System Finput u(t) output y(t)System Finput u(t)System Goutput y(t)2 LINEAR SYSTEMS 2 2 LINEAR SYSTEMS We will discuss what we mean by a linear time-invariant system, and then consider several useful transforms. 2.1 Definition of a System In short, a system is any process or entity that has one or more well-defined inputs and one or more well-defined outputs. Examples of systems include a simple physical object obeying Newtonian mechanics, and the US economy! Systems can be physical, or we may talk about a mathematical description of a system. The point of modeling is to capture in a mathematical representation the behavior of a physical system. As we will see, such representation lends itself to analysis and design, and certain restrictions such as linearity and time-invariance open a huge set of available tools. We often use a block diagram form to describe systems, and in particular their interconnec-tions: In the second case shown, y(t) = G[F [u(t)]]. Looking at structure now and starting with the most abstracted and general case, we may write a system as a function relating the input to the output; as written these are both functions of time: y(t) = F [u(t)] The system captured in F can be a multiplication by some constant factor - an example of a static system, or a hopelessly complex set of differential equations - an example of a dynamic system. If we are talking about a dynamical system, then by definition the mapping from u(t) to y(t) is such that the current value of the output y(t) depends on the past history of u(t). Several examples are: � t y(t) = u 2(t1)dt1, t−3 Ny(t) = u(t) + � u(t − nδt). n=1 In the second case, δt is a constant time step, and hence y(t) has embedded in it the current input plus a set of N delayed versions of the input.u(t)y(t)u(t-W)y(t-W)2 LINEAR SYSTEMS 3 2.2 Time-Invariant Systems A dynamic system is time-invariant if shifting the input on the time axis leads to an equivalent shifting of the output along the time axis, with no other changes. In other words, a time-invariant system maps a given input trajectory u(t) no matter when it occurs: y(t − τ) = F [u(t − τ)]. The formula above says specifically that if an input signal is delayed by some amount τ , so will be the output, and with no other changes. An example of a physical time-varying system is the pitch response of a rocket, y(t), when the thrusters are being steered by an angle u(t). You can see first that this is an inverted pendulum problem, and unstable without a closed-loop controller. It is time-varying because as the rocket burns fuel its mass is changing, and so the pitch responds differently to various inputs throughout its flight. In this case the ”absolute time” coordinate is the time since liftoff. To assess whether a system is time-varying or not, follow these steps: replace u(t) with u(t − τ) on one side of the equation, replace y(t) with y(t − τ ) on the other side of the equation, and then check if they are equal. Here are several examples. y(t) = u(t)3/2 This system is clearly time-invariant, because it is a static map. Next example: � t �y(t) = u(t1)dt1 0 Replace u(t1) with u(t1 − τ ) in the right hand side and carry it through: � t �u(t1 − τ )dt1 = � t−τ �u(t2)dt2. 0 −τ The left hand side is simply y(t − τ) = � t−τ �u(t1)dt1 02 LINEAR SYSTEMS 4 Clearly the right and left hand sides are not equal (the limits of integration are different), and hence the system is not time-invariant. As another example, consider � t y(t) = u 2(t1)dt1 t−5 The right-hand side becomes with the time shift � t u 2(t1 − τ )dt1 = � t−τ u 2(t2)dt2, t−5 t−5−τ whereas the left-hand side is y(t − τ) = � t−τ u 2(t1)dt1; t−5−τ the two sides of the defining equation are equal under a time shift τ, and so this system is time-invariant. A subtlety here is encountered when considering inputs that are zero before time zero - this is the usual assumption in our work, namely u(t) = 0 for t ≤ 0. While linearity is not affected by this condition, time invariance is, because the assumption is inconsistent with advancing a signal in time. Clearly part of the input would be truncated! Restricting our discussion to signal delays (the insertion of −τ into the argument, where strictly τ > 0) resolves the issue, and preserves time invariance as needed. 2.3 Linear Systems Next we consider linearity. Roughly speaking, a system is linear if its behavior is scale-independent; a result of this is the superposition principle. More precisely, suppose that y1(t) = F [u1(t)] and y2(t) = F [u2(t)]. Then linearity means that for any two constants α1 and α2, y(t) = α1y1(t) + α2y2(t) = F [α1u1(t) + α2u2(t)]. A simple special case is seen by setting α2 = 0: y(t) = α1y1(t) = F [α1u1(t)], making clear the scale-invariance. If the input is scaled by α1, then so is the output. Here are some examples of linear and nonlinear systems: du y(t) = c (linear and time-invariant) dt � t y(t) = u(t1)dt1 (linear but not time-invariant) 0 y(t) = 2u 2(t) (nonlinear but time-invariant) y(t) = 6u(t) (linear and time-invariant).width Hheight 1/Ht00G(t) = limit asHÆ0t2 LINEAR SYSTEMS 5 Linear, time-invariant (LTI) systems are of special interest because of the powerful tools we can apply to them. Systems described by sets of linear, ordinary or differential differential equations having constant coefficients are LTI. This is a large class! Very useful examples include a mass m on a spring k, being driven by a force u(t): my��(t) + ky(t) = u(t), where the output y(t) is interpreted as a position. A classic case of an LTI partial differential equation is transmission of lateral waves down a half-infinite string. Let m be the mass per unit length, and T be the tension (constant on the length). If the motion of the end is u(t), then the lateral motion satisfies ∂2y(t, x) ∂2y(t, x) m = T ∂t2 ∂x2 with y(t, x = 0) = u(t). Note that the system output y is not only a function of time but also of space in this case. 2.4 The Impulse Response and Convolution A fundamental property of LTI systems is that they obey the convolution operator. This operator is defined by � ∞ � ∞y(t) = u(t1)h(t − t1)dt1 = u(t − t1)h(t1)dt1. −∞ −∞ The function h(t) above is a particular