problem8

# problem8 - G s = 1 s 2 1 Is this system BIBO stable 1...

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EE221A Linear System Theory Problem Set 8 Professor C. Tomlin Department of Electrical Engineering and Computer Sciences, UC Berkeley Fall 2011 Issued 11/10; Due 11/18 Problem 1: BIBO Stability. T H T C f f H C T H T C i i , , V H V C Figure 1: A simple heat exchanger, for Problem 1. Consider the simple heat exchanger shown in Figure 1, in which f C and f H are the ±ows (assumed constant) of cold and hot water, T H and T C represent the temperatures in the hot and cold compartments, respectively, T Hi and T Ci denote the temperature of the hot and cold in±ow, respectively, and V H and V C are the volumes of hot and cold water. The temperatures in both compartments evolve according to: V C dT C dt = f C ( T Ci - T C ) + β ( T H - T C ) (1) V H dT H dt = f H ( T Hi - T H ) - β ( T H - T C ) (2) Let the inputs to this system be u 1 = T Ci , u 2 = T Hi , the outputs are y 1 = T C and y 2 = T H , and assume that f C = f H = 0 . 1 ( m 3 / min), β = 0 . 2 ( m 3 /min) and V H = V C = 1 ( m 3 ). (a) Write the state space and output equations for this system in modal form. (b) In the absence of any input, determine y 1 ( t ) and y 2 ( t ). (c) Is the system BIBO stable? Show why or why not. Problem 2: BIBO Stability Consider a single input single output LTI system with transfer function

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Unformatted text preview: G ( s ) = 1 s 2 +1 . Is this system BIBO stable? 1 Problem 3: Exponential stability of LTI systems. Prove that if the A matrix of the LTI system ˙ x = Ax has all of its eigenvalues in the open left half plane, then the equilibrium x e = 0 is asymptotically stable. Problem 4: Characterization of Internal (State Space) Stability for LTI systems. (a) Show that the system ˙ x = Ax is internally stable if all of the eigenvalues of A are in the closed left half of the complex plane (closed means that the jω-axis is included), and each of the jω-axis eigenvalues has a Jordan block of size 1. (b) Given A = -3 1-3 1-3-4 1-4 Is the system ˙ x = Ax exponentially stable? Is it stable? 2...
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problem8 - G s = 1 s 2 1 Is this system BIBO stable 1...

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