Model Predictive Control Structures in the Non-Minimal State Space

This thesis is concerned with constraint handling for systems described by a Non-Minimal State Space (NMSS) form. Such NMSS models are formulated directly from the measured input and output signals of the controlled process, without resort to the design and implementation of an observer. The thesis largely focuses on the application of Model Predictive Control (MPC) methods, a very common technique for dealing with system constraints. It is motivated by earlier research into both NMSS and MPC systems, with features of both combined in this thesis to yield improved control. The main contribution lies in the development of new methods for constraint handling of NMSS/MPC systems that contrasts with the ad hoc approach previously used for NMSS design based on the Proportional-Integral-Plus (PIP) algorithm. Structural aspects of NMSS/MPC design are considered, that result from mathematical manipulation of the closed-loop block diagram or from the definition of the state space description. The properties of these structures are investigated to provide an insight on features of the proposed strategies. More specifically, a Reference Governor scheme is utilised as a supervisory controller to account for constraints. This can lead to constraint handling in cases where a controller is already available. Furthermore, the use of an internal model is considered in the case of the `Forward Path' NMSS/MPC controller that is shown to have improved robustness properties in comparison to the conventional `Feedback' structure. In contrast to existing internal model approaches, this technique utilises the NMSS structure of the state vector and estimates only the elements of the state vector that are related to past values of the output. In addition, an optimal tuning technique is presented for MPC controllers. This approach allows for multiple objectives to be specified, whilst satisfying any system constraints. It is also shown that a specific NMSS/MPC structure that is proposed in this thesis, namely the NMSS/MPC controller with an integral-of-error state, provides the designer with additional freedom when using this tuning method. New NMSS/MPC methods are presented for both linear and non-linear systems, with the latter case being described by State Dependent Parameter (SDP) models. The development and analysis of MPC/SDP control in this thesis represents the first constraint handling control system and associated stability results for this class of non-linear models. Simulation examples are used to illustrate the advantages and potential limitations of the various control structures in comparison to existing solutions.Abstract i Acknowledgements iii Declaration iv Contents v List Of Tables x List Of Figures xi 1 Introduction 1 1.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 NMSS and Proportional–Integral–Plus Control . . . . . . . . . . . . 2 1.3 Model Predictive Control . . . . . . . . . . . . . . . . . . . . . . . . 4 1.4 Academic Contribution . . . . . . . . . . . . . . . . . . . . . . . . . 5 1.5 Thesis Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 2 Background Concepts 8 2.1 System identification . . . . . . . . . . . . . . . . . . . . . . . . . . 9 2.2 Optimisation techniques . . . . . . . . . . . . . . . . . . . . . . . . 10 2.2.1 The general constrained optimisation problem . . . . . . . . 11 2.2.2 The MPC optimisation problem . . . . . . . . . . . . . . . . 14 2.2.3 Active set methods . . . . . . . . . . . . . . . . . . . . . . . 15 2.2.4 Interior point methods . . . . . . . . . . . . . . . . . . . . . 16 2.3 Receding Horizon Control (RHC) . . . . . . . . . . . . . . . . . . . 18 2.4 Lyapounov theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 2.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 3 Reference governor over PIP control 23 3.1 System description and closed loop control . . . . . . . . . . . . . . 25 3.2 Control Description . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 3.3 Stability and Reference Tracking . . . . . . . . . . . . . . . . . . . 32 3.4 Simulation example . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 3.5 Concluding remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 4 MPC with an integral–of–error state 39 4.1 The controller of Wang and Young (2006) . . . . . . . . . . . . . . 40 4.1.1 System Description . . . . . . . . . . . . . . . . . . . . . . . 41 4.1.2 Control Description . . . . . . . . . . . . . . . . . . . . . . . 42 4.2 The alternative NMSS/MPC controller . . . . . . . . . . . . . . . . 44 4.3 Properties of the control scheme . . . . . . . . . . . . . . . . . . . . 48 4.3.1 Set–point following and disturbance rejection . . . . . . . . 48 4.3.2 Stability analysis . . . . . . . . . . . . . . . . . . . . . . . . 51 4.4 Simulation Examples . . . . . . . . . . . . . . . . . . . . . . . . . . 52 4.4.1 Double integrating plant . . . . . . . . . . . . . . . . . . . . 52 4.4.2 The IFAC ’93 benchmark . . . . . . . . . . . . . . . . . . . 54 4.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58 5 Multi–objective optimisation 60 5.1 Predictive Control Descriptions . . . . . . . . . . . . . . . . . . . . 62 5.1.1 MPC with an integral–of–error state variable . . . . . . . . . 62 5.1.2 The controller of Wang and Young (2006) . . . . . . . . . . 63 5.1.3 The minimal controller . . . . . . . . . . . . . . . . . . . . . 64 5.2 Tuning the MPC controllers . . . . . . . . . . . . . . . . . . . . . . 64 5.2.1 The goal attainment method . . . . . . . . . . . . . . . . . . 65 5.2.2 Application to MPC . . . . . . . . . . . . . . . . . . . . . . 66 5.3 Simulation Examples . . . . . . . . . . . . . . . . . . . . . . . . . . 67 5.3.1 SISO case: The IFAC ’93 benchmark . . . . . . . . . . . . . 68 5.3.2 MIMO case: The Shell Heavy Oil Fractionator Simulation . 72 5.3.2.1 Dynamic Decoupling . . . . . . . . . . . . . . . . . 74 5.3.2.2 ‘Designed For’ Response . . . . . . . . . . . . . . . 78 5.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82 6 The Forward Path MPC 83 6.1 System and Control Description . . . . . . . . . . . . . . . . . . . . 84 6.1.1 Stability analysis . . . . . . . . . . . . . . . . . . . . . . . . 85 6.1.2 Feasibility . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86 6.2 Control structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87 6.2.1 Feedback structure . . . . . . . . . . . . . . . . . . . . . . . 88 6.2.2 Forward path form . . . . . . . . . . . . . . . . . . . . . . . 89 6.2.3 Constraints . . . . . . . . . . . . . . . . . . . . . . . . . . . 91 6.3 Performance tests . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91 6.3.1 Monte Carlo Simulation (MCS) based uncertainty analysis . 92 6.3.2 Marginally stable system . . . . . . . . . . . . . . . . . . . . 94 6.3.3 The IFAC ’93 Benchmark . . . . . . . . . . . . . . . . . . . 97 6.3.4 The ALSTOM Benchmark . . . . . . . . . . . . . . . . . . . 100 6.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107 7 Disturbance Handling 109 7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109 7.2 System description . . . . . . . . . . . . . . . . . . . . . . . . . . . 110 7.2.1 Augmenting the state vector . . . . . . . . . . . . . . . . . . 111 7.2.2 Direct use of the disturbance model . . . . . . . . . . . . . . 113 7.3 Control Description . . . . . . . . . . . . . . . . . . . . . . . . . . . 115 7.4 Simulation Examples . . . . . . . . . . . . . . . . . . . . . . . . . . 118 7.4.1 SISO Example . . . . . . . . . . . . . . . . . . . . . . . . . . 119 7.4.1.1 Augmenting the state vector . . . . . . . . . . . . . 120 7.4.1.2 Direct use of the disturbance model . . . . . . . . . 120 7.4.1.3 Control Simulation . . . . . . . . . . . . . . . . . . 120 7.4.2 Control of a temperature control installation . . . . . . . . . 122 7.4.2.1 Description of the installation . . . . . . . . . . . . 123 7.4.2.2 System Identification . . . . . . . . . . . . . . . . . 125 7.4.2.3 Control and simulation results . . . . . . . . . . . . 126 7.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129 8 MPC for State Dependent Parameter Models 132 8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132 8.2 The Reference Governor approach . . . . . . . . . . . . . . . . . . . 134 8.3 MPC with an explicit integral–of–error state . . . . . . . . . . . . . 136 8.3.1 The basic algorithm . . . . . . . . . . . . . . . . . . . . . . 136 8.3.2 Improving the predictions . . . . . . . . . . . . . . . . . . . 137 8.3.3 Stability analysis . . . . . . . . . . . . . . . . . . . . . . . . 140 8.4 MPC Formulation using an increment in the control action . . . . . 142 8.4.1 The basic algorithm . . . . . . . . . . . . . . . . . . . . . . 142 8.4.2 Improving the predictions . . . . . . . . . . . . . . . . . . . 145 8.4.3 Stability analysis . . . . . . . . . . . . . . . . . . . . . . . . 146 8.5 Simulation Examples . . . . . . . . . . . . . . . . . . . . . . . . . . 147 8.5.1 SDP/PIP control and Reference Management . . . . . . . . 147 8.5.2 SDP/MPC control . . . . . . . . . . . . . . . . . . . . . . . 149 8.6 Concluding remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . 153 9 Conclusion 156 9.1 Summary of results . . . . . . . . . . . . . . . . . . . . . . . . . . . 156 9.1.1 Constraint handling with supervisory control . . . . . . . . . 156 9.1.2 The importance of structure . . . . . . . . . . . . . . . . . . 157 9.1.3 Tuning of MPC controllers . . . . . . . . . . . . . . . . . . . 158 9.1.4 NMSS/MPC control of Non–linear systems . . . . . . . . . . 159 9.2 Directions for future research . . . . . . . . . . . . . . . . . . . . . 160 9.2.1 Control structure . . . . . . . . . . . . . . . . . . . . . . . . 160 9.2.2 SDP/MPC control . . . . . . . . . . . . . . . . . . . . . . . 160 9.2.3 NMSS/MPC control applications . . . . . . . . . . . . . . . 161 Appendices 163 A Notation 164 A.1 Acronyms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164 A.2 Nomenclature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165 B Convex sets and functions 167 B.1 Convex Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167 B.2 Convex Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168 C Stability analysis 171 C.1 Optimisation problem . . . . . . . . . . . . . . . . . . . . . . . . . 171 C.2 Lyapounov stability . . . . . . . . . . . . . . . . . . . . . . . . . . . 172 References 175nrpages: 187status: publishe