Recursive LTI Systems Evaluation by transfer functions.

Table of Contents

• Continous-time transfer functions
  • Example: Evaluate the given continuous transfer function
• Discrete-time transfer functions
  • Example: Evaluate the given discrete transfer function

qlibs::ltisys is a class that evaluates single-input, single-output (SISO) transfer function models in real-valued systems. A transfer function is a model that describes the frequency-dependent response of a linear time-invariant system, and this class can handle both continuous-time and discrete-time systems. qlibs::ltisys can be used for simulating dynamic systems and implementing filters, compensators, or controllers.

Continous-time transfer functions

Here, the transfer function \( G(s) \) is the linear mapping of the Laplace transform of the input, \( U(s)= \mathcal{L}[u(t)]\), to the Laplace transform of the output \( Y(s)= \mathcal{L}[y(t)]\).

\( G(s) = \frac{N(s)}{D(s)} = \frac{ b_{0}s^{n} + b_{1}s^{n-1} + b_{2}s^{n-2} + ... + b_{n} }{ s^{n} + a_{1}s^{n-1} + a_{2}s^{n-2} + ... + a_{n} } \)

\( N(s) \) and \( D(s) \) are the numerator and denominator polynomials in descending powers of \(s\), respectively.

To instantiate a continuous transfer function, you should define a variable of type qlibs::continuousSystem, two arrays of N+1 elements with the coefficients of the polynomials for both, the numerator and denominator, and finally, an array of type qlibs::continuousStates to hold the N states of the system. Then, you can call qlibs::continuousSystem::setup() to construct the system and set initial conditions. Subsequently, you can evaluate the system with a given input-signal by just calling qlibs::continuousSystem::excite() .

Attention

The user must ensure that the evaluation of the system is executed periodically at the required time step.

Example: Evaluate the given continuous transfer function

\( G(s) = \frac{ 2s^{2} + 3s + 6 }{ s^{3} + 6s^{2} + 11s + 16 } \)

#include <iostream>
#include <chrono>
#include <thread>
#include <qlibs.h>
 
using namespace qlibs;
 
void xTaskSystemSimulate( void )
{
    const real_t dt = 0.05f; // Time step
    std::chrono::milliseconds delay(static_cast<long long>( dt*1000 ) );
    real_t num[] = { 0.0f, 2.0f, 3.0f, 6.0f };
    real_t den[] = { 1.0f, 6.0f, 11.0f, 16.0f };
    continuousStates<3> x = { 0.0f, 0.0f, 0.0f }; // n = 3
    continuousSystem gc( num, den, xC, dt );
    real_t ut, yt;
 
    for( ;; ) {
        ut = BSP_InputGet();
        yt = gc.excite( ut );
 
        std::this_thread::sleep_for(delay);
        std::cout << "u(t) = " << ut << " y(t) = " << yt << std::endl;
    }
}

Alternatively, you can also use a simpler definition by employing the qlibs::continuousTF structure as follows:

#include <iostream>
#include <chrono>
#include <thread>
#include <qlibs.h>
 
using namespace qlibs;
 
void xTaskSystemSimulate( void )
{
    const real_t dt = 0.05f; // Time step
    std::chrono::milliseconds delay(static_cast<long long>( dt*1000 ) );
    continuousTF<3> ctf= {
        { 0.0f, 2.0f, 3.0f, 6.0f },
        { 1.0f, 6.0f, 11.0f, 16.0f },
    };
 
    continuousSystem gc( ctf, dt );
    real_t ut, yt;
    for( ;; ) {
        ut = BSP_InputGet();
        yt = gc.excite( ut );
 
        std::this_thread::sleep_for(delay);
        std::cout << "u(t) = " << ut << " y(t) = " << yt << std::endl;
    }
}

Discrete-time transfer functions

The z-transform is used in discrete-time systems to deal with the relationship between an input signal \( u(t) \) and an output signal \( y(t) \) , so the transfer function is similarly written as \( G(z^{-1}) \) and is often referred to as the pulse-transfer function.

\( G(z^{-1}) = \frac{N(z^{-1})}{D(z^{-1})} = \frac{ b_{0} + b_{1}z^{-1} + b_{2}z^{-2} + ... + b_{m}z^{-m} }{ 1 + a_{1}z^{-1} + a_{2}z^{-2} + ... + a_{n}z^{-n} } \)

Discrete systems are instantiated in a similar way to continuous systems, but there are some differences. States should be stored in an array of type qlibs::discreteStates. The size of polynomials can vary according to their order. Please take a look at the following example:

Example: Evaluate the given discrete transfer function

\( G(z^{-1}) = \frac{ 0.1 + 0.2z^{-1} + 0.3z^{-2} }{ 1 - 0.85z^{-1} + 0.02z^{-2} } \)

#include <iostream>
#include <chrono>
#include <thread>
#include <qlibs.h>
 
using namespace qlibs;
 
void xTaskSystemSimulate( void )
{
    real_t num[] = { 0.1f 0.2f, 0.3f };
    real_t den[] = { 1.0f, -0.85f, 0.02f };
    discreteStates<3> xd = { 0.0f, 0.0f, 0.0f };
    discreteSystem gc( num, den, xd );
    real_t uk, yk;
 
    for( ;; ) {
        uk = BSP_InputGet();
        yk = gc.exite( uk );
        std::this_thread::sleep_for( std::chrono::milliseconds( 100 ) ); /*100mS sample time*/
        std::cout << "u(k) = " << uk << " y(k) = " << yk << std::endl;
    }
}

Alternatively, you can also use a simpler definition by employing the qlibs::discreteTF structure as follows:

#include <iostream>
#include <chrono>
#include <thread>
#include <qlibs.h>
 
using namespace qlibs;
 
void xTaskSystemSimulate( void )
{
    discreteTF<3,3> dtf= {
        { 0.1f, 0.2f, 0.3f },
        { 1.0f, -0.85f, 0.02f },
    };
    discreteSystem gc( dtf );
    real_t uk, yk;
 
    for( ;; ) {
        uk = BSP_InputGet();
        yk = gc.excite( ut );
        std::this_thread::sleep_for( std::chrono::milliseconds( 100 ) ); /*100mS sample time*/
        std::cout << "u(k) = " << uk << " y(k) = " << yk << std::endl;
    }
}