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1.2: LTI Systems and ODEs

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  • We consider physical systems that can be modeled with reasonable engineering fidelity as linear, time-invariant (LTI) systems. Such a system is represented mathematically by an ordinary differential equation (ODE), or by a set of coupled ODEs, for which the single independent variable is time, denoted as \(t\). These ODEs are linear, and they have constant coefficients, so we describe them as linear, time-invariant (LTI), the same as the systems they represent1. For example, suppose we denote a dependent variable as \( x(t) \), here a general symbol representing some physical dynamic response quantity for which we want to solve. Then an LTI ODE that models an LTI physical system might have the form

    \[\dfrac{dx}{dt} - a\,x =b\,u(t)\label{eqn:1}\]

    in which \(a\) and \(b\) are constant multiplying coefficients, and known function \(u(t)\) is the excitation and is independent of the response. In the study of systems, an independent excitation \(u(t)\) is often called an input, and a dependent response \(x(t)\) is often called an output.

    Hereafter, we will usually employ the common shorthand dot notation for denoting derivatives with respect to time: \( \dfrac{dx}{dt} = \dot{x} \), \( \dfrac{d^2 x}{dt^2} = \ddot{x} \) etc., so that Equation \(\ref{eqn:1}\) can be written more simply as

    \[ \dot{x} - a\,x = b\,u(t) .\]

    The linearity of Equation \(\ref{eqn:1}\) is manifested by the linear appearance of \( x(t) \) and all of its derivatives in the ODE. The following are some similar ODEs that are not linear (they are nonlinear) for obvious reasons: \( \dot{x} - a\,x^2 = b\,u(t) \); \( sin(\dot{x}) - a\,x = b\,u(t) \); \( \sqrt(\dot{x}) - a\,tan(x) = b\,u(t) \). Linear ODEs are almost always easier to solve (at least in closed form, i.e., as equations involving standard functions) than nonlinear ODEs. Moreover, the important principle of superposition applies to linear ODEs, but not to nonlinear ODEs. An example of the application of this principle is: let the response to input \( u_1(t) \) be \( \ x_1(t) \), and let the response to another input \( u_2(t) \) be \( x_1 (t) \); if a third input is the sum of multiplied terms \( u_3(t) = c_1\,u_1(t) + c_2\,u_2 (t) \), in which \( c_1 \) and \( c_2 \) are constants, then the response to \( u_3(t) \) is \( x_3 (t) = c_1\,x_1 (t) + c_2\,x_2 (t) \). This result is easy to derive just by multiplying two ODEs such as Equation \(\ref{eqn:1}\) by the constants, then adding the multiplied ODEs. The principle of superposition allows us to solve accurately for the responses of linear systems to any physically realistic inputs. (See Section 8.10 for a derivation of system response to an arbitrary physically realistic input by direct application of superposition.)

    The time invariance of Equation \(\ref{eqn:1}\) is manifested by the constant coefficients of \( x(t) \) and all of its derivatives in the ODE. ODEs with time-invariant coefficients model the behavior of systems assumed to have physical properties that either remain constant in time or vary so slowly and/or slightly that the variation is negligible for engineering purposes. But many practically important systems have time-varying physical properties. For example, a vehicle such as a space shuttle between liftoff and achievement of orbital position has rapidly varying (decreasing) mass as propellant is burned and external fuel tanks and boosters are released. The following is a linear equation somewhat similar to Equation \(\ref{eqn:1}\), but with an obviously time-varying coefficient: \( \dot{x} -3\,x\,(1-e^(-2\,t))\,x = b\,u(t) \). The study of systems with time-varying physical properties is generally more complicated, not fundamental, so only time-invariant systems and ODEs are considered in this book.

    The form of Equation \(\ref{eqn:1}\), \( \dot{x} - a\,x = b\,u(t) \), is widely regarded as the standard form for a first order LTI ODE, and we will use it as such in this book. Beginning in the next section, we will study idealized physical systems whose dynamic behaviors are described by equations that are directly analogous to Equation \(\ref{eqn:1}\). We will express the mathematical constants \(a\) and \(b\) in terms of specific physical constants. Also, the roles of input \( u(t) \) and output \( x(t) \) in Equation \(\ref{eqn:1}\) will be assumed by some specific physical quantities, such as force, velocity, voltage, etc., and we will denote them with relevant symbols [often different than \( u(t) \) and \( x(t) \)] when appropriate.

    Although only first order ODEs are discussed in this section, we certainly will encounter and study systems and ODEs of second and higher orders.

    1LTI ODEs are also sometimes described as linear, constant-coefficient, or LCC.