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Unformatted text preview: System F input u(t) output y(t) System F input u(t) System G output 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 ( t 1 ) dt 1 , t − 3 N y ( 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....
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- Spring '05
- Linear Systems, LTI system theory, Impulse response