Motor dc

Robust Generalized Proportional Integral control of a DC-Motor
Hebertt Sira-Ram´ ırez Jes´s Linares-Flores u October 15, 2010

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The DC motor model

Consider the popular model of a DC motor dia L = (uV − Ria − Kw) dt dw J = Kia − Bw − τ (t) dt where ia is the armature current, w is the angular velocity of the motor shaft τ (t) is the load torque applied to the shaft, The parameters L, Rand K represent respectively the inductance in the armature circuit, the armature winding resistance and the current gain of the motor. V is the fixed input voltage amplitude, modulated by the control input u, taking values on the segment [−1, 1]. On the other hand J and B represent the motor inertia and the viscous friction coefficient. The torque gain is assumed to be identical to the current gainK.
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Flatness of the DC motor model The unperturbed motor system (τ (t) = 0, ∀ t) is represented by a linear controllable system model. Hence, the system is flat with flat output given by the angular velocity w. Indeed, all variables can be written in terms of the angular velocity and its time derivatives: 1 ia = ˙ [J w + Bw] K BL JR JL w+ ¨ + u = KV KV KV

RB +K w w+ ˙ K

The perturbedversion of the above differential parametrization is simply given by ia = J B τ (t) w+ w+ ˙ K K K BL RJ K RB JL w+ ¨ + + w+ ˙ w u = KV KV KV KV V R L τ (t) + ˙ τ (t) + KV KV
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Sensor aided control problem formulation We first present the GPI observer-based solution to the problem of robust angular velocity reference trajectory tracking when the angular velocity is gathered from the motor with theaid of a tachometer. We formulate the problem as follows: Given a desired angular reference trajectory w∗(t) device, on the basis of the measured w and the relevant system parameters, a feedback control law which, irrespectively of the uniformly absolutely bounded values of the torque input τ (t), forces the motor angular velocity, w to asymptotically exponentially track the given referencesignal, w∗(t), modulo a small as desired tracking error neighborhood around the origin of the tracking error phase space.
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Robust angular velocity feedback control A simplified non-phenomenological relation linking the angular velocity to the control input is readily obtained from the above perturbed differential parametrization of the control input. We have, KV JL

w= ¨

u + ϕ(t)

where ϕ(t)is, both, an exogenous and a state dependent unknown disturbance input given by: B R RB K2 1 R ϕ(t) = + w+ ˙ + w+ τ (t)+ ˙ τ (t) J L JL JL J JL We assume ϕ(t) is an unknown time signal, but otherwise known to be uniformly absolutely bounded combined state dependent and exogenous disturbance input signal. We also assume that the time derivatives: ϕ(j) (t), j = 1, 2, 3, 4 are also uniformlyabsolutely bounded.
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A GPI observer based feedback controller is readily proposed to be u = u∗(t) + JL −ϕs(t) − k1(w2s − w∗(t)) ˆ ˙ KV −k0(w − w∗(t))

where ϕs(t) and w2s are, respectively, a smoothed ˆ estimate of the unknown signal ϕ(t) and a smoothed estimate of the angular acceleration w, given by the following GPI linear observer ˙ w1 = w2 + λ4(w − w1) ˙ KV w2 = ˙ u + z1 + λ3(w − w1) JL z1 =z2 + λ2(w − w1) ˙ z2 = z3 + λ1(w − w1) ˙ z3 = λ0(w − w1) ˙ where w1 is the redundant estimate of the measured angular velocity w. The signal w2 is the estimate of the angular acceleration w and ˙ z1 represents an internal, self-updating, time polynomial model (also called Taylor polynomial) of just second degree, estimating the perturbation input ϕ(t).
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The estimation error vector (e1, e2)= (w − w1, w − w2) satisfies ˙ e1 = e2 − λ4e1 ˙ e2 = ϕ(t) − z1 − λ3e1 ˙ z1 = z2 + λ2e1 ˙ z2 = z3 + λ1e1 ˙ z3 = λ0e1 ˙ The estimation error e = e1 = w − w1 obeys, upon elimination of the variables z1, z2 and z3, the following perturbed linear differential equation e(5) + λ4e(4) + · · · + λ0e = ϕ(3) (t) We have the following result:

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Theorem. The linear feedback control law JL −ϕ(t) − k1(w2...