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10.2: Linear Time-Invariant Models
https://eng.libretexts.org/Bookshelves/Industrial_and_Systems_Engineering/Book%3A_Dynamic_Systems_and_Control_(Dahleh_Dahleh_and_Verghese)/10%3A_Discrete-Time_Linear_State-Space_Models/10.02%3A_Linear_Time-Invariant_Models
What this series establishes, on comparison with the definition of the $\mathcal{Z}$ -transform, is that the inverse transform of $z(zI - A)^{-1}$ is the matrix sequence whose value at time $k$ i...What this series establishes, on comparison with the definition of the $\mathcal{Z}$ -transform, is that the inverse transform of $z(zI - A)^{-1}$ is the matrix sequence whose value at time $k$ is $A^{k}$ for $k \geq 0$ the sequence is 0 for time instants $k < 0$ . Also since the inverse transform of a product such as $(z I-A)^{-1} B U(z)$ is the convolution of the sequences whose transforms are $(z I-A)^{-1} B$ and $U(z)$ respectively, we get
4.4: Relationship to Matrix Norms
https://eng.libretexts.org/Bookshelves/Industrial_and_Systems_Engineering/Book%3A_Dynamic_Systems_and_Control_(Dahleh_Dahleh_and_Verghese)/04%3A_Matrix_Norms_and_Singular_Value_Decomposition/4.04%3A_Relationship_to_Matrix_Norms
\sup _{x \neq 0} \frac{\|A x\|_{2}}{\|x\|_{2}} &=\sup _{x \neq 0} \frac{\left\|U \Sigma V^{\prime} x\right\|_{2}}{\|x\|_{2}} \\ &=\sup _{y \neq 0} \frac{\left(\sum_{i=1}^{r} \sigma_{i}^{2}\left|y_{i}\...\sup _{x \neq 0} \frac{\|A x\|_{2}}{\|x\|_{2}} &=\sup _{x \neq 0} \frac{\left\|U \Sigma V^{\prime} x\right\|_{2}}{\|x\|_{2}} \\ &=\sup _{y \neq 0} \frac{\left(\sum_{i=1}^{r} \sigma_{i}^{2}\left|y_{i}\right|^{2}\right)^{\frac{1}{2}}}{\left(\sum_{i=1}^{r}\left|y_{i}\right|^{2}\right)^{\frac{1}{2}}} \\ \|\Sigma y\|_{2} &=\left(\sum_{i=1}^{n}\left|\sigma_{i} y_{i}\right|^{2}\right)^{\frac{1}{2}} \\
5.6: Exercises
https://eng.libretexts.org/Bookshelves/Industrial_and_Systems_Engineering/Book%3A_Dynamic_Systems_and_Control_(Dahleh_Dahleh_and_Verghese)/05%3A_Matrix_Perturbations/5.06%3A_Exercises
[The result in (b), and some extensions of it, give rise to the following sound (and widely used) procedure for estimating the rank of some underlying matrix $A$ , given only the matrix $A + E$ and...[The result in (b), and some extensions of it, give rise to the following sound (and widely used) procedure for estimating the rank of some underlying matrix $A$ , given only the matrix $A + E$ and knowledge of $\|E\|_{2}$ : Compute the SVD of $A + E$ , then declare the "numerical rank" of $A$ to be the number of singular values of $A + E$ that are larger than the threshold $\|E\|_{2}$ .
19: Robust stability in SISO systems
https://eng.libretexts.org/Bookshelves/Industrial_and_Systems_Engineering/Book%3A_Dynamic_Systems_and_Control_(Dahleh_Dahleh_and_Verghese)/19%3A_Robust_stability_in_SISO_systems
A feedback controller can be designed so as to maintain stability of the closed-loop and an acceptable level of performance in the presence of uncertainties in the plant description, i.e., so as to ac...A feedback controller can be designed so as to maintain stability of the closed-loop and an acceptable level of performance in the presence of uncertainties in the plant description, i.e., so as to achieve robust stability and robust performance respectively. For the study of robust stability and robust performance, we assume that the dynamics of the actual plant are represented by a transfer function that belongs to some uncertainty set $\Omega$ .
Back Matter
https://eng.libretexts.org/Bookshelves/Industrial_and_Systems_Engineering/Book%3A_Dynamic_Systems_and_Control_(Dahleh_Dahleh_and_Verghese)/zz%3A_Back_Matter
5.3: Perturabtions Measured in the Frobenius Norm
https://eng.libretexts.org/Bookshelves/Industrial_and_Systems_Engineering/Book%3A_Dynamic_Systems_and_Control_(Dahleh_Dahleh_and_Verghese)/05%3A_Matrix_Perturbations/5.03%3A_Perturabtions_Measured_in_the_Frobenius_Norm
Suppose we have a matrix $A \in \mathbb{C}^{n \times n}$ , and we are interested in finding the closest matrix to $A$ of the form $cW$ where $c$ is a complex number and $W$ is a unitary matri...Suppose we have a matrix $A \in \mathbb{C}^{n \times n}$ , and we are interested in finding the closest matrix to $A$ of the form $cW$ where $c$ is a complex number and $W$ is a unitary matrix. $|\operatorname{Tr}(Z \Sigma)|^{2}=\left|\sum_{i=1}^{n} \sigma_{i} z_{i i}\right|^{2} \leq\left(\sum_{i=1}^{n} \sigma_{i}\right)^{2}\nonumber$ $\min _{c, W}\|A-c W\|_{F}^{2}=\sum_{i=1}^{n} \sigma_{i}^{2}-\frac{1}{n}\left(\sum_{i=1}^{n} \sigma_{i}^{2}\right)^{2}\nonumber$
11.1: The Time Varying Case
https://eng.libretexts.org/Bookshelves/Industrial_and_Systems_Engineering/Book%3A_Dynamic_Systems_and_Control_(Dahleh_Dahleh_and_Verghese)/11%3A_Continuous-time_linear_state-space_models/11.01%3A_The_Time_Varying_Case
\frac{d}{d t} \operatorname{det}\left[\Phi\left(t, t_{0}\right)\right] &=\lim _{\epsilon \rightarrow 0} \frac{1}{\epsilon}\left(\operatorname{det}\left[\Phi\left(t+\epsilon, t_{0}\right)\right]-\opera...\frac{d}{d t} \operatorname{det}\left[\Phi\left(t, t_{0}\right)\right] &=\lim _{\epsilon \rightarrow 0} \frac{1}{\epsilon}\left(\operatorname{det}\left[\Phi\left(t+\epsilon, t_{0}\right)\right]-\operatorname{det}\left[\Phi\left(t, t_{0}\right)\right]\right) \\ &=\lim _{\epsilon \rightarrow 0} \frac{1}{\epsilon}\left(\operatorname{det}\left[\Phi\left(t, t_{0}\right)+\epsilon A(t) \Phi\left(t, t_{0}\right)\right]-\operatorname{det}\left[\Phi\left(t, t_{0}\right)\right]\right) \\
12.2: Similarity Transformations
https://eng.libretexts.org/Bookshelves/Industrial_and_Systems_Engineering/Book%3A_Dynamic_Systems_and_Control_(Dahleh_Dahleh_and_Verghese)/12%3A_Modal_decomposition_of_state-space_models/12.02%3A_Similarity_Transformations
The matrix $T$ embodies the details of the transformation from $x$ coordinates to $r$ coordinates - it is easy to see from (12.2) that the columns of $T$ are the representations of the standar...The matrix $T$ embodies the details of the transformation from $x$ coordinates to $r$ coordinates - it is easy to see from (12.2) that the columns of $T$ are the representations of the standard unit vectors of $r$ in the coordinate system of $x$ , which is all that is needed to completely define the new coordinate system.
20.2: Additive Representation of Uncertainty
https://eng.libretexts.org/Bookshelves/Industrial_and_Systems_Engineering/Book%3A_Dynamic_Systems_and_Control_(Dahleh_Dahleh_and_Verghese)/20%3A_Stability_Robustness/20.02%3A_Additive_Representation_of_Uncertainty
In the above situation, with a nominal plant model given by the proper rational matrix $P_{0}(s)$ , the actual plant represented by $P (s)$ , and the difference $P (s) - P_{0}(s)$ assumed to be st...In the above situation, with a nominal plant model given by the proper rational matrix $P_{0}(s)$ , the actual plant represented by $P (s)$ , and the difference $P (s) - P_{0}(s)$ assumed to be stable, we may be able to characterize the model uncertainty via a bound of the form
4.3: Singular Value Decomposition
https://eng.libretexts.org/Bookshelves/Industrial_and_Systems_Engineering/Book%3A_Dynamic_Systems_and_Control_(Dahleh_Dahleh_and_Verghese)/04%3A_Matrix_Norms_and_Singular_Value_Decomposition/4.03%3A_Singular_Value_Decomposition
Define $V_{1}^{\prime} \in C^{m \times n}$ has orthonormal rows as can be seen from the following calculation: $V_{1}^{\prime} V_{1}=\Sigma_{1}^{-1} U^{\prime} A A^{\prime} U \Sigma_{1}^{-1}=I$ . w...Define $V_{1}^{\prime} \in C^{m \times n}$ has orthonormal rows as can be seen from the following calculation: $V_{1}^{\prime} V_{1}=\Sigma_{1}^{-1} U^{\prime} A A^{\prime} U \Sigma_{1}^{-1}=I$ . which is a weighted sum of the $u_{i}$ , where the weights are the products of the singular values and the projections of $x$ onto the $v_{i}$ .
22.2: The Reachability Problem
https://eng.libretexts.org/Bookshelves/Industrial_and_Systems_Engineering/Book%3A_Dynamic_Systems_and_Control_(Dahleh_Dahleh_and_Verghese)/22%3A_Reachability_of_DT_LTI_systems/22.02%3A_The_Reachability_Problem
To show that $R a\left(R_{n}\right)=R a\left(R_{\ell}\right)$ for $\ell \geq n$ , note from the Cayley-Hamilton theorem that $A^{i}$ for $i \geq n$ can be written as a linear combination of \(A...To show that $R a\left(R_{n}\right)=R a\left(R_{\ell}\right)$ for $\ell \geq n$ , note from the Cayley-Hamilton theorem that $A^{i}$ for $i \geq n$ can be written as a linear combination of $A^{n-1}, \cdots, A, I$ , so all the columns of $R_{\ell}$ for $\ell \geq n$ are linear combinations of the columns of $R_{n}$ .

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