Multivariable Control Systems
Field
Field is a triplet \((\mathbb{F}, +, \cdot)\) that satisfies- Additive identity \(a + 0 = a\)
- Multiplicative identity \(a \cdot 1 = a\)
- Additive inverse \(a + b = 0\)
- Multiplicative inverse \(a \cdot c = 1\) (except for \(0 \in \mathbb{F}\))
Examples
- Set of real numbers \(\mathbb{R}\), rational numbers \(\mathbb{Q}\), complex numbers\(\mathbb{C}\) form a field. Note that the set of integers \(\mathbb{Z}\) is not a field.
- \(\{0,1,2,3,4\}\) is a field with modular arithmetic.
- Binary numbers \(\mathbb{F}_2 = \{0,1\}\)
- \(\mathbb{Z} \% p\) where \(p\) is a prime numbers is a field.
- Surds: \( \mathbb{F}_{\sqrt{2}}=\{a + b\sqrt{2}: a,b\in\mathbb{Q}\}, \mathbb{F}_{\sqrt{3}} = \{a + b\sqrt{3}: a,b\in\mathbb{Q}\}\), \( \{a + b\sqrt{2} + c\sqrt{3} + d\sqrt{6}: a,b,c,d \in \mathbb{Q}\} \)
Note that \( \mathbb{Q} \subset \mathbb{F}_{\sqrt{2}} \subset \mathbb{R}\)
Vector Space
\( V_{\mathbb{F}}\) denotes a collection of vectors with vector addition and scalar multiplication
Let \(\oplus, \cdot\) denote vector addition and scalar multiplication, respectively. Then, \[ V_1 \oplus V_2 \in V_{\mathbb{F}}, \quad \alpha \cdot v \in V_{\mathbb{F}}; \ \alpha \in \mathbb{F} \]
Properties: distributive, associative, commutative, ...
Linear independence
A set of vectors \(v_1, v_2, \ldots, v_n\) are said to be linearly independent if
\[ a_1 v_1 + a_2 v_2 + \ldots + a_n v_n = \overline{O} \]implies
\[ a_1 = a_2 = \ldots = a_n = 0 \]Dimension of a vector space
The cardinality of a maximal set of independent vectors is the dimension of the vector space.
Basis of the vector space
Maximal linearly independent vectors is a basis of the vector space.
Example
- \[ \begin{aligned} &\left( \begin{array}{c} a \\ b \\ c \\ d \end{array} \right), \ a, b, c, d \in \mathbb{R}, \quad \overline{O} := \text{zero vector} \\ &\left( \begin{array}{c} a_1 \\ b_1 \\ c_1 \\ d_1 \end{array} \right) + \left( \begin{array}{c} a_2 \\ b_2 \\ c_2 \\ d_2 \end{array} \right) = \left( \begin{array}{c} a_1 + a_2 \\ b_1 + b_2 \\ c_1 + c_2 \\ d_1 + d_2 \end{array} \right), \quad \alpha \cdot \left( \begin{array}{c} a \\ b \\ c \\ d \end{array} \right) = \left( \begin{array}{c} \alpha a \\ \alpha b \\ \alpha c \\ \alpha d \end{array} \right) \end{aligned} \]
- \( \mathbb{R}[x] \) = set of all polynomials with real coefficients is also a vector space. Dimension is infinite. \( \{1, x, x^2, \ldots\} \) is a basis.
- \( \mathbb{C}_{\mathbb{R}} \): dimension is 2. \( \{1, i\} \) is a basis.
- Collection of all functions from \( [0, 1] \rightarrow \mathbb{R} \). Dimension is infinite.
- \( \mathbb{C}_{\mathbb{C}} \) has dimension 1. Basis is \( \{1\} \)
Basis transformation/change
Let basis 1 for the vector space be
\[ \{v_1, v_2, v_3, v_4\} \]Let standard basis for the vector space be
\[ \{e_1, e_2, e_3, e_4\} \] \[ \begin{bmatrix} v_1 & v_2 & v_3 & v_4 \\ \end{bmatrix} \begin{bmatrix} \text{vector in basis 1 coordinate} \end{bmatrix} = \begin{bmatrix} \text{vector in standard basis coordinate} \end{bmatrix} \]Linear map
\[ S: V_{\mathbb{F}} \rightarrow W_{\mathbb{F}} \]For linear maps, both fields must be same or the domain \({\mathbb{F}}\) is subset of domain of image \({\mathbb{F}}\).
A linear map satisfies following properties.
\[ v_1 \rightarrow w_1, v_2 \rightarrow w_2 \Rightarrow v_1 + v_2 \rightarrow w_1 + w_2 \] \[ v_1 \rightarrow w_1 \Rightarrow \alpha v_1 \rightarrow \alpha w_1 \]We just need to define mapping for basis vectors to encapsulate entire linear map.
\[ \dim(V_{\mathbb{F}}) = 4 \quad \dim(W_{\mathbb{F}}) = 4 \]Basis \( (V_{\mathbb{F}}) = \{ v_1, v_2, v_3 \} \)
\[ S(v_1) = w_1, \quad S(v_2) = w_2, \quad S(v_3) = w_3 \]It could be many-to-one mapping.
Let coordinates in \( V_{\mathbb{F}} \) be
\[ \begin{pmatrix} \alpha \\ \beta \\ \gamma \end{pmatrix} \]\( W_{\mathbb{F}} \) could be \( 2 \times 2 \) matrix space.
\[ S(v_1) = \begin{pmatrix} 3 & 5 \\ 1 & 2 \end{pmatrix}, \quad S(v_2) = \begin{pmatrix} 1 & 2 \\ 1 & 3 \end{pmatrix}, \quad S(v_3) = \begin{pmatrix} -2 & 1 \\ 0 & 3 \end{pmatrix} \]We have assumed standard basis vectors for \( W_{\mathbb{F}} \):
\[ \therefore v_1 \rightarrow \begin{pmatrix} 3 \\ 5 \\ 1 \\ 2 \end{pmatrix} \text{ (in coordinate format)} \] \[ \begin{pmatrix} 3 & 1 & -2 \\ 5 & 2 & 1 \\ 1 & 1 & 0 \\ 2 & 3 & 3 \end{pmatrix} : \begin{pmatrix} \alpha \\ \beta \\ \gamma \end{pmatrix} \rightarrow \begin{pmatrix} a \\ b \\ c \\ d \end{pmatrix} \rightarrow \begin{pmatrix} a & b \\ c & d \end{pmatrix} \]This is a matrix representation of the linear map
Kernel of a linear map
\[ \text{ker}(S) = \{ v \in V_{\mathbb{F}} : S(v) = 0_W \} \]
then \( v_i + q_i \rightarrow w_i \) where \( q_i \in \text{ker}(S) \)
Therefore, many-to-one function if \( |\text{ker}(S)| > 0 \)
Scanned notes
These scanned lecture notes are from the course Multivariable Control Systems at Indian Institute of Technology, Bombay. Use them at your own discretion. If you would like to help digitize these lecture notes, contact the editor.multivariable control systems_1.jpg
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Author
Anurag Gupta is an M.S. graduate in Electrical and Computer Engineering from Cornell University. He also holds an M.Tech degree in Systems and Control Engineering and a B.Tech degree in Electrical Engineering from the Indian Institute of Technology, Bombay.
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