Linear Systems Theory A Structural Decomposition Approach

Structural properties play an important role in our understanding of linear systems in the state space representation. The structural canonical form representation of linear systems not only reveals the structural properties but also facilitates the design of feedback laws that meet various control objectives. In particular, it decomposes the system into various subsystems. These subsystems, along with the interconnections that exist among them, clearly show the structural properties of the system. The simplicity of the subsystems and their explicit interconnections with each other lead us to a deeper insight into how feedback control would take effect on the system, and thus to the explicit construction of feedback laws that meet our design specifications. The discovery of structural canonical forms and their applications in feedback design for various performance specifications has been an active area of research for a long time. The effectiveness of the structural decomposition approach has also been extensively explored in nonlinear systems and control theory in the recent past. The aim of this book is to systematically present various canonical represen­ tations of the linear system, that explicitly reveal different structural properties of the system, and to report on some recent developments on its utilization in sys­ tem analysis and design.

1 Introduction and Preview.- 1.1 Motivation.- 1.2 Preview of Each Chapter.- 1.3 Notation.- 2 Mathematical Background.- 2.1 Introduction.- 2.2 Vector Spaces and Subspaces.- 2.3 Matrix Algebra and Properties.- 2.4 Norms.- 3 Review of Linear Systems Theory.- 3.1 Introduction.- 3.2 Dynamical Responses.- 3.3 System Stability.- 3.4 Controllability and Observability.- 3.5 System Invertibilities.- 3.6 Normal Rank, Finite Zeros and Infinite Zeros.- 3.7 Geometric Subspaces.- 3.8 Properties of State Feedback and Output Injection.- 3.9 Exercises.- 4 Decompositions of Unforced and/or Unsensed Systems.- 4.1 Introduction.- 4.2 Autonomous Systems.- 4.3 Unforced Systems.- 4.4 Unsensed Systems.- 4.5 Exercises.- 5 Decompositions of Proper Systems.- 5.1 Introduction.- 5.2 SISO Systems.- 5.3 Strictly Proper Systems.- 5.4 Nonstrictly Proper Systems.- 5.5 Proofs of Properties of Structural Decomposition.- 5.6 Kronecker and Smith Forms of the System Matrix.- 5.7 Discrete-time Systems.- 5.8 Exercises.- 6 Decompositions of Descriptor Systems.- 6.1 Introduction.- 6.2 SISO Descriptor Systems.- 6.3 MEMO Descriptor Systems.- 6.4 Proofs of Theorem 6.3.1 and Its Properties.- 6.5 Discrete-time Descriptor Systems.- 6.6 Exercises.- 7 Structural Mappings of Bilinear Transformations.- 7.1 Introduction.- 7.2 Mapping of Continuous- to Discrete-time Systems.- 7.3 Mapping of Discrete- to Continuous-time Systems.- 7.4 Proof of Theorem 7.2.1.- 7.5 Exercises.- 8 System Factorizations.- 8.1 Introduction.- 8.2 Strictly Proper Systems.- 8.3 Nonstrictly Proper Systems.- 8.4 Discrete-time Systems.- 8.5 Exercises.- 9 Structural Assignment via Sensor/Actuator Selection.- 9.1 Introduction.- 9.2 Simultaneous Finite and Infinite Zero Placement.- 9.3 Complete Structural Assignment.- 9.4 Exercises.- 10 Time-Scale and Eigenstructure Assignment via State Feedback.- 10.1 Introduction.- 10.2 Continuous-time Systems.- 10.3 Discrete-time Systems.- 10.4 Exercises.- 11 Disturbance Decoupling with Static Output Feedback.- 11.1 Introduction.- 11.2 Left Invertible Systems.- 11.3 General Multivariable Systems.- 11.4 Exercises.- 12 A Software Toolkit.- 12.1 Introduction.- 12.2 Descriptions of m-Functions.