By Luc Jaulin

With the intention to allow a greater realizing of the major techniques of automation, this ebook develops the elemental points of the sector whereas additionally providing a number of concrete routines and their suggestions. The theoretical process that it offers essentially makes use of the kingdom house and makes it attainable to procedure normal and intricate structures in an easy method, concerning numerous switches and sensors of alternative forms. This technique calls for using built theoretical instruments reminiscent of linear algebra, research and physics, often taught in preparatory periods for professional engineering classes.

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8 (Hamilton’s method) 1) The Hamiltonian is written as: H (q, p) = 1 m ( q) ˙ 2 + mg (1 − cos q) 2 Kinetic energy = Potential energy 1 p2 + mg (1 − cos q) 2m 2 Modeling 31 since the amount of movement of the pendulum (or rather the kinetic moment in this case) is p = J q˙ = m 2 q˙ and thus 2 2 = mp 2 . The state equations of the m ( q) ˙ 2 = m mp 2 pendulum are therefore: q˙ = p˙ = ∂H(q,p) ∂p − ∂H(q,p) = ∂q = p m 2 −mg sin q where the state vector here is x = (q, p)T . Let us note that: q¨ = −mg sin q −g sin q p˙ = = 2 2 m m and we come back to the differential equation of the pendulum.

The total energy of a ﬂuid element of mass m is conserved if we consider this latter to be falling freely in the tube ﬂow. Thus, for the two points A and B, where A is at the surface and B in the tube, we have: 1 1 2 2 = mghB + mvB mghA + mvA 2 2 =0 and therefore: vB = 2g (hA − hB ) We can then deduce Toricelli’s relation. This reasoning is only possible in the case of a perfect ﬂuid, where, given the absence of friction, we assume that the forces of tangential pressure are nil. 2) The state equations are obtained by writing that the volume of water in a container is equal to the sum of the incoming ﬂows minus the sum of the outgoing ﬂows, in other words: h˙ 1 = −Q1ext − Q12 + u1 h˙ 2 = Q12 − Q23 h˙ 3 = −Q3ext + Q23 + u2 or: √ h˙ 1 = −a.

The model therefore becomes: ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ x˙ x 0 ⎜ ⎟ ⎜ θ˙ ⎟ ⎜ 0 ⎟ ⎟ ⎜ ⎟ ⎜ d ⎜θ ⎟ ⎟=⎜ ⎟+⎜ ⎟a dt ⎜ ⎝ x˙ ⎠ ⎝ 0 ⎠ ⎝ 1 ⎠ g sin θ cos θ θ˙ This model, referred to as kinetic, only involves positions, velocities and accelerations. It is much more simple than the dynamic model and contains less coefﬁcients. On the other hand, it corresponds less to reality since the real input is a force and not an acceleration. 17). 17. The inverted rod pendulum, looped by a high gain K, behaves like a kinematic model The acceleration x ¨ can be measured by an accelerometer.