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MIT 13 42 - Dynamical Systems

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13.42 Design Principles for Ocean Vehicles Reading #1 ©2004, 2005 A. H. Techet 1 Version 3.1, updated 2/2/2005 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: 3113()()ttytutdt−=∫ 1()()()Nnytututnδ==+−∑ 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:13.42 Design Principles for Ocean Vehicles Reading #1 ©2004, 2005 A. H. Techet 2 Version 3.1, updated 2/2/2005 Step 1: Replace ()ut by ()utτ+ (Change of variables) Step 2: Replace ()yt by ()ytτ+ (Replace all occurrences of t with tτ+) Step 3: Are the results from steps 1 and 2 equal? To illustrate this procedure we can use a few simple examples of basic systems with input, ()xt, and output, ()yt. Example 1: 34()[()]ytut/= System is clearly time invariant: [ ]34()()yt utτ τ/+= + Example 2: 110()()tytutdt=∫ Check time invariance: Step (1): Plug in 1tτ+ for 1t on the RHS and perform a change of variables (let 1tζτ=+). Note that the limits of integration must also shift with this change of variables. 110()()ttutdtudτττζζ++=∫∫ Step (2): Plug in tτ+ 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 tτ+ we do not shift the limits of integration as we did in step 1. 110()()tytutdtττ++=∫ Step (3): Compare results from steps (1) and (2). They are not equal, therefore this system is not time invariant. 110()()ttudutdtτττζζ++≠∫∫13.42 Design Principles for Ocean Vehicles Reading #1 ©2004, 2005 A. H. Techet 3 Version 3.1, updated 2/2/2005 Example 3: 4115()()ttytutdt−=∫ Step (1): Plug in 1tτ+ for 1t on the RHS and perform a change of variables (let1tζτ=+): 441155()()ttttutdtudτττζζ+−−++=∫∫ Step (2): Plug in tτ+ for t on the LHS, and again, note the shift in integration limits: 4115()()ttytutdtτττ+−++=∫ Step (3): Compare steps (1) and (2). They are equivalent, therefore system is time invariant! 441155()()ttttudutdtττττζζ++−+−+=∫∫ 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,()xt, and output,()yt. Figure 1 illustrates typical notation for a linear system,L, where the function ()xt is input into the system, shown as a box, and the system returns the output signal ()yt. The arrows indicate whether the function is being input or output from the system. Figure 1. Block diagram of linear system with input()xt and output()yt. In general, given a linear system L, as shown in figure 1, and some input, 1()xt, the system would result in an output, 1()yt, conversely some other input, 2()xt, into the same system would simply yield the output, 2()yt, such that the inputs and outputs obey the following properties:13.42 Design Principles for Ocean Vehicles Reading #1 ©2004, 2005 A. H. Techet 4 Version 3.1, updated 2/2/2005 Linear Superposition: 1212()()()()xtxtytyt+→+ Scaling: 11()()axtayt→ Superposition and Scaling: 11221122()()()()axtaxtaytayt+→+ A system must satisfy both the superposition and the scaling criteria for it to be considered linear. Example 1: ()dudtytC= . This system is linear. Example 2: 110()()tytutdt=∫. This system is linear. (But it is not time invariant!) Example 3: 3()()ytaut=. 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,()out, otherwise known as the delta function (see fig 2), is an idealization of a pulse which is so short that its duration, tδ is inconsequential for any real system.13.42 Design Principles for Ocean Vehicles Reading #1 ©2004, 2005 A. H. Techet 5 Version 3.1, updated 2/2/2005 Figure 2. Delta (impulse) function with height 1ε/ between times 2ε−/ and 2ε/ as tδε= goes to zero. Any continuous single-valued function, ()ft, can be represented as a sum of scaled and time shifted unit impulses: 1||2()0||2otuttεεε/;≤/=;>/ (1) The integral of an impulse from minus infinity to infinity is 1 and ()out is an even function:()()ooutut=−. Impulses can be scaled, shifted and summed to represent a function()ft, see figure 3. Figure 3. A function ()ft represented as a sum of scaled and time-shifted impulses.13.42 Design Principles for Ocean Vehicles Reading #1 ©2004, 2005 A. H. Techet 6 Version 3.1, updated 2/2/2005 The impulse function has the following properties: ()1outdt+∞−∞=∫ (2) ()()()oftfutdτττ+∞−∞=−∫ (3) ()()()oftutadtfa+∞−∞−=∫ (4) Let’s take a closer look at equation (4) from above. Here the value of the constant a is set to zero and we see that the integral simply equals that


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