Analysis of improved fractional backstepping and lyapunov strategies for stabilization of inverted pendulum

Abstract

Controlling an inverted pendulum towards an upright position is a difficult task. Backstepping control is an emerging tool for assisting this extremely nonlinear system to stabilize. Since several studies demonstrated fractional modern strategies with Oustaloup approximation, this current work proposes a novel fractional backstepping rule with improved biquadratic equiripple approximation method to stabilize the system with superior accuracy. On the basis of study in the frequency domain, a suitable fractional order is established. Closed-loop performances and control efforts between proposed fractional and conventional backstepping controllers are illustrated based on time domain analysis from a real-time perspective. By abruptly changing the system's parameters, the effectiveness of the proposed controller is also verified. A further fractional Lyapunov improved architecture is proposed to investigate control efficacy with proposed fractional backstepping strategy. The selection of tuning parameters of all control strategies is addressed analytically in depth. It is explored that the suggested fractional backstepping control scheme outperforms the conventional backstepping and fractional Lyapunov stability rules by effectively tracking desired position. This enhanced performance is achieved with relatively smooth control action. On the basis of error measurements, quantitative performance analysis is also subjected to all control strategies.

Appendices

Appendix A

Computation of Lyapunov function:

Second order process and ideal model are chosen as follows

$$ \frac{{d^{2} Y\left( t \right)}}{{dt^{2} }} = - c\frac{dY\left( t \right)}{{dt}} - dY\left( t \right) + dU\left( t \right) $$

(A.1)

$$ \frac{{d^{2} y_{m} \left( t \right)}}{{dt^{2} }} = - c_{m} \frac{{dy_{m} \left( t \right)}}{dt} - d_{m} y_{m} \left( t \right) + d_{m} U\left( t \right) $$

(A.2)

Similarly, first order process and ideal model are chosen as follows

$$ \frac{dY\left( t \right)}{{dt}} = - cY\left( t \right) + dU\left( t \right) $$

(A.3)

$$ \frac{{dy_{m} \left( t \right)}}{dt} = - c_{m} y_{m} \left( t \right) + d_{m} U\left( t \right) $$

(A.4)

To construct FOLY error function is regarded as

$$ \ddot{e}\left( t \right) = \ddot{Y}_{p} \left( t \right) - \ddot{y}_{m} \left( t \right) $$

(A.5)

$$ \ddot{e}\left( t \right) = - e\left( {c_{m}^{2} - d_{m} } \right) + Y_{p} \left( {cd\theta_{2} - d - d\theta_{2} + c_{m}^{2} - d_{m} } \right) - u_{c} \left( {K_{p} + K_{d} s} \right)\left( {cd\theta_{1} - d\theta_{1} + c_{m} d_{m} - d_{m} } \right) $$

(A.6)

First order derivative of Lyapunov function (43) is derived as

$$ \dot{V}\left( t \right) = - 1.5e^{2} \left( {c_{m}^{2} - d_{m} } \right) + \frac{3}{2\gamma }\left[ {\left( {cd - d} \right)\theta_{2} + c_{m}^{2} - d - d_{m} } \right]\left( {\frac{2}{3}\frac{{d\theta_{2} }}{dt} + eY_{p} \gamma } \right) + \frac{3}{2\gamma }\left[ {\left( {cd - d} \right)\theta_{1} + c_{m} d_{m} - d_{m} } \right]\left( {\frac{2}{3}\frac{{d\theta_{1} }}{dt} + eu_{c} \left( {K_{p} + K_{d} s} \right)\gamma } \right) $$

(A.7)

Abbreviations

MRAC:

Model reference adaptive control

FO:

Fractional order

CFE:

Continued fractional expansions

FOPI:

Fractional order proportional integral

FOPID:

Fractional order proportional integral derivative

PID:

Proportional integral derivative

FOMRAC:

Fractional order model reference adaptive control

DOF:

Degree of freedom

PD:

Proportional derivative

MIT:

Massachusetts Institute of Technology

FOLY:

Fractional order Lyapunov

CB:

Conventional backstepping

FOB:

Fractional order backstepping

IAE:

Integral absolute error

ITAE:

Intgral time absolute error

ISE:

Integral square error

R–L:

Riemann–Liouville

TV:

Total variation of control effort

List of symbols

$F$:

Force

$V$:

Horizontal direction of force

$U$:

Vertical direction of force

$\varphi$:

Pendulum angle

$\ddot{x}$:

Acceleration of cart

$p_{1} ,p_{2}$:

Design constants

$\alpha$:

Extra degree of freedom

$\gamma$:

Adaptive gain

$e$:

Error signal between model and plant of MRAC scheme

$u_{c}$:

Reference input of model reference adaptive scheme

$U\left( t \right)$:

MRAC control law

$K_{p} ,K_{P1} ,K_{P2}$:

Proportional gains

$K_{D}$:

Derivative gain

$K_{I1} ,\;K_{I2}$:

Integral gains

$b$:

Constant

$\lambda$:

Extra degree of freedom of integral gain

$y_{m} \left( t \right)$:

Reference model

$Y_{P} \left( t \right)$:

Actual plant

$m$:

Constant

$\omega_{c}$:

Corner frequency

$q_{0}$:

Coefficient

$q_{1}$:

Coefficient

$q_{2}$:

Coefficient

$\theta_{2}$:

Vector of adaptive scheme

$\theta_{2}$:

Vector of adaptive scheme

$c,d$:

Coefficients of plant

$c_{m} d_{m}$:

Coefficients of model

$u$:

Backstepping control input