Test at 6 pm, 75 min, closed notes, closed books. no cheat sheet. Talked little about what can be on the exam. Does there exist \(T\) that takes \(\sum _{1}\rightarrow \sum _{2}\,\)? depends on controllability and observability. There are canonical forms: controllable (good for feedback) and observable (good for state estimation). We talked about duality and observer design. State feedback control.
Vector spaces preliminaries:
Most common vector space is \(\Re ^{n}\). \(x=\left \{ x_{1},x_{2},\cdots ,x_{n}\right \} \). Need vector spaces where vectors are functions. For this, the function space must have these three operators defined on it: \(+,\times ,0\). The first \(+\) is addition, as in \(\vec{x}+\vec{y}\) or \(f_{1}\left ( t\right ) +f_{2}\left ( t\right ) \). Second is scalar multiplication, as in \(5\vec{x}=5\left \{ x_{1},x_{2},\cdots ,x_{n}\right \} =\left \{ 5x_{1},5x_{2},\cdots ,5x_{n}\right \} \) and the third is the zero vector \(\left \{ 0,0,0,\cdots \right \} \).
Examples: We can have spaces of vectors that are infinite dimension sequences.
Reader: Consider the continuous functions on \(\left [ 0,1\right ] \) as vector spaces.
Reader: Generalize to n-dimensional continuous functions on time interval \(\left [ 0,T\right ] \)
Now we will talk about another important function space. This is the space of bounded functions. A function \(f\left ( t\right ) \) is said to be bounded on \(\left [ 0,T\right ] \) if \(\exists \) some \(\beta >0\) s.t. \(\left \vert f\left ( t\right ) \right \vert \leq \beta \) for all \(t\in \left [ 0,T\right ] .\) Bounded functions need not be continuous.
Reader: Consider associated vector spaces \(B\left ( \left [ 0,T\right ] \right ) \) where \(B\) means bounded (I do not understand this)
Reader: Generalize to \(B\left ( \left [ 0,T\right ] ,\Re ^{n}\right ) \)
A critical point in solving \(x^{\prime }=Ax\) is to know where solution sequence converges to actual solution. Convergence in function spaces. Need \(\left \Vert{}\right \Vert \) defined so we can say \(\left \Vert x^{\left ( k\right ) }-x^{\ast }\right \Vert \rightarrow 0\) as \(k\rightarrow \infty \). So need notion of norm. Vector spaces with norm are called normed vector spaces. A norm is mapping from \(X\) to the reals, where \(X\) is the vector space. Norm must satisfy the following
On \(\Re ^{n}\) we typically use the Euclidean norm defined as \(\left \Vert \vec{x}\right \Vert =\sqrt{x_{1}^{2}+x_{2}^{2}+\cdots +x_{n}^{2}}\). We should write this as \(\left \Vert x\right \Vert _{2}\). Other norms are possible, such as \(\left \Vert x\right \Vert _{\infty }=\max \left \{ \left \vert x_{1}\right \vert ,\left \vert x_{2}\right \vert ,\cdots ,\left \vert x_{n}\right \vert \right \} \) and also \(\left \Vert x\right \Vert _{1}={\displaystyle \sum \limits _{i=1}^{n}} \left \vert x_{i}\right \vert \) which is used in control theory. Reader: Verify these are norms. Need to check the triangle inequality.
Reader: Verify that \(\max \) is norm. i.e given two vectors, say \(a=\left \{ 3,6,8\right \} \) ,\(b=\left \{ 4,8,18\right \} \) then show that \(\max \left \{ a+b\right \} \leq \max \left \{ a\right \} +\max \left \{ b\right \} \)
Now consider vector spaces of \(m\times n\) matrices. A norm \(\left \Vert M\right \Vert =\sqrt{\lambda _{\max }\left ( M^{T}M\right ) }\). Reader: Verify for \(n=1\) this reduces to \(L^{2}\) norm above. i.e. this become normal vector \(\left \Vert{}\right \Vert _{2}\) norm. Proof: For example, let \(M=\begin{pmatrix} 1\\ 2 \end{pmatrix} \), then \(\left \Vert M\right \Vert =\sqrt{\lambda _{\max }\begin{pmatrix} 1 & 2\\ 2 & 4 \end{pmatrix} }\), but \(\lambda _{\max }\begin{pmatrix} 1 & 2\\ 2 & 4 \end{pmatrix} =5\), hence \(\left \Vert M\right \Vert =\sqrt{5}\). Now \(\left \Vert M\right \Vert _{2}=\sqrt{1^{2}+2^{2}}=\sqrt{5}.\) The same.