Types of systems in the Laplace domain

System order

• Most systems that we will be dealing with can be characterised as first or second order systems

Derivatives with respect to time

• Derivatives with respect to time• Time derivatives have their own notation :

• and so on...

Differential equation

• Definition (DE, Order, ODE, System)• A differential equation (DE) is an equation with

derivatives in it.• Its order is the highest order of the derivatives in

it.• If the derivative is with respect to only one

variable, then it is an ordinary differential equation (ODE).

• Many DE all together are called a system of DEs.

Examples

Differential equation.

• Definition (Linear DE)• An (O)DE is linear if it is a linear combination

of the derivatives.• Definition (Homogenous ODE)• An ODE is homogenous if it involves solely the

derivatives of a variable and the variable itself

Examples

Consider the motor• A DC motor can be described as a first or

second order system, the final differential equations relates the motor output (with respect to time) to the motor input (with respect to time).

DC Motor systems• The DC motor can be described as first or second order

systems depending on the complexity of the model. • Generalized second order linear differential equation for a

motor• aw’’(t) + bw’(t) + cw(t) = V(t) -second order differential

equation where a, b, c are determined from data pertaining to the motor under investigation and V is voltage and w is motor speed

• Generalized first order linear differential equation for a motor

• aw’(t) + bw(t) = V(t) -first order differential equation where a, b are determined from data pertaining to the motor under investigation and V is voltage and w is motor speed

Movement into the laplace domain

• For derivatives when we perform the laplace transform because the differential equations are easier to understand and use in the laplace domain.

• It is this reason that some motor manufacturers supply transfer function models of their motors within the laplace domain.

The Laplace transform

• properties & formulas• linearity• the inverse Laplace transform• time scaling• exponential scaling• time delay• derivative• integral• multiplication by t• convolution

Laplace transform• The signal is defined for t > 0• the Laplace transform of a signal (function) f is the

function F = L(f) defined by

• F is a complex-valued function of complex numbers• s is called the (complex) frequency variable, with

units sec-1; t is called the time variable (in sec); st is unitless

• for now, we assume f contains no impulses at t = 0

derivatives• The signal f is continuous at t = 0, then

• time-domain differentiation becomes multiplication by frequency variable s (as with phasors) plus a term that includes initial condition (i.e., -f(0))

• higher-order derivatives: applying derivative formula twice yields

• similar formulas hold for

Solving the derivative equation• In regards to the term• on the previous slide, if we are to solve a first or

second order differential equation we must have more information pertaining to the value of f(0) and f’(0)

• We call this information boundary conditions• In the case of a motor with a step input. We

assume the acceleration, velocity and initial position of motor rotor to be zero at t=0.

• Making this assumption makes life very easy and our equations in the laplace domain much simpler.

Integral

• let g be the running integral of a signal f, i.e.,

then

Laplace Transform and solutions for Differential equations

• Laplace Transform can be used for solving differential equations by converting the differential equation to an algebraic equation and is particularly suited for differential equations with initial conditions.

• The solution requires the use of the Laplace of the derivative

Solving a first order ODE• Consider the first order differential equation for y(t)

below:- • + ay(t) = V(t)• Taking the Laplace Transform (Y(s) is the transform

of y(t) and V(s) is the transform of V(t)) and assuming zero initial condition gives:-

• sY(s)+aY(s)=V(s)• Solving for Y(s) the Laplace Transform of y(t) gives:-• If this was a model for a motor we could enter this

into matlab (assuming we knew the value of a)

Solving a second order ODE

• For second order equations:- + b + cy(t) = V(t)

• and assuming no initial condition gives:- Gives s2Y(s) + bsY(s) + cY(s) = V(s)• Y(s)= • If this was a model for a motor we could enter

this into matlab (assuming we knew the value of b and c)

Model of a motor

• Ibrahim pp 42-44