1) Continuing Exercise MSD System Stability

EGR 326 HW 10 Due April 21, 2017

1) Continuing Exercise – MSD System Stability Analyze the stability of our CE1 problem – the 3 mass, 4 spring example. a) Analyze the stability of this system (using Matlab) to determine stability in terms of eigenanalysis. Explain why you find that the system is, or is not, stable. b) Identify the two dominant eigenvalues and their associated eigenvectors. c) Use phase portraits to confirm your stability analysis (see example 6.1 for definition of these, as well as Romeo and Juliet HW solutions and class slides). Note that phase portraits are graphs that plot x1 versus x2, for

example, rather than x1 versus time.

2) National (or your personal) Economy example

An alternate model (different from the one you used previously) for national economics is based on the following assumptions. National income, Y[k], is equal to the sum of consumption C[k], investment I[k], and government expenditure G[k]. Consumption, C[k], is proportional to the national income of the preceding year; and investment, I[k], is proportional to the change in consumer spending between the current year and the preceding year.

NOTE that we will have class on Monday to work on parts (c) through (e), so you will probably want to have created your model before Monday class.

a) Develop a dynamic state space model (matrix form for the equations) for this system. Clearly state your assumptions and show all your work. i. Let a = 1.2, b = 0.5 b) Diagonalize this system. Show your work for this process (which can be results from Matlab.)

c) Create a detailed Simulink model (using integrator blocks and not the state- space block) of either the system model from part (a), or the diagonalized system model from part (b) i. Simulate the system using a Matlab script. ii. Briefly interpret the system behavior (and the modes if part (b)), in terms of the original system behavior and original state variables (C[k] and I[k]). (think about the role of the right eigenvectors as well as the eigenvalues regardless of which system you model in Simulink) iii. You will want to have multiple data streams going to the “to workspace” blocks: C[k], I[k], Y[k] (and your two modal state variables if modeling part (b)) d) Analyze the controllability of this system (show your work) i. Is the system controllable? How do you know? EGR 326 HW 10 Due April 21, 2017

ii. Develop TWO input sequences that will move the state vector to the origin in a finite number of time steps – using a different number of time steps for the two different input sequences. iii. Interpret your solutions in an interesting and insightful manner e) Analyze the observability of this system (show your work) i. Is your system observable? How do you know? ii. Estimate the initial state vector using the system matrices, inputs, and output observations recorded below. iii. Perform the estimate using at least two different sequences of output observations GIVEN BELOW (i.e., 1st use just a few Y[k], 2nd: add additional Y[k] observations, and estimate x[0] again) iv. Interpret your results f) Analyze the stability of this system (show your work) i. First use the eigenvalues to determine stability ii. Also create and interpret phase portraits for this system (see parts of figures 6.4 – 6.6)

National Economy Output Observations

86.4790000000000 137.139200000000 208.184700000000 300.828600000000 426.262100000000 596.323800000000 829.103740000000 1144.38293200000 1570.94323360000 2153.85316128000 2942.31985014400 4015.36509349120 5472.27763819776 7449.31431266124 10136.9410598716 13787.2315401721 18742.2130763868 25478.1952933930 34624.8818422754 47045.0870400599 63915.6395467425 86832.6716401006 117957.428264136 160237.767751384 217660.463874009 EGR 326 HW 10 Due April 21, 2017

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