A New Approach to the Tracking Control of Uncertain Nonlinear Multi-body Mechanical Systems

Abstract

This chapter presents a new approach for the tracking control of uncertain mechanical systems. Real-life multi-body systems are in general highly nonlinear and modeling them is intrinsically error prone due to uncertainties related to both their description and the description of the various forces that they may be subjected to. As such, in the modeling of such systems one only has in hand the so-called nominal system—a model based upon our best assessment of the system and our best assessment of the generalized forces acting on it. Uncertainties that are time-varying, unknown but bounded, are assumed in this chapter, and a new approach to the development of a closed-form controller is developed. The approach uses the concept of a generalized sliding surface. Its closed-form approach can guarantee, regardless of the uncertainty, that the uncertain system can track a desired reference trajectory that the nominal system is required to follow. An example of a simple multi-body system whose description is known only imprecisely is illustrated showing the simplicity of the approach and its efficacy in tracking the trajectory of the nominal system. The approach is easily implemented for a wide range of complex multi-body mechanical systems.

Key Symbols

M

The n by n nominal mass matrix

Q

The n-vector nominal given force

q

The n-vector generalized coordinate of the nominal mechanical system

$\dot{q}$

The n-vector generalized velocity of the nominal mechanical system

a

The n-vector unconstrained generalized acceleration of the nominal mechanical system ($a={M^{-1 }}Q$)

$\varphi$

The m-vector constraints (holonomic and/or nonholonomic)

A

The left-handed side, m by n matrix of constraint equations, see Eq. (4.5)

b

The right-handed side, m-vector of constraint equations, see Eq. (4.5)

r

The rank of matrix A

Q ^c

The n-vector generalized constraint force

+

The Moore–Penrose (MP) inverse of a matrix

g

The gravitational acceleration

${m_i}$

The ith mass of the system

${L_i}$

The length of the ith massless rod of the pendulum, see Fig. 4.1

${\theta_i}$

The angle between the ith massless rod of the pendulum and the vertical axis, see Fig. 4.1

${E_i}$

The total energy of the mass ${m_i}$

α

The arbitrary, nowhere-zero, sufficiently smooth real function of time

${M_a}$

The n by n actual mass matrix (${M_a}=M+\delta M$)

$\delta M$

The n by n matrix that characterizes uncertainties in the actual mass matrix

${Q_a}$

The n-vector given force of the actual system (${Q_a}=Q+\delta Q$)

$\delta Q$

The n-vector of uncertainties in given force of the actual system

$\tilde{q}$

The n-vector generalized coordinate of the actual mechanical system

$\dot{\tilde{q}}$

The n-vector generalized velocity of the actual mechanical system

${q_a}$

The n-vector generalized coordinate of the actual mechanical system, which is obtained by using the correct control force that the actual system is required to be subjected to, because of the control requirements

${{\dot{q}}_a}$

The n-vector generalized velocity of the actual mechanical system, which is obtained by using the correct control force that the actual system is required to be subjected to, because of the control requirements

${a_a}$

The n-vector unconstrained generalized acceleration of the actual mechanical system (${a_a}=M_a^{-1 }{Q_a}$)

${Q^u}$

The n-vector additional generalized control force that compensates for uncertainties (${Q^u}=M\ddot{u}$)

$\ddot{u}$

The n-vector additional generalized acceleration to compensate for uncertainties

${q_c}$

The n-vector generalized coordinate of the controlled actual mechanical system

${{\dot{q}}_c}$

The n-vector generalized velocity of the controlled actual mechanical system

e

The tracking error between the controlled actual system’s response and the nominal system’s response ($e={q_c}-q$)

$\delta \ddot{q}$

The n-vector of uncertainties in acceleration of the actual system

$\bar{M}$

The n by n matrix of uncertainties in the compensating controller ($\bar{M}\approx I-{{\left( {I+{M^{-1 }}\delta M} \right)}^{-1 }}$)

$\varGamma$

The arbitrary positive function of time that characterizes the bound on $\delta \ddot{q}$ ($\left\| {\delta \ddot{q}} \right\|\leq \varGamma (t)$)

${\varGamma_m}$

The constant upper bound on the function $\varGamma (t)$

s

The n-vector sliding surface

${k_1}$

The arbitrary small positive number

${\varOmega_{\varepsilon }}$

The surface of the n-dimensional cube around the point $s=0$

${\beta_0}$

The arbitrary positive constant (${\beta_0}>{k_1}\left\| {\bar{M}} \right\|\left\| {\dot{e}} \right\|$)

$\gamma$

The small positive constant ($\gamma =\frac{{\left\| s \right\|\left\| {f(s)} \right\|}}{{{s^T}f(s)}}$)

$\sigma$

The arbitrary small positive constant ($\gamma \leq \sigma \leq 1$)

${\alpha_0}$

The arbitrary small positive constant ($0<{\alpha_0}<1-n\sigma \left\| {\bar{M}} \right\|$)

$\beta$

The positive function of time ($\beta (t)>\frac{{n(\varGamma +{\beta_0})}}{{{\alpha_0}}}$)

$\varepsilon$

The arbitrary small positive constant

${\alpha_c}$

The arbitrary positive constant

$f(s)$

The arbitrary monotonic increasing odd continuous function of $s$

${G_{ss }}$

The generalized sliding surface controller

$V$

The Lyapunov function

${L_{\varepsilon }}$

The length of each side of the cubical surface ${\varOmega_{\varepsilon }}$

${Q^T}$

The n-vector total control force of the mechanical system (${Q^T}={Q^c}+{Q^u}$)