Linear systems theory Joao P Hespanha (original) (raw)

Linear systems theory is the cornerstone of control theory and a prerequisite for essentially all graduate courses in this area. It is a well-established discipline that focuses on linear differential equations from the perspective of control and estimation. CONTENT The first set of lectures (1-17) covers the key topics in linear systems theory: system representation, stability, controllability and state feedback, observability and state estimation, and realization theory. The main goal of these chapters is to provide the background needed for advanced control design techniques. Feedback linearization and the LQR problem are also briefly introduced to increase the design component of this set of lectures. The preview of optimal LQR control facilitates the introduction of notions such as controllability and observability, but is pursued in much greater detail in the second set of lectures. MATLAB R Computational tools such as the MATLAB r software environment offer a significant step forward in teaching linear systems because they allow students to solve numerical problems without being exposed to a detailed treatment of numerical computations. PREAMBLE xv By systematically annotating the theoretical developments with marginal notes that discuss the relevant commands available in MATLAB r , this textbook helps students learn to use these tools. An example of this can be found, e.g., in MATLAB r Hint 9 in page 12, which is further expanded on page 51. The commands discussed in the "MATLAB r Hints" assume that the reader has version R2007b of MATLAB r with Simulink r , the Symbolic Math Toolbox, and the Control System Toolbox. However, essentially all these commands have been fairly stable for several versions so they are likely to work with previous and subsequent versions for several years to come. Lecture 25 assumes that the reader has installed CVX version 1.2, which is a MATLAB r package for Disciplined Convex Programming, distributed under the GNU General Public License 2.0 [7]. MATLAB r and Simulink r are registered trademarks of The MathWorks Inc. and are used with permission. The MathWorks does not warrant the accuracy of the text or exercises in this book. This book's use or discussion of MATLAB r , Simulink r , or related products does not constitute an endorsement or sponsorship by The Math-Works of a particular pedagogical approach or particular use of the MATLAB r and Simulink r software. 1.2.3 SYSTEM INTERCONNECTIONS WITH MATLAB R MATLAB R Hint 4 (series). The command sys=series(sys1,sys2) or, alternatively, sys=sys2*sys1 creates a system sys from the cascade connection Attention! Note the different order in which sys1 and sys2 appear in the two forms of this command. 3.4 LAPLACE TRANSFORM (REVIEW) Given a continuous-time signal x(t), t ≥ 0 its unilateral Laplace transform is given by MATLAB R Hint 10. laplace(F,t,s) symbolically computes the Laplace transform of F. p. 52 MATLAB R Hint 11. ilaplace(F,s,t) symbolically computes the inverse Laplace transform of F. p. 53