Ee263 Homework 8 Solutions Free

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How to Solve EE263 Homework 8 Problems

EE263 is a course on Introduction to Linear Dynamical Systems offered by Stanford University. The course covers topics such as applied linear algebra, linear dynamical systems, circuits, signal processing, communications, and control systems. The homework assignments are designed to help students practice the concepts and skills learned in the lectures and recitations.

In this article, we will provide some hints and tips on how to solve the problems in homework 8, which focuses on state feedback and observer design. The homework 8 solutions can be found here, but we encourage you to try to solve them on your own first.

Problem 1: State feedback for a FIR filter

This problem asks you to design a state feedback controller for a finite impulse response (FIR) filter with small feedback. The filter has a cascade of 100 one-sample delays and a feedback gain of 10. You are given the state-space model of the filter and asked to find the eigenvalues of the system matrix A with and without feedback, the impulse response of the system with feedback, and a state feedback gain K that places all eigenvalues at 0.9.

To find the eigenvalues of A without feedback, you can use the fact that A is a lower triangular matrix with all ones on the diagonal, so its eigenvalues are all equal to one. To find the eigenvalues of A with feedback, you can use the fact that A is similar to a diagonal matrix with entries 10e, where k = 0,...,99. These are complex numbers with magnitude 0.8913 and uniformly spaced angles on the unit circle. To find the impulse response of the system with feedback, you can apply a unit pulse input at t = 0 and observe the output at different times. You will see that the output is a delayed unit pulse at t = 100 followed by very faint echoes with amplitude 10 at t = 100n, where n = 1,2,... To find the state feedback gain K that places all eigenvalues at 0.9, you can use the Ackermann's formula or any other method for pole placement. One possible solution is K = [0.1 -0.1 ... -0.1].

Problem 2: State feedback for an unstable system

This problem asks you to design a state feedback controller for an unstable system with two states and one input. You are given the state-space model of the system and asked to find a state feedback gain K that stabilizes the system and achieves a unit step response with settling time less than 5 seconds and overshoot less than 10%.

To find a state feedback gain K that stabilizes the system, you need to place the eigenvalues of A - BK in the left half of the complex plane. To achieve a unit step response with desired specifications, you need to choose K such that A - BK has eigenvalues with negative real part and small imaginary part. One possible method is to use root locus or bode plot techniques to find suitable values of K. One possible solution is K = [-4 -3].

Problem 3: Observer design for a mass-spring-damper system

This problem asks you to design an observer for a mass-spring-damper system with two states and one output. You are given the state-space model of the system and asked to find an observer gain L that makes the observer error converge to zero exponentially fast.

To find an observer gain L that makes the observer error converge to zero exponentially fast, you need to place the eigenvalues of A - LC in the left half of the complex plane. One possible method is to use pole placement or LQR techniques to find suitable values of L. One possible solution is L = [-10; -20]. aa16f39245