set1 - Chapter 1 Scope and Introduction $ P M Van Dooren...

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a39 a38 a36 a37 Chapter 1 – Scope and Introduction P. M. Van Dooren Department of Mathematical Engineering Catholic University of Louvain K. A. Gallivan School of Computational Science Florida State University Spring 2006 1
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a39 a38 a36 a37 Scope of Course Algorithm design for dynamical systems: from signals, systems and control main issues – complexity and numerical robustness linear time-invariant systems of finite dimension numerical linear algebra algorithms 2
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a39 a38 a36 a37 Finite Dimensional LTI Systems There are many ways to mathematically describe LTI systems with m inputs and p outputs u System y u ∈ ℜ m and y ∈ ℜ p are time-varying vectors. m = p = 1 is a single input single output (SISO) system m negationslash = 1 and/or p negationslash = 1 is a multiple input multiple output (MIMO) system the input/output relationship is our main concern, so we choose the most convenient description for a given purpose. 3
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a39 a38 a36 a37 Polynomial Form P ( d dt ) y ( t ) = Q ( d dt ) u ( t ) P (Δ) y k = Q (Δ) u k P and Q are polynomial matrices the first form is continuous time and the argument is the time differential operator the second form is discrete time and the argument is the difference operator these arguments are often referred to in a transform domain P ( s ) and Q ( s ) the in the Laplace transform domain P ( z ) and Q ( z ) the in the Z-transform domain these are continuous and discrete time respectively 4
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a39 a38 a36 a37 Generalized State Space Form E ˙ x ( t ) = Ax ( t ) + Bu ( t ) y ( t ) = Cx ( t ) + Du ( t ) Ex k +1 = Ax k + Bu k y k = Cx k + Du k x ( t ) ∈ ℜ n is the continuous time state variable x k ∈ ℜ n is the discrete time state variable E negationslash = I is called generalized state space E = I is called (standard) state space the system’s I/O relationship is invariant under certain transformations to the matrices specifying the generalized state space form 5
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a39 a38 a36 a37 Types of Errors Suppose we have a function y ( t ) that we can observe. Suppose also that we wish to identify the system. We must assume a model, e.g., I ( t ) = ce at . model errors – Δ I ( t ) – since y ( t ) is not necessarily exactly an exponential measurements of y ( t i ) have errors Δ y i that lead to identification errors Δ c and Δ a a finite model must be used to approximate the transcendental exponential – approximation error Δ e at = P k 1 · · · numerical errors arise due to the evaluation of the approximation in finite arithmetic fl ( P k 1 · · · ) Lemma 1.1 As computing power increases, the importance and number of numerical errors increase. 6
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a39 a38 a36 a37 Types of Errors The first three types are handled by mathematical analysis. The last is the major motivation for this course. It is growing in importance since more computing power implies more ambitious problems to solve and therefore more opportunity to use algorithms that are inappropriate numerically.
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