# A Two-Mass System

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# Coupled electric drives

## Model description:

This particular laboratory-scale process simulates the actual industrial problems in tension and speed controls as they occur in magnetic tape drives, textile machines, paper mills, strip metal production plants, etc. To simulate these problems. the coupled electric drives consists of two similar servo-motors which drive a jockey pulley via a continuous flexible belt (see attached image).The jockey pulley assembly constitutes a simulated work station. The basic control problem is to regulate the belt speed and tension by varying the two servo-motor torque.

The structure of the transfer function matrix model (numerator and denominator orders of each transfer function) have been determined from the theoretical modelling of the coupled electric drive system which has given

$$\begin{bmatrix} Y_1(s)\\ Y_2(s) \end{bmatrix} =G(s) \begin{bmatrix} U_1(s)\\ U_2(s) \end{bmatrix}$$

with

$G(s) = \begin{bmatrix} \dfrac{b_{1,1,0}}{s^2+a_{11}s+a_{12}} & \dfrac{b_{1,2,0}}{s^2+a_{11}s+a_{12}}\\ \dfrac{b_{2,1,0}s+b_{2,1,1}}{s^3+a_{21}s^2+a_{22}s+a_{23}} & \dfrac{b_{2,2,0}}{s^3+a_{21}s^2+a_{22}s+a_{23}} \end{bmatrix}$

The inputs $U(s)$ to the system are the drive voltages to the servo-motor power amplifiers. The outputs $Y_1(s)$ and $Y_2(s)$ are the jockey pulley velocity and the belt tension respectively.

## Publication details:

 Title A new bias-compensating least-squares method for continuous-time MIMO system identification applied to a laboratory-scale process Publication Type Conference Paper Authors Garnier, H., Sibille P., and Nguyen H.L.

# A linear system

## Model description:

Consider a linear system represented by the transfer function

$$G(s)=\dfrac{c}{s(s+a)}$$

where $a$ and $c>0$ are unknowns constants, and the reference model

$$G_m(s)=\dfrac{\omega^2}{s^2 + 2\zeta\omega s + \omega^2}.$$

2

## Publication details:

 Title Adaptive output feedback control of nonlinear systems represented by input-output models Publication Type Journal Article Authors Khalil, H.K.

# A Three-Mass System

## Model description:

The transfer function of the three-mass-system is much more complex than it is for one dominant elasticity (two-mass-system).

$${G_{\rm mech}}(s) = \underbrace{{\dfrac{1} {T_{ \Sigma} \cdot s}}}_{G_{\rm rs}(s)} \cdot \underbrace{ \dfrac{ a_{7} \cdot s^{4} + a_{6} \cdot s^{3}+a_{5} \cdot s^{2} + a_{4} \cdot s + 1}{a_{3} \cdot s^{4} + a_{2} \cdot s^{3}+ a_{1} \cdot s^{2} + a_{4} \cdot s + 1}} _{G_{\rm nrs}(s)}$$

with

$T_{\Sigma} = T_{\rm M} + T_{{\rm L}1} + T_{{\rm L}2}$

and

\begin{align*} a_{1}&=d_{1}d_{2}T_{{\rm C}1}T_{{\rm C}2}+T_{{\rm L}2}\left(T_{\rm M}+T_{{ \rm L}1}\right) \cdot \frac{T_{C2}}{T_{\Sigma }}+T_{\rm M}\left(T_{{\rm L}1}+T_{{\rm L}2}\right) \cdot \frac{T_{{\rm C}1}}{T_{\Sigma}} \\ a_{2}&=\frac{T_{{\rm C}1}T_{{\rm C}2}}{T_{\Sigma}}\cdot\left(d_{1}T_{{\rm L}2}\left(T_{\rm M}+T_{{\rm L}1}\right)+d_{2}T_{\rm M} \left(T_{{\rm L}1}+T_{{\rm L}2}\right)\right) \\ a_{3}&=\frac{T_{\rm M}T_{{\rm L}1}T_{{\rm L}2}T_{{\rm C}1}T_{{\rm C}2}}{T_{\Sigma}} \\ a_{4}&=d_{1}T_{{\rm C}1}+d_{2}T_{{\rm C}2} \\ a_{5}&=d_{1}d_{2}T_{{\rm C}1}T_{{\rm C}2}+T_{{\rm L}2}T_{{\rm C}2}+\left(T_{{\rm L}1}+T_{{\rm L}2}\right)\cdot T_{{\rm C}1} \\ a_{6}&=\left(\left(d_{1}+d_{2}\right)T_{{\rm L}2}+d_{2}T_{{\rm L}1}\right)\cdot T_{{\rm C}1}T_{{\rm C}2} \\ a_{7}&=T_{{\rm L}1}T_{{\rm L}2}T_{{\rm C}1}T_{{\rm C}2}. \end{align*}

## Publication details:

 Title Application of the Welch-Method for the Identification of Two- and Three-Mass-Systems Publication Type Journal Article Authors Villwock, S., and Pacas M.

# A Two-Mass System

## Model description:

The transfer function of a nonrigid mechanical system with two concentrated masses is given by

$$G_{\rm mech}(s) = \underbrace{ \dfrac{1}{s \cdot \left(T_{\rm M} + T_{\rm L}\right)} }_{G_{\rm rs}(s)} \cdot \underbrace{ \dfrac{T_{ \rm L} \cdot T_{\rm C} \cdot s^{2} + d \cdot T_{\rm C} s + 1} {\dfrac{ T_{\rm L} \cdot T_{\rm C} \cdot T_{\rm M}}{T_{\rm M} + T_{\rm L}} \cdot s^{2} + d \cdot T_{\rm C} \cdot s + 1}}_{G_{\rm nrs}(s)}.$$

$T_M$ and $T_L$ are the run-up times of the motor and the load. The nonrigid shaft of the two-mass-configuration is modeled as a damper-spring-system. $T_C$ is the normalized spring-constant and $d$ is the related damping of the spring.

## Publication details:

 Title Application of the Welch-Method for the Identification of Two- and Three-Mass-Systems Publication Type Journal Article Authors Villwock, S., and Pacas M.