Abstract
This chapter focuses on how to obtain dynamic linear models from nonlinear physical models through linearization using a Taylor series expansion around a nominal operating point. Several examples are included, so that linear systems with different characteristics are obtained: the tank level system, variable section tank level system, ball & beam system, inverted pendulum on a cart system, and DC motor position system. Main concepts and basic definitions are included.
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Notes
- 1.
It is the property of a system whereby a scaled input results in an equally scaled output, that is, the response of a system to a signal bu(t) is equal to b times the response to u(t).
- 2.
Superposition will apply if and only if the system is linear.
- 3.
In the scope of transfer functions modeling the system dynamics, this concept will be related to the difference or relative degree between the number of poles and finite zeros. If the number of poles and zeros is the same, the system is called causal and there is an instantaneous transmission from the input signal to the output, while if the transfer function is strictly proper (more poles than finite zeros), the system is called strictly causal.
- 4.
Notice that as usual, \(\dot{y}\) represents the time derivative \(\frac{dy(t)}{dt}\), \(\ddot{y}=\frac{d^2y(t)}{dt^2}\) and so on, where t is the independent variable (time), y(t) is the dependent variable (output), and u(t) is the input. Notation \(d^ny(t)/dt^n\) is used to indicate the n-th derivative of y with respect to time. The dependence on time is often obviated to facilitate understanding.
- 5.
The gap metric is an extension of the common measure of the \(\infty \)-norm of the difference between two systems and it is used to measure the gap between a linearization of a nonlinear system at its operating point and a fixed linear system [12].
- 6.
Actuators are the devices that provide the driving power to the process, and therefore, are in charge of converting the control signal into the manipulated variable.
- 7.
If each fluid element at each point in the flow has no net angular velocity about that point, the flow is termed irrotational.
- 8.
The velocity at a given point does not change with time.
- 9.
Bernoulli’s principle can be derived from the principle of conservation of energy. It states that, in a steady flow, the sum of all forms of energy of a fluid along a streamline is the same at all points along the streamline. This requires that the sum of kinetic energy, potential energy, and internal energy remain constant. The hydrostatic pressure across the orifice is \(\varDelta P= \rho g h\). In applications using valves, the volumetric flow is multiplied by a coefficient of discharge.
- 10.
Notice that \((\overline{q},\overline{h})\) is used here to denote the actual operating point representing a steady state of the system. Other texts use different equivalent notations, as \((q_0,h_0)\), \((q_e,h_e)\), ...The notation \((q_0,h_0)\) is sometimes preferred because it also represents the initial state from which the response to changes in the input is studied (initial condition).
- 11.
\(I_b=\frac{2}{5} m R^2\), R being the radius of the ball (see Fig. 2.7).
- 12.
As pointed out by [46], the validity of these assumptions is questionable, depending upon the relative physical features of the system. For many laboratory equipments, the approximations made are felt to be quite justifiable.
- 13.
As pointed out by [49], initially, in the 60s of the last century, this system was present in the control laboratories of the most prestigious universities. The demonstration consisted of initially placing the pendulum manually in the inverted vertical position, then releasing it and autonomously, by feeding back its position, the pendulum would continue in the inverted position by means of the appropriate action on the cart. The control problem, thus considered, is local and its interest lies in the fact that it involves stabilizing an open-loop unstable position. This problem, because of its local character, can be solved with linear methods, as discussed in this book. The problems arise when the pendulum has to be swung up from its natural equilibrium, and that the path of the cart is bounded, so that if one of the ends of the horizontal support is reached, the system stops working.
- 14.
- 15.
In [29] (Sect. 2.6.1), the model of the motor is obtained also for the case when the excitation voltage \(e_f\) is varying through a linearization process.
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Guzmán, J.L., Costa-Castelló, R., Berenguel, M., Dormido, S. (2023). From Nonlinear Physical Models to Linear Models. In: Automatic Control with Interactive Tools. Springer, Cham. https://doi.org/10.1007/978-3-031-09920-5_2
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