13 42 Design Principles for Ocean Vehicles Reading 1 13 42 Design Principles for Ocean Vehicles Prof A H Techet Spring 2005 1 Dynamical Systems Dynamical systems are representations of physical objects or behaviors such that the output of the system depends on present and past values of the input to the system For example y t u 3 t 1 dt1 t t 3 y t u t n 1 u t nd N In order to model dynamical systems we need to build a set of tools and guidelines that can be used to analyze systems such as a ship in waves This section will introduce tools for analyzing linear systems System Recognize a set of physical objects behaviors of interest Modeling Representing the behavior of this system through a set of equations that approximate the original physical system Inputs Identify external actions influencing the system behavior Outputs Identify the outputs of interest 1 1 Time Invariant System Systems are time invariant if their behavior and characteristics do not vary over time In other words if the input to a system is shifted in time the resulting output experiences an identical time shift In order to determine whether the system is time invariant we use the following procedure in three steps 2004 2005 A H Techet 1 Version 3 1 updated 2 2 2005 13 42 Design Principles for Ocean Vehicles Reading 1 Replace u t by u t t Change of variables Replace y t by y t t Replace all occurrences of t with t t Step 1 Step 2 Are the results from steps 1 and 2 equal Step 3 To illustrate this procedure we can use a few simple examples of basic systems with input x t and output y t Example 1 y t u t 3 4 System is clearly time invariant y t t u t t 3 4 Example 2 y t t 0 u t1 dt1 Check time invariance Step 1 Plug in t1 t for t1 on the RHS and perform a change of variables let z t1 t Note that the limits of integration must also shift with this change of variables u t1 t dt1 t t t t 0 u z dz Step 2 Plug in t t for t on the LHS Notice that the limits of integration do not change in the same fashion as in step 1 The original integral on the RHS is bounded from zero to t and since we are simply replacing all occurrences of t with t t we do not shift the limits of integration as we did in step 1 y t t t t u t1 dt1 0 Step 3 Compare results from steps 1 and 2 They are not equal therefore this system is not time invariant t t t 2004 2005 A H Techet u z dz t t 0 2 u t1 dt1 Version 3 1 updated 2 2 2005 13 42 Design Principles for Ocean Vehicles Reading 1 Example 3 y t u 4 t1 dt1 t t 5 Step 1 Plug in t1 t for t1 on the RHS and perform a change of variables let z t1 t t t 5 u 4 t1 t dt1 t t t 5 t u 4 z d z Step 2 Plug in t t for t on the LHS and again note the shift in integration limits y t t t t u 4 t1 dt1 t 5 t Step 3 Compare steps 1 and 2 They are equivalent therefore system is time invariant t t t 5 t u 4 z dz t t t 5 t u4 t1 dt1 1 2 Linear Dynamical System A subset of dynamical systems is linear dynamical systems A system is considered to be linear if it satisfies properties of linear superposition and scaling Typically we can represent mathematically a system with some input x t and output y t Figure 1 illustrates typical notation for a linear system L where the function x t is input into the system shown as a box and the system returns the output signal y t The arrows indicate whether the function is being input or output from the system Figure 1 Block diagram of linear system with input x t and output y t In general given a linear system L as shown in figure 1 and some input x1 t the system would result in an output y1 t conversely some other input x2 t into the same system would simply yield the output y2 t such that the inputs and outputs obey the following properties 2004 2005 A H Techet 3 Version 3 1 updated 2 2 2005 13 42 Design Principles for Ocean Vehicles Reading 1 Linear Superposition x1 t x2 t y1 t y2 t Scaling ax1 t ay1 t Superposition and Scaling a1 x1 t a2 x2 t a1 y1 t a2 y2 t A system must satisfy both the superposition and the scaling criteria for it to be considered linear Example 1 y t C du dt This system is linear Example 2 y t u t1 dt1 This system is linear But it is not time invariant t 0 Example 3 y t au3 t This system is not linear But it is time invariant 1 3 Linear Time Invariant LTI Systems Systems that satisfy both the linear and the time invariant criteria are considered Linear Time invariant or LTI systems The property of superposition makes LTI systems easier to analyze By representing complex inputs as the superposition of basic signals such as an impulse we can then use superposition to determine the system output 1 4 Unit Impulse We can characterize a time continuous LTI system by understanding its response to a unit impulse A unit impulse uo t otherwise known as the delta function see fig 2 is an idealization of a pulse which is so short that its duration d t is inconsequential for any real system 2004 2005 A H Techet 4 Version 3 1 updated 2 2 2005 13 42 Design Principles for Ocean Vehicles Reading 1 Figure 2 Delta impulse function with height 1 e between times e 2 as d t e goes to zero e 2 and Any continuous single valued function f t can be represented as a sum of scaled and time shifted unit impulses 1 e t e 2 uo t 0 t e 2 1 The integral of an impulse from minus infinity to infinity is 1 and uo t is an even function uo t uo t Impulses can be scaled shifted and summed to represent a function f t see figure 3 Figure 3 A function 2004 2005 A H Techet f t represented as a sum of scaled and time shifted impulses 5 Version 3 1 updated 2 2 2005 13 42 Design Principles for Ocean Vehicles Reading 1 The impulse function has the following properties f t 2 f t uo t t dt 3 f t uo t a dt f a 4 uo t dt 1 Let s take a closer …