\(\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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