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linearsystems copy - Linear Systems Notes for CDS-140a 1...

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Unformatted text preview: Linear Systems Notes for CDS-140a October 27, 2008 1 Linear Systems Before beginning our study of linear dynamical systems, it only seems fair to ask the question “why study linear systems?” One might hope that most/all real- world systems are linear systems, so that our development of linear systems theory is directly applicable to physical problems. Unsurprisingly, however, most real systems are nonlinear. However, developing a robust theory of linear systems is advantageous for a number of reasons: • It gives us practice in analyzing dynamical system. • It builds up a set of general techniques that can be used to analyze both linear and nonlinear systems. • In nonlinear systems, we often are interested in local behaviors, which done using a linear approximation of the system near the point of interest. 1.1 Definition An autonomous linear dynamical system (simply a linear system from now on) is a collection of autonomous linear ordinary differential equations. Such systems can be written in first-order form : braceleftBigg ˙ x ( t ) = Ax ( t ) x (0) = x (1.1.1) where x maps R to R n , A is an n × n matrix, and b ∈ R n . We call x ( t ) the state of the system at time t . Whenever b = 0, we call the system homogeneous , otherwise it is inhomogeneous . If A is diagonal, the equations in the system 1 1 LINEAR SYSTEMS 2 decouple into n independent equations, each one in a single variable. In this case we say the system is uncoupled . We will consider only homogeneous systems until 1.12. Also, we will restrict our analysis to the reals, and will explicitely note whenever we need to utilize complex numbers. 1.2 Formal Solution Consider the ordinary differential equation d dt x ( t ) = αx ( t ) with x (0) = x . We know that this problem has x ( t ) = e αt x as its solution. Motivated by this, we can consider formally writing the solution to our problem (Equation 1.1.1) in the same way: x ( t ) = e At x (1.2.1) However, as yet we do not have a definition for the exponential of a matrix. For a square matrix A , we define the matrix exponential to be the Taylor-series expansion of the real-valued exponential: e A = I + A + A 2 2! + A 3 3! + ··· (1.2.2) where I is, as usual, the n × n identity matrix. As we will see in Section 1.5, this series converges for all matrices. Although the series is defined and converges for all square matrices A, it is computationally intractible to calculate the matrix exponential by using the series expansion. Instead, we will develop techniques to exponentiate matrices without resorting to the series definition. To explore these methods, let’s first look at some special cases and examples. 1.3 Diagonal matrices Let A be a diagonal matrix with diagonal elements λ 1 ,...,λ n : A = λ 1 ....
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