august 28

\(\def\Real{\mathbb{R}}\def\Comp{\mathbb{C}}\def\Rat{\mathbb{Q}}
\def\Field{\mathbb{F}}
\)

What this course deals with?

Linear control systems. A continuous-time state-space linear system is defined by the following equations:
The signals
\[
x'(t) = A(t)x(t) + B(t)u(t), y(t) = C(t)x(t) + D(t)u(t),
\]
where
\[
x\in \Real^n, u\in\Real^k, y\in\Real^m;\\
y:[0,\infty)\to\Real^m,
u :[0,\infty)\to\Real^k,
x :[0,\infty)\to\Real^n,
\]
are the (vector-valued) functions called input, state, and output of the system.

The first-order differential equations is called the state and the output equation.

When all the matrices \(A(t), B(t), C(t), D(t)\) are constant, the system
is called a linear time-invariant (LTI) system.

In the general case, it is called a linear time-varying (LTV) system.

To keep formulas short, in the following we abbreviate the state/output equations to
\[
x’=A(t)x+B(t)u, y=C(t)x+D(t)u, x\in \Real^n, u\in\Real^k, y\in\Real^m;
\mathrm{(CLTV)}
\]
and in the time-invariant case, we further shorten this to
\[
x’= Ax + Bu, y = Cx + Du, x\in \Real^n, u\in\Real^k, y\in\Real^m.\mathrm{(CLTI)}
\]

Linear systems come typically from the linearizations of (far more prevalent) nonlinear ones. Physical pendulum is the standard example.

Linearization around equilibria leads to LTIs; linearization around a trajectory leads to LTVs.

Fields, vector spaces, subspaces, linear operators, range space, null space

A field \((F,+,·)\) is a set with two commutative associative operations with \(0\) and\(1\), additive and multiplicative inverses and subject to the distributive law.

Examples (Common fields)
\((\Real,+,·)\) is a field. Rational, Complex numbers. Matrices (when they are fields?)

Vector spaces – modules over a field.

Examples

Fun(\(A,\Field\)), any \(A\) – vector space over \(\Field\). (For example, sequences…)

Subclasses of functions: continuous, integrable, …

Linear independence. Bases. Dimension.

Linear subspace (of a linear space)- subset that is a linear space (over the same field, of course).

Linear operators, their range (image), null-space (kernel), rank.

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