base for an interaction measure has been proposed by Birk and Medvedev (23) as an alternative to the HIIA. In HIIA the Hankel norm forms the basis. In this chapter this H 2 norm based interaction measure is investigated. Further motivations for incorporating the concept of output controllability in an interaction measure are also given. This chapter is organized in the following way: Section 2.2 gives a brief description of the systems that will be analysed and some general assumptions. In Section 2.3 state controllability and output controllability are defined. Section 2.4 introduces the reader to the RGA. Section 2.5 presents the Gramian based interaction measures the Hankel Interaction Index Array and the Participation Matrix and their theoretical foundations. Section 2. defines the H 2 norm based interaction measure and investigates its relation to the concept of output controllability. In Section 2.7 different interaction measures are compared in the analysis of the interactions present in some MIMO systems. A simulation study is also performed. Finally, the conclusions are drawn in Section Systems description Consider a continuous-time linear time-invariant system, with inputs at time t given by the N 1 vector u(t) and outputs at time t given by the p 1 vector y(t). The system can be described as a state-space realization ẋ(t) = Ax(t) + Bu(t), y(t) = Cx(t) + Du(t) (2.1) where A, B, C and D are matrices of dimension N N, N q, p N and p q, respectively. x(t) is the state vector. Furthermore, a discrete-time system x(t + 1) = Ax(t) + Bu(t), y(t) = Cx(t) + Du(t) (2.2) will be considered as well. Note that (A,B,C,D) both denote the continuoustime system matrices and the discrete-time system matrices. These do not generally coincide; what quadruple of matrices that are referred to will be clear from the context. 2

25 2.3 Controllability State controllability for continuous-time systems Most often, the term controllability refers to the property of a system as being state controllable. The concepts of state controllability and state observability were introduced by Kalman, see for example Kalman et al. (193), Kreindler and Sarachik (194) and the references therein. With an initial state x(t ) and an input u(t), the solution of (2.1) for t t is given by x(t) = e A(t t ) x(t ) + t t e A(t τ) Bu(τ)dτ. (2.3) This is a standard result found in many text books such as (Skogestad and Postlethwaite, 199; Zhou et al., 199). Since the system is time-invariant t can be set to. A system with an arbitrary initial state x() = x is said to be state controllable if there exists a piecewise continuous input u(t) such that x(t 1 ) = x 1 for any final state x 1 and t 1 >. Equivalently, a state controllable system can be transferred from any initial state x(t ) to any final state x(t 1 ) in finite time. It can be verified using (2.3) that one input that satisfies this criterion is given by (ibid.) u(t) = B T e AT (t 1 t) W c (t 1 ) 1 (e At 1 x x 1 ) (2.4) where W c (t) is a Gramian matrix defined as W c (t) = t e Aτ BB T e AT τ dτ. (2.5) Clearly, for the solution in (2.4) to exist, the inverse of W c (t) needs to exist, i.e. W c (t) must have full rank for every t >. For a stable timeinvariant system it is enough to require W c ( ) to have full rank. Hence, state controllability can be investigated by considering the controllability Gramian, P, defined for stable time-invariant systems as P e Aτ BB T e AT τ dτ. (2.) If P has full rank the system is state controllable. Similarly, a stable system will be state observable if the observability Gramian, Q, defined as Q e AT τ C T Ce Aτ dτ (2.7) has full rank. These Gramians can be obtained by solving the following continuous-time Lyapunov equations (Skogestad and Postlethwaite, 199): AP + PA T + BB T =, (2.8a) 21

26 A T Q + QA + C T C =. (2.8b) The rank of P is the dimension of the controllable subspace corresponding to the given system, and correspondingly, the rank of Q is the dimension of the observable subspace of the same system. State controllability and state observability can also be examined by considering the matrices 1 W c [B AB... A N 1 B], (2.9a) W o C CA. CA N 1. (2.9b) The system (A,B) is then state controllable if W c has full rank N where N is the number of states. Similarly, the system (A,C) is state observable if W o has full rank N. Even though a system is state controllable, it should be noted that there is no guarantee that the system can remain in its final state x 1 as t. Furthermore, nothing is said about the behaviour of the required inputs. These can both be very large and change suddenly. Therefore, state controllability is rather a result of theoretical interest than a result of practical importance State controllability for discrete-time systems The discrete-time case can be treated similarly. The discrete controllability Gramian is given by (Weber, 1994) and the discrete observability Gramian by P = W c W T c (2.1) Q = W T o W o. (2.11) Similarly to the continuous-time case, these Gramians can also be obtained as the solutions to the (discrete-time) Lyapunov equations APA T P + BB T =, (2.12a) 1 It can be verified that the controllable states can be expressed as linear combinations of the matrices B, AB,..., A n 1 B by considering the solution to (2.1) given in (2.3) (let x = ) and expressing e At as a power series and using the Cayley-Hamilton theorem. This motivates the introduction of W c. The Cayley-Hamilton theorem says that every quadratic matrix satisfies its own characteristic equation. For details, see for instance Glad and Ljung (1989) where also further motivations for the definition of W o are given. 22

27 A T QA Q + C T C =. (2.12b) Once again, the same symbols are used for both the continuous-time quantities and the discrete-time counterparts. Note also that in the continuoustime case, P and Q cannot be obtained from W c and W o as in (2.1) and (2.11) for discrete-time systems Output controllability Whereas state controllability considers the ability of affecting the states of a given system by manipulating the inputs, output controllability rather considers the situation of affecting the outputs by means of manipulating the inputs. In practical control problems it is often more relevant to be able to control the outputs rather than the states (see e.g. Kreindler and Sarachik (194)). State controllability is neither necessary nor sufficient to be able to control the outputs as pointed out by Kreindler and Sarachik (194). According to Skogestad and Postlethwaite (199), state controllability is rather a system theoretical concept and it does not imply that the system is controllable from a practical point of view. For this reason the concept of output controllability was introduced. Kreindler and Sarachik (194) discuss time-varying plants of the form given in (2.1) and defines a plant as being completely output-controllable on [t o,t f ] if for given t and t f any final output y(t f ) can be attained starting with arbitrary initial conditions in the plant at t = t. For a plant without a direct term, i.e. D(t) =, this holds if and only if the Gramian P oc (t,t f ) tf t H y (t f,τ)h T y (t f,τ)dτ (2.13) is non-singular (Kreindler and Sarachik, 194) where H y (t,τ) is the impulse response matrix (Skogestad and Postlethwaite, 199), For linear timeinvariant stable plants with t set to the Gramian in (2.13) transforms to the output controllability Gramian given by P oc = = C Ce Aτ BB T e AT τ C T dτ e Aτ BB T e AT τ dτ C T = CPC T. (2.14) For plants including direct transmission (i.e. D ) the matrix DD T has to be added to the Gramian in (2.14) for the output controllability criteria to be valid. In contrast to the state controllability Gramian, P, P oc is independent of the selected state-space realization. To see this, change the state coordinates 23

28 by multiplying the state vector x(t) with a linear non-singular transformation matrix T. This is a similar transformation that transforms the state vector x(t) to z(t) = Tx(t). The plant can now be described by ż(t) = TAT 1 z(t) + TBu(t), y(t) = CT 1 z(t) + Du(t). (2.15) For the new realization, the output controllability Gramian becomes P oc = = = C = CPC CT 1 e TAT 1t TBB T e (T 1 ) A T t (T 1 ) C dt CT 1 Te At T 1 TBB T (T 1 ) e A t T (T 1 ) C dt e At BB e A t dt C = P oc (2.1) where it has been utilized that e TAT 1t = {e At = I + At + A2 t 2 2! } +... =... = Te At T 1 and that the plant is assumed to be time-invariant so that C is independent of time. Clearly, P oc = P oc and thus P oc is independent of the selected state-space realization. 2.4 The Relative Gain Array (RGA) The static RGA for a quadratic plant is given by RGA(G) = G(). (G() 1 ) T (2.17) where G() is the steady-state transfer function matrix and. denotes the Hadamard or Schur product (i.e. elementwise multiplication). Each element in the RGA can be regarded as the quotient between the open-loop gain and the closed-loop gain. Hence, the RGA element (i, j) is the quotient between the gain in the loop between input j and output i when all other loops are open and the gain in the same loop when all other loops are closed. For a full derivation of the RGA, see e.g. Bristol (19), Kinnaert (1995) or Skogestad and Postlethwaite (199). In the case of a 2 2 system, the following RGA matrix is obtained: [ ] λ 1 λ RGA(G) =. (2.18) 1 λ λ Depending on the value of λ, five different cases occur (Kinnaert, 1995): 24

29 λ = 1: This is the ideal case when no interaction between the loops is present. The pairing should be along the diagonal, i.e. u 1 y 1 and u 2 y 2 ; λ = : This is the same situation as above, except that now the suggested pairing is along the anti-diagonal, i.e. u 1 y 2, u 2 y 1 < λ < 1: This case is not desirable since the gain increases (i.e. ĝ ij increases) when the loops are closed, hence, there is interaction; λ > 1: Now, the gain decreases when the loops are closed. This situation is therefore also undesirable. λ < : This situation corresponds to the worst case scenario since now, even the sign changes when the loops are closed and this is highly undesirable. The conclusion is that u 1 should only be paired with y 1 when λ >.5, otherwise it should be paired with y 2. For the higher-dimensional case, the rule should be to choose pairings that have an RGA-element close to one. Negative pairings should definitely be avoided. 2.5 Gramian based interaction measures The Hankel norm The controllability and observability Gramians as defined in (2.) and (2.7) can be seen as measures of how hard it is to control and to observe the states of the given system. Unfortunately, both of these Gramians depend on the chosen state-space realization. However, as can be verified, the eigenvalues of the product of these will not. The Hankel norm for a system with transfer function G (continuous-time or discrete-time) can be calculated as G H = λ max (PQ) = σ H 1 (2.19) where σ1 H is the maximum Hankel singular value (HSV). Clearly, this measure is invariant with respect to the state-space realization and it is therefore well suited as a combined measure for controllability and observability. In fact, the Hankel singular values can be interpreted as a measure of the joint controllability and observability of the states of the considered system, see for instance Farsangi et al. (24), Skogestad and Postlethwaite (199) and Lu and Balas (1998). Furthermore, the HSV:s of G can be regarded as measures of the gain between past inputs and future outputs since these are the singular values of the Hankel matrix (defined below) for discretetime systems, or equivalently, for the Hankel operator (defined below) for 25

30 continuous-time systems (Zhou et al., 199; Skogestad and Postlethwaite, 199; Wilson, 1989; Antoulas, 21; Weber, 1994; Glover, 1984; Wittenmark and Salgado, 22). To see this, consider the discrete-time time-invariant system given in (2.2) and let D =. Assume starting from zero initial state, x( L) =, the influence of the L past inputs on the state x() is given by (Weber, 1994; Glover, 1984) x() = [B AB... A L 1 B] u( 1) u( 2). u( L) = W c u( 1) u( 2). u( L) (2.2) and the influence of the initial state x() on the L future outputs is given by (Weber, 1994; Glover, 1984) y() y(1). y(l 1) = C CA. CA L 1 x() = W ox() (2.21) where it is assumed that u(t) = for t. When L = N, W c and W o are the controllability and observability matrices, respectively. For L > N these are the extended controllability matrix and the extended observability matrix. To be able to reconstruct all of the states in the state vector at time, i.e. x(), from the past inputs according to (2.2), W c must have full rank N. Similarly, W o must have full rank N so that the outputs can be found from (2.21). For a derivation of (2.2) and (2.21), see Weber (1994). Combining (2.2) and (2.21) the result is the following expression that links the past inputs to the future outputs via the state x() at time zero (Weber, 1994; Antoulas, 21) y() y(1). y(l 1) = W ox() = W o W c u( 1) u( 2). u( L) = Γ u( 1) u( 2). u( L). (2.22) Γ is the Hankel matrix which in the considered time-invariant case is defined as (Antoulas, 21; Weber, 1994) Γ = S 1 S 2... S L S 2 S 3... S L+1... S L S L+1... S 2L 1 2 (2.23)

31 where {S k } are the Markov 2 parameters of the system. For multiple-input multiple-output (MIMO) systems the Markov parameters are matrices and consequently, the Hankel matrix is a block matrix. The impulse response in the discrete-time case for the system given in (2.2) with D = is given by (Antoulas, 21) { CA h(t) = t 1 B t > (2.24) t <. For this reason the Hankel matrix can be expressed as CB CAB... CA L 1 B CAB CA 2 B... CA L B Γ =.... (2.25) CA L 1 B CA L B... CA 2L 2 B Clearly, it follows that Γ = W o W c as stated in (2.22). The Hankel singular values equal the non-zero singular values of the Hankel matrix. In the continuous-time case, the counterpart to the Hankel matrix is the Hankel operator Γ given by (see e.g. Antoulas (21)) y(t) = (Γu)(t) = g(t + τ)u( τ)dτ t (2.2) where g(t) is the continuous-time impulse response matrix given by g(t) = { t < Ce At B t (2.27) when D in (2.2) is assumed to be. Similarly to the discrete-time case, the Hankel operator relates the past inputs to the future outputs. The Hankel singular values are the same as the singular values of the Hankel operator. However, note that these do not coincide with the singular values of the corresponding Markov parameters as in the discrete-time case. The Hankel norm can also be regarded as an induced norm 3. In fact, it is the induced operator norm of the Hankel operator. For a stable system 2 A strictly proper continuous transfer function G(s) can be expressed as a power series in the Laplace variable s as G(s) = k=1 g ks k where {g k } = {S k } are the Markov parameters. In the discrete-time case, the Markov parameters are the impulse response. (Weber, 1994) 3 Let be some vector norm. Then the norm A i = max x Ax x is said to be an induced norm of the current vector norm. A i can be interpreted as the maximum gain for all possible input directions of a system with amplification A and input x. See Skogestad and Postlethwaite (199), Zhou et al. (199) and Horn and Johnson (1985) for a more detailed description of induced norms. 27