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April4Lecture_Done - CIS 540 Principles of Embedded...

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CIS 540 Principles of Embedded Computation Spring 2017 http://www.seas.upenn.edu/~cis540/ Instructor: Rajeev Alur [email protected]
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Control Design Problem Plant model as Continuous-time Component H Uncontrolled Inputs Observable Outputs Design a controller C so that the composed system C || H is stable Is there a mathematical way to check when a system is stable? Is there a way to design C so that C||H is stable ? Yes, if the plant model is linear ! Controllable inputs Controller C CIS 540 Spring 2017; Lecture April 4
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Response of Linear Systems General case: S contains n variables, system has no inputs, and dynamics is given by: dS/dt = A S; Initial state s 0 Execution is given by the signal S (t) = e At s 0 Matrix exponential e A = I + A + A 2 /2 + A 3 /3! + A 4 /4! + … What if we have an input signal? dS = A S + B I S(t) = e At s 0 + Integral 0 t [ e A(t- ) B I ( ) d ] CIS 540 Spring 2017; Lecture April 4
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Matrix Exponential Matrix exponential e A = I + A + A 2 /2 + A 3 /3! + A 4 /4! + … Is there a way to compute this? Observe that each term in the sum is (n x n) matrix If A k = 0 for some k, then sum is finite and can be computed directly In general case, sum of first k terms will give approximation (with quality of approximation improving with increasing k) What if A is a diagonal matrix ? Notation: D (a 1 , a 2 , … a n ) represents a matrix whose (i,i)-th entry is a i , and (i,j)-th entry, for i != j, is 0 Claim: if A is diagonal matrix D (a 1 , a 2 , … a n ) then e A is the diagonal matrix D (e a1 , e a2 , …e an ) Analytical method based on eigenvalues and similarity transformation CIS 540 Spring 2017; Lecture April 4
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Eigenvalues and Eigenvectors For an (n x n) matrix A, if the equation A x = x holds, for an n- dimensional non-zero vector x and scalar , then x is called an eigenvector of A, and is called a corresponding eigenvalue How to compute eigenvalues ? Consider the set of equations with unknown variables 1 , 1 , … n determinant (A – I ) = 0 Determinant of a 2 x2 matrix A is A 1,1 A 2,2 - A 1,2 A 2,1 Calculate eigenvalues and eigenvectors of the matrix [[4 6] [1 3]] Are eigenvectors linearly independent? What are the eigenvalues of a diagonal matrix? Calculate eigenvalues and eigenvectors of the matrix [[1 2] [0 1]] Calculate eigenvalues and eigenvectors of the matrix [[0 1] [-1 0]] CIS 540 Spring 2017; Lecture April 4
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Eigenvalues and Eigenvectors Eigenvalues are the roots of the characteristic polynomial An eigenvalue can be a complex number Multiplicity of eigenvalue can be more than 1 If A is a diagonal matrix then diagonal entries are the eigenvalues For a given eigenvalue , compute the corresponding eigenvector(s) by solving system of linear equations A x = x, with unknown vector x If all eigenvalues are distinct then corresponding eigenvectors are linearly independent CIS 540 Spring 2017; Lecture April 4
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