Flatness-based longitudinal vehicle control with embedded torque constraint

Flatness-based longitudinal vehicle control with embedded torque constraint Abstract This paper aims at establishing a simple yet efficient solution to the problem of trajectory tracking with input constraint of a non-linear longitudinal vehicle model. We make use of differential flatness by embedding the constraint into the reference trajectory design. 1. Introduction The aim of this paper is to come up with simple yet efficient control techniques for vehicle longitudinal speed control with constraint on its torque. More precisely, we consider a longitudinal non-linear model including a simple adherence/friction law (see, e.g. Ellis, 1969; Gillespie, 1992; Kiencke & Nielsen, 2000; Mitschke & Wallentowitz, 2004; Rajamani, 2011). For this model, we consider the problem of tracking a reference speed trajectory with constraint on the input torque. Traditional treatment of such a problem includes model predictive control (Li et al., 2011) and use of optimization techniques (Hsu & Chen, 2013; Hsu et al., 2010). Some other works use adaptive anti-windup techniques (Kahveci & Ioannou, 2010; Tarbouriech & Turner, 2009), saturated inputs (Valmorbida et al., 2013), to name a few. The constraint is embedded in the flat output trajectory design. Thus, the closed-loop tracking controller naturally satisfies the required constraint, without the recourse to costly optimization procedure. A key advantage of the advocated technique is that the physical meaning is kept throughout the whole process, a feature often lost in MPC (Model Predictive Control) or other optimization-based techniques. More precisely, a dynamical system with m inputs is differentially flat (Fliess et al., 1995) if there exists a so-called flat output ω with m components ω = (ω1, … , ωm) such that: first, these components are functions of the system’s variables (endogenous character); second, the ωis are differentially independent, i.e. they do not satisfy a differential equation involving themselves only (independent character); third, all the system’s variables can be expressed as non-linear functions of the ωis and of a finite number of their derivatives (parametrization property). Thus, when a system variable is subject to a constraint, the latter is directly translated into a flat output constraint, thanks to the parametrization property. The tracking problem with constraints is thus elaborated in two steps: first, design a flat output reference trajectory ωr satisfying all the required constraints; second, design a closed-loop feedback control law ensuring the tracking of ωr with stability. The constraints satisfaction is ensured by design, since it is embedded in the reference trajectory elaboration process. The involved constraints can be given on any system variable, since all of the system is parametrized by the flat output. The constraint is enforced on the reference variables and is ensured practically on the actual variables since the tracking error is meant to tend to zero in general exponentially. Ensuring the constraints on the flat output is simplified by specializing the flat output reference trajectory to specific classes of functions with convenient properties, such as closedness wrt differentiation or being a solution of a differential equation. To the best of the author’s knowledge, almost all the current works on differentially flat systems with constraints are managed through optimization procedures (Chamseddine et al., 2013; Faiz et al., 2001; Flores & Milam, 2006; Keck et al., 2015; Petit & Sciarretta, 2011; Ross & Fahroo, 2004; Tsuei & Milam, 2016; Walambe et al., 2016). In the study by Löwis & Rudolph (2003), no optimization technique is used, but the flat output trajectory is not known in advance; thus, the trajectory is built step by step, by concatenating pieces. The only work partially related to our approach is the study by Ruppel et al. (2011), where the constraints appear solely on derivatives of the flat output, which is specialized to piecewise polynomial functions. Preliminary results related to the present one have been presented for linear systems with delays in the study by Bekcheva et al. (2017) and for an Euler–Bernoulli beam in the study by Bekcheva et al. (2015). Other works related to the present theme include differential flatness-based techniques for longitudinal and lateral vehicle dynamics (Menhour et al., 2014). The paper is organized as follows. In the next section, the model is recalled. In Section 3, the flatness of the model is established, and a closed-loop feedback tracking controller is given in Section 4. The torque constraint management is dealt with in Section 5. 2. Longitudinal model The equations of the vehicle dynamics can be written as follows (see, e.g. Ellis, 1969; Gillespie, 1992; Kiencke & Nielsen, 2000; Mitschke & Wallentowitz, 2004; Rajamani, 2011):   \begin{align} m\dot{V}_{x} = F_{x} \qquad\quad \end{align} (2.1a)  \begin{align} \qquad\ I_{w}\dot{\omega} = RT-r F_{x} - F_{o} \end{align} (2.1b)with the following slip ratio and forces:   \begin{align} F_{x} = \mu_{x} (\lambda) F_{z}, \quad \lambda = \frac{V_{x}-r{\omega}}{\max(V_{x}, r{\omega})} \end{align} (2.2a)  \begin{align} F_{z} = mg\qquad\qquad\qquad\qquad\qquad\ \ \end{align} (2.2b)  \begin{align} F_{o} = - F_{a} - F_{s} \qquad\qquad\qquad\qquad\! \end{align} (2.2c)  \begin{align} \qquad\qquad\quad\ R_{x} = mg C_{r}, \quad F_{a} = \frac{\rho\, C_{a}A{V_{x}^{2}}}{2}, \quad F_{s} = m g \sin \alpha. \end{align} (2.2d) The notations for the models (2.1a) and (2.1b) are as follows: Vx is the longitudinal speed of the vehicle, m its mass, Fx the longitudinal tyre force, Iw the inertia moment of the wheel, ω the angular wheel speed, R the damping coefficient of the driveline, T the engine torque, r the effective tyre radius, Fo the other forces exerted on the car body. The expressions of the forces are given in Equations (2.2a)–(2.2d), with the following notations: μx is the adherence function, Fz the normal force on the tyre, λ the slip ratio, g the gravity constant, Rx the rolling resistance force, Fa the longitudinal aerodynamic drag force, ρ is the air volumic mass, A is frontal area of the vehicle, Ca is the drag coefficient, Fs the force due to the road slope and α the road slope angle. A possible model for μx(λ) introduced by Kiencke & Daiss, (1994) and depicted in Fig. 1 is given by the function  \begin{align} \mu_{x} (\lambda) = \frac{a \lambda}{b + c |\lambda| + \lambda^{2}}. \end{align} (2.3) One easily obtains that the maximum μ* of such a curve occurs at λ* with   $$ \lambda^{\ast} = \sqrt{b}, \quad \mu^{\ast} = \frac{a}{c + 2 \sqrt{b}}. $$ Conversely, the constants a, b, c can be expressed as functions of μ*, λ* and μ1  $$ a = \frac{\mu^{\ast} \mu_{1} \left(1-\lambda^{\ast}\right)^{2}}{\mu^{\ast} - \mu_{1}}, \quad b ={\lambda^{\ast}}^{2}, \quad c = \frac{\mu_{1} \left(1+{\lambda^{\ast}}^{2}\right) - 2 \mu^{\ast} \lambda^{\ast}}{\mu^{\ast} - \mu_{1}} $$where μ1 = μ(1) is the value of the function μ at λ = 1 (i.e. at wheel lock). Note that a and b are strictly positive constants. Fig. 1. View largeDownload slide Adherence function μ(λ). Fig. 1. View largeDownload slide Adherence function μ(λ). Remark 2.1 Another, quite popular, model is the Pacejka one (Bakker et al., 1987; Pacejka, 2006). We have not used the latter for simplicity reasons, but a similar, although more complex, analysis could be made with Pacejka’s model. The measured outputs are traditionally the wheel speed (e.g. through ABS (Anti Blocking System) encoders). We shall here suppose that the speed Vx of the vehicle’s centre of gravity is either measured or reconstructed via an observer or an estimator (see, e.g. the study by Villagra et al., 2008, a previous work of some author of the present paper). 3. Differential flatness of the model The model (2.1) is trivially flat, with flat output Vx. Indeed,   \begin{align} \dot{V}_{x} = g \mu_{x} (\lambda). \end{align} (3.1) Then,   \begin{align} \lambda = \mu^{-1} \left( \frac{\dot{V}_{x}}{g} \right)\hskip-2pt. \end{align} (3.2) Now one has to distinguish two acceleration and deceleration cases (implied by the form of λ in (2.2a)). Acceleration case, where $$r\omega \geqslant V_{x}$$  $$ \lambda = \frac{V_{x}}{r\omega} - 1 = \mu^{-1} \left( \frac{\dot V_{x}}{g}\right)\hskip-2pt. $$ Hence   \begin{align} \omega = \frac{V_{x}}{r \left[ 1 + \mu^{-1} \left( \frac{\dot V_{x}}{g}\right)\right]}. \end{align} (3.3) And thus   $$ \ddot{V}_{x} = g \mu_{x}^{\prime} (\lambda) \dot \lambda = g \mu_{x}^{\prime} (\lambda)\frac{1}{r\omega^{2}} \left( {\omega} \dot V_{x} - \dot{\omega} V_{x}\right)\hskip-2pt. $$ Deceleration case, where $$r\omega \leqslant V_{x}$$  $$ \lambda = 1 - \frac{r\omega}{V_{x}} = \mu^{-1} \left( \frac{\dot V_{x}}{g}\right)\hskip-2pt. $$ Hence   \begin{align} \omega = \frac{V_{x}}{r}\, \left[ 1 - \mu^{-1} \left( \frac{\dot V_{x}}{g}\right)\right]\hskip-2pt. \end{align} (3.4) And thus   $$ \ddot{V}_{x} = g \mu_{x}^{\prime} (\lambda) \dot \lambda = g \mu_{x}^{\prime} (\lambda)\frac{r}{{V_{x}^{2}}} \left({\omega} \dot V_{x} - \dot{\omega} V_{x}\right)\hskip-2pt. $$ Thus, one has the following dynamics in Vx:   \begin{align} \ddot V_{x} = \frac{g\mu_{x}^{\prime}}{\max \left(r\omega^{2}, \frac{{V_{x}^{2}}}{r}\right)} \left[ \omega \dot V_{x} + \frac{V_{x}}{I_{w}}\, \left( mr\dot V_{x} + F_{o} + RT \right) \right] \end{align} (3.5)and the control input T is then obtained as   \begin{align} T = \frac{1}{R}\, \left[ \left( mr + \frac{I_{w} \omega}{V_{x}} \right) \dot V_{x} + F_{o} - \frac{{I_{w}}\max \left(r^{2}\omega^{2}, {V_{x}^{2}}\right)}{gr{V_{x}}\mu_{x}^{\prime}}\, \ddot V_{x} \right]\hskip-2pt. \end{align} (3.6) Remark 3.1 The reader could have the (quite normal) feeling that the laws (3.3) and (3.4) yield a discontinuity when the vehicle switches from acceleration to deceleration (leading to a chattering-like phenomenon). First note that this can only occur at extremely low slip, i.e. when rω − Vx ≪ 1, where the μ() curve is in the linear zone (and thus the μ−1 also); thus   $$ \mu^{-1} \left( \frac{\dot V_{x}}{g}\right) \approx \beta \frac{\dot V_{x}}{g}. $$ Moreover, when the vehicle switches from acceleration to deceleration (or vice versa), one has $$|\dot V_{x} | \ll 1$$. Thus, in (3.3), one has   $$ \frac{1}{1 + \mu^{-1} \left( \frac{\dot V_{x}}{g}\right)} = 1 - \mu^{-1} \left( \frac{\dot V_{x}}{g}\right) + o \left( \left( \frac{\dot V_{x}}{g}\right)^{\!\!2} \right)\hskip-2pt. $$ Thus, the expression of ω is   $$ \omega = \frac{V_{x}}{r \left[ 1 + \mu^{-1} \left( \frac{\dot V_{x}}{g}\right)\right]} = \frac{V_{x}}{r} \left[ 1 - \mu^{-1} \left( \frac{\dot V_{x}}{g}\right) + o \left( \left( \frac{\dot V_{x}}{g}\right)^{\!\!2} \right) \right] $$whose term in $$o\left((V_{x}/g)^{2}\right)$$ is exactly the one of (3.4). Thus, in case of acceleration–deceleration switching, the expression of ω is continuous and differentiable. 4. Trajectory tracking 4.1. Trajectory tracking control law Recalling the flat output dynamics (3.5) and setting the right member equal to a new input v, one obtains the linearizing feedback   $$ \omega \dot V_{x} + \frac{V_{x}}{I_{w}}\, \left( mr\dot V_{x} + F_{o} + RT \right) = \frac{\max \left(r^{2}\omega^{2}, {V_{x}^{2}}\right)}{gr\mu_{x}^{\prime}}\, v $$transforming the flat output dynamics (3.5) to   $$ \ddot V_{x} = v. $$ Setting the new input v to   $$ v = \ddot V_{xr} - K_{p} e_{V_{x}} - K_{d} \dot e_{V_{x}}, \quad e_{V_{x}} = V_{x} - V_{xr} $$with Kp, Kd > 0 yields an exponentially stable error dynamics. The original input is then obtained as   \begin{align} T = \frac{1}{R}\, \left[ \left( mr + \frac{I_{w} \omega}{V_{x}} \right) \dot V_{x} + F_{o} - \frac{{I_{w}}\max \left(r^{2}\omega^{2}, {V_{x}^{2}}\right)}{gr{V_{x}}\mu_{x}^{\prime}}\, v \right] \end{align} (4.1)  \begin{align} v = \ddot V_{xr} - K_{p} e_{V_{x}} - K_{d} \dot e_{V_{x}}.\qquad\qquad\qquad\qquad\qquad\qquad\! \end{align} (4.2) Remark 4.1 Note that, in (4.2), one could have used equally a second order sliding mode or a model free control law, for instance, in order to gain in robustness. 4.1.1. Open- and closed-loop tracking Let Vxr be a reference trajectory for the flat output Vx. Denoting by Tr, the following open-loop control law, one has by direct substitution from (3.6):   \begin{align} T_{r} = \frac{1}{R}\, \left[ \left( mr + \frac{I_{w} \omega_{r}}{V_{xr}} \right) \dot V_{xr} + F_{o} - \frac{{I_{w}}\max \left(r^{2}{\omega_{r}^{2}}, V_{xr}^{2}\right)}{gr{V_{x}}\mu_{xr}^{\prime}}\, \ddot V_{xr} \right] \end{align} (4.3) Thus, the Equations (4.1) and (4.2) can be rewritten as   \begin{align} T = T_{r} - \frac{1}{R}\, \left( K_{p} e_{V_{x}} + K_{d} \dot e_{V_{x}} \right)\hskip-2pt. \end{align} (4.4) We thus see that, if the error $$e_{V_{x}}$$ and its derivative $$\dot e_{V_{x}}$$ remain small (which is the case when the tracking performance is good), the closed-loop torque T remains close to the open loop one Tr. 4.2. Trajectory tracking scenario 4.2.1. Trajectory form We shall choose a trajectory Vxr(t) of the following form:   \begin{align} V_{xr} (t) = \Omega_{\,{\boldsymbol{p}}_{u},\ {\boldsymbol{p}}_{d}} (t) = \Theta_{\,{\boldsymbol{p}}_{u}} (t) - \Theta_{\,{\boldsymbol{p}}_{d}} (t)\qquad\qquad\qquad\qquad\qquad\qquad\ \end{align} (4.5)  \begin{align} \Theta_{\,{\boldsymbol{p\ast}}} (t) &= \frac{V_{h \ast} - V_{l\ast}}{2 \left(t_{e\ast} - t_{b\ast}\right)}\, \left(\textrm{logCh}_{\sigma_{\ast}} \!(t-t_{b\ast}) + \textrm{logCh}_{-\sigma_{\ast}} \!(t-t_{e\ast}) \right) + \frac{V_{h\ast} - V_{l\ast}}{2}\\ \textrm{logCh}_{\sigma} (t) &= \frac{1}{\sigma}\, \log \left(\cosh (\sigma t)\right) \nonumber \\ \nonumber{\boldsymbol{p}}_{\ast} &\in \left\{{\boldsymbol{p}}_{u}, {\boldsymbol{p}}_{d} \right\}, \quad{\boldsymbol{p}}_{u} = (t_{bu}, t_{eu}, V_{lu}, V_{hu}, \sigma_{u}),\ \ {\boldsymbol{p}}_{d} = (t_{bd}, t_{ed}, V_{ld}, V_{hd}, \sigma_{d}) \end{align} (4.6) The forms of $$ \Omega_{\,{\boldsymbol{p}}_{u},\ {\boldsymbol{p}}_{d}}$$ and $$ \Theta_{\,\boldsymbol{p}}$$ are depicted in Figs 2 and 3. The speeds Vl* and Vh* are the beginning and reached speeds, respectively; tb* and te* are the beginning and ending times of speed change. The real σ* is a stiffness parameter: the higher σ*, the closer $$\textrm{logCh}_{\sigma _{\ast }}(t)$$ is from |t|. Remark 4.2 One could have chosen a $$\tanh $$-like trajectory for Vxr. The chosen form (which amounts to a combination of primitives of $$\tanh $$) is a smooth (in fact entire) approximation of a trajectory yielding a piecewise constant acceleration. The difference te*− tb* is related to the acceleration, while the stiffness σ is related to the jerk. A $$\tanh $$-like trajectory would furnish only a single design parameter (the stiffness). Fig. 2. View largeDownload slide An example of $$ V_{xr}(t)=\Omega_{\,{\boldsymbol{p}}_{u},\ {\boldsymbol{p}}_{d}}(t) $$ trajectory. Fig. 2. View largeDownload slide An example of $$ V_{xr}(t)=\Omega_{\,{\boldsymbol{p}}_{u},\ {\boldsymbol{p}}_{d}}(t) $$ trajectory. Fig. 3. View largeDownload slide An example of $$ \Theta_{\,\boldsymbol{p}}(t)$$ function. Fig. 3. View largeDownload slide An example of $$ \Theta_{\,\boldsymbol{p}}(t)$$ function. The associated trajectory parameters are: σ = 0.5, tbu = 20s, teu = 35s, tbd = 70s, ted = 85s. 4.2.2. A physical constraint Using Equation (2.1a), we have   $$ \dot V_{x} = g \mu_{x} (\lambda). $$ Since the μx curve is imposed by the tyre/ground physics, we should ensure that $$ \dot{V}_{x}$$ does not exceed the maximum (resp. minimum) of gμx. In other words, the chosen trajectory will be such that the physical constraint   \begin{align} \left|\dot V_{xr}\right| \leqslant g \max\limits_{\lambda \in [-1,1]} (\mu_{x} (\lambda)) \end{align} (4.7)is met, where   $$ \max\limits_{\lambda \in [-1,1]}(\mu_{x} (\lambda)) = \mu_{x} (\lambda^{\ast}) = \mu^{\ast} $$is given by (see Equation (2.3) and below)   $$ \mu^{\ast} = \frac{a}{c + 2 \sqrt{b}}, \quad \textrm{with} \quad \lambda^{\ast} = \sqrt{b}. $$ We shall consider the following   \begin{align} \max\limits_{t \in{\mathbb{R}}} \left|\dot V_{xr} (t)\right| = g\left(\mu^{\ast} - \epsilon_{\mu_{x}}\right) \:{\triangleq}\: g \mu_{M} \end{align} (4.8)where $$\epsilon _{\mu _{x}}$$ is such that $$\epsilon _{\mu _{x}} / \mu ^{\ast } \ll 1$$. This corresponds to   \begin{align} \lambda_{M} = \mu^{-1} (\mu_{M}) = \lambda^{\ast} - \epsilon_{\lambda} \end{align} (4.9)where ϵλ is such that ϵλ/λ*≪ 1. 4.2.3. Trajectory tracking The trajectory tracking of $$ V_{xr}=\Omega_{\,{\boldsymbol{p}}_{u},\ {\boldsymbol{p}}_{d}} $$ is depicted in Figs 4 and 5. Fig. 4. View largeDownload slide Speed trajectory tracking; in blueVx and in Vxr, standard trajectory. Fig. 4. View largeDownload slide Speed trajectory tracking; in blueVx and in Vxr, standard trajectory. Fig. 5. View largeDownload slide Speed trajectory tracking error Vx − Vxr, standard trajectory. Fig. 5. View largeDownload slide Speed trajectory tracking error Vx − Vxr, standard trajectory. The chosen parameters are the following: initial conditions Vx0 = 5 m/s, ω0 = 16.67 rad/s, starting speed Vlu = Vld = 5 m/s, reached speed Vhu = Vhd = 15 m/s. We see on Figs 4 and 5 that the trajectory tracking is achieved with a very good precision, since the maximum error Vx − Vxr in Fig. 5 is 2.055.10−5. The slip ratio λ and the adherence function μ(λ) are plotted in Figs 6 and 7. Remark that this slip ratio λ remains very small (the maximum of λ is 4.613.10−4). The parameters of the function μ(λ) are a = 3.661, b = 0.022, c = 5.153. Fig. 6. View largeDownload slide Slip ratio λ, standard trajectory. Fig. 6. View largeDownload slide Slip ratio λ, standard trajectory. Fig. 7. View largeDownload slide Adherence μx, standard trajectory. Fig. 7. View largeDownload slide Adherence μx, standard trajectory. The control law T and the error T − Tr are depicted in Figs 8 and 9. The chosen feedback gains are Kp = 200, Kd = 10. Finally, the closed-loop torque T is very close to the open-loop torque Tr, as can be seen on Fig. 9: the maximum error (in absolute value) T − Tr is − 1.4.10−6. Fig. 8. View largeDownload slide Closed-loop control T, standard trajectory. Fig. 8. View largeDownload slide Closed-loop control T, standard trajectory. Fig. 9. View largeDownload slide Control error T − Tr, standard trajectory. Fig. 9. View largeDownload slide Control error T − Tr, standard trajectory. 5. Torque constraint management Since the constraints will be expressed in terms of the flat output Vx and its derivatives, we have to compute analytically the first derivatives of Vx. 5.1. Trajectory first derivatives The derivatives of $$ \Omega$$ are the following:   \begin{align} \dot V_{xr} =&\, \dot \Omega_{\,{\boldsymbol{p}}_{u},\ {\boldsymbol{p}}_{d}} (t) = \frac{V_{hu} - V_{lu}}{2 (t_{eu} - t_{bu})}\, \left( \tanh \left( \sigma_{u} (t-t_{bu}) \right) + \tanh \left( -\sigma_{u} (t-t_{eu}) \right) \right)\qquad\ \nonumber\\ &-\frac{V_{hd} - V_{ld}}{2 (t_{ed} - t_{bd})}\, \left( \tanh \left( \sigma_{d} (t-t_{bd}) \right) + \tanh \left( -\sigma_{d} (t-t_{ed}) \right) \right) \end{align} (5.1)  \begin{align} \ddot V_{xr} =&\, \ddot{\Omega}_{\,{\boldsymbol{p}}_{u},\ {\boldsymbol{p}}_{d}} (t) = \frac{\sigma_{u} (V_{hu} - V_{lu})}{2 (t_{eu} - t_{bu})}\, \left( \tanh^{2} \left( -\sigma_{u} (t-t_{eu}) \right) - \tanh^{2} \left( \sigma_{u} (t-t_{bu}) \right) \right) \nonumber\\ &-\frac{\sigma_{d} (V_{hd} - V_{ld})}{2 (t_{ed} - t_{bd})}\, \left( \tanh^{2} \left( \sigma_{d} (t-t_{ed}) \right) - \tanh^{2} \left( -\sigma_{d} (t-t_{bd}) \right) \right) \end{align} (5.2)For the example depicted in Fig. 2, we get the derivatives in Figs 10 and 11. The maximum and minimum of $$\dot{\Omega}_{\,{\boldsymbol{p}}_{u},\ {\boldsymbol{p}}_{d}}$$ and $$\ddot{\Omega}_{\,{\boldsymbol{p}}_{u},\ {\boldsymbol{p}}_{d}}$$ are   \begin{align} \max \left(\dot{\Omega}_{\,{\boldsymbol{p}}_{u},\ {\boldsymbol{p}}_{d}} (t)\right) = \frac{V_{hu} - V_{lu}}{2 (t_{eu} - t_{bu})}, \quad \min \left(\dot{\Omega}_{\,{\boldsymbol{p}}_{u},\ {\boldsymbol{p}}_{d}} (t)\right) = -\frac{V_{hd} - V_{ld}}{2 (t_{ed} - t_{bd})} \end{align} (5.3)  \begin{align} \max \left(\ddot{\Omega}_{\,{\boldsymbol{p}}_{u},\ {\boldsymbol{p}}_{d}} (t)\right) = \max \left(\frac{\sigma_{u} (V_{hu} - V_{lu})}{2 (t_{eu} - t_{bu})}, \frac{\sigma_{d} (V_{hd} - V_{ld})}{2 (t_{ed} - t_{bd})}\right)\qquad\qquad\ \end{align} (5.4)  \begin{align} \min \left(\ddot{\Omega}_{\,{\boldsymbol{p}}_{u},\ {\boldsymbol{p}}_{d}} (t)\right) = -\max \left(\frac{\sigma_{u} (V_{hu} - V_{lu})}{2 (t_{eu} - t_{bu})}, \frac{\sigma_{d} (V_{hd} - V_{ld})}{2 (t_{ed} - t_{bd})}\right).\qquad\ \ \ \end{align} (5.5) Fig. 10. View largeDownload slide The first derivative $$\dot V_{xr} = \dot{\Omega}_{\,{\boldsymbol{p}}_{u},\ {\boldsymbol{p}}_{d}}$$ (acceleration). Fig. 10. View largeDownload slide The first derivative $$\dot V_{xr} = \dot{\Omega}_{\,{\boldsymbol{p}}_{u},\ {\boldsymbol{p}}_{d}}$$ (acceleration). Fig. 11. View largeDownload slide The first derivative $$\ddot V_{xr} = \ddot{\Omega}_{\,{\boldsymbol{p}}_{u},\ {\boldsymbol{p}}_{d}}$$ (jerk). Fig. 11. View largeDownload slide The first derivative $$\ddot V_{xr} = \ddot{\Omega}_{\,{\boldsymbol{p}}_{u},\ {\boldsymbol{p}}_{d}}$$ (jerk). We then set   \begin{align*} \Delta_{V\!u} &= V_{hu} - V_{lu}, \quad \Delta_{V\!d} = V_{hd} - V_{ld}, \quad \Delta_{tu} = t_{eu} - t_{bu}, \quad \Delta_{td} = t_{ed} - t_{bd} \\[-2pt] \dot V_{xm} &= -\frac{\Delta_{V\!d}}{\Delta_{td}}, \quad \dot V_{xM} = \frac{\Delta_{V\!u}}{\Delta_{tu}}, \quad \ddot V_{xm} = -\max \left(\frac{\sigma_{u} \Delta_{V\!u}}{\Delta_{tu}}, \frac{\sigma_{d} \Delta_{V\!d}}{\Delta_{td}}\right), \quad \ddot V_{xM} = - V_{xm}. \end{align*} 5.2. Torque expression and simple bounds We shall give in this subsection various bounds, postponing a discussion about them to Subsection 5.3. 5.2.1. Torque expression amenable to be bounded Recall the expression obtained for the trajectory tracking feedback law in Equation (4.1).   $$ T = \frac{1}{R}\, \left[ \left( mr + \frac{I_{w} \omega}{V_{x}} \right) \dot V_{x} + F_{o} - \frac{{I_{w}}\max \left(r^{2}\omega^{2}, {V_{x}^{2}}\right)}{gr{V_{x}}\mu_{x}^{\prime}}\, \ddot V_{x} \right] $$ Then, we have In the acceleration case, where $$r\omega \geqslant V_{x}$$, $$\lambda \leqslant 0$$  $$ \frac{\omega}{V_{x}} = \frac{1}{r(1+\lambda)}. $$ In the deceleration case, where $$r\omega \leqslant V_{x}$$, $$\lambda \geqslant 0$$  $$ \frac{\omega}{V_{x}} = \frac{1-\lambda}{r}. $$ Thus, the expression for the torque is In the acceleration case   $$ T = \frac{1}{R}\, \left[ \left( mr + \frac{I_{w}}{r(1+\lambda)} \right) \dot V_{x} + F_{o} + \frac{{I_{w}}}{gr\mu_{x}^{\prime} {(1+\lambda)^{2}}}\, {V_{x}} \ddot V_{x} \right]\!. $$ In the deceleration case   $$ T = \frac{1}{R}\, \left[ \left( mr + \frac{I_{w}(1-\lambda)}{r} \right) \dot V_{x} + F_{o} - \frac{{I_{w}}}{gr\mu_{x}^{\prime}}\, {V_{x}} \ddot V_{x} \right]\!. $$ 5.2.2. Generic bound We have the following bounds for |T|: In the acceleration case   \begin{align} |T| \leqslant \frac{1}{R}\, \left[ \left( mr + \frac{I_{w}}{r(1+\lambda)} \right) \dot V_{x} + |F_{o}| + \frac{{I_{w}}}{gr|\mu_{x}^{\prime}| {(1+\lambda)^{2}}}\, {V_{x}} |\ddot V_{x}| \right]\!. \end{align} (5.6) In the deceleration case   \begin{align} |T| \leqslant \frac{1}{R}\, \left[ \left( mr + \frac{I_{w}(1-\lambda)}{r} \right) |\dot V_{x}| + |F_{o}| + \frac{{I_{w}}}{gr\mu_{x}^{\prime}}\, {V_{x}} |\ddot V_{x}| \right]\!. \end{align} (5.7) 5.2.3. A simplistic bound A simplistic bound is given by considering minimum (in denominators) and maximum (in numerators) values for the various expressions in the bounding formulas (5.6) and (5.7). In the acceleration case   \begin{align} |T| \leqslant \frac{1}{R}\, \left[ \left( mr + \frac{I_{w}}{r(1+\lambda_{m})} \right) \dot V_{x\hskip.4pt M} + \frac{{I_{w}}}{gr|\mu_{x\hskip.4ptm}^{\prime}| {(1+\lambda_{m})^{2}}}\, {V_{x\hskip.4ptM}} |\ddot V_{x\hskip.4ptM}| \right]. \end{align} (5.8) In the deceleration case   \begin{align} |T| \leqslant \frac{1}{R}\, \left[ \left( mr + \frac{I_{w}(1-\lambda_{m})}{r} \right) |\dot V_{x\hskip.4pt M}| + \frac{{I_{w}}}{gr\mu_{x\hskip.4pt m}^{\prime}}\, {V_{x\hskip.4pt M}} |\ddot V_{x\hskip.4pt M}| \right] \end{align} (5.9) with the following notations (see, in particular, Equation (4.9))   \begin{align} \dot V_{x\hskip.4pt M} = \max \left(\frac{\Delta_{V\hskip.4pt u}}{\Delta_{t\hskip.4pt u}}, \frac{\Delta_{V\!\hskip.4pt d}}{\Delta_{t\hskip.4pt d}} \right), \quad \ddot V_{xM} = \max \left(\frac{\sigma_{u} \Delta_{V\!u}}{\Delta_{tu}}, \frac{\sigma_{d} \Delta_{V\!d}}{\Delta_{td}}\right) \end{align} (5.10)  \begin{align} \quad \lambda_{m} = -\lambda_{M} = \mu^{-1} (\mu_{M}) = -\lambda^{\ast} + \epsilon_{\lambda}, \quad \mu^{\prime}_{m} = \mu^{\prime} (\lambda^{\ast} - \epsilon_{\mu^{\prime}}). \end{align} (5.11) 5.2.4. A simple but realistic bound We shall then consider the following more realistic bounding function: In the acceleration case   \begin{align} |T| \leqslant \left( \frac{mr}{R} + \frac{I_{w}}{rR(1+\lambda_{m})} \right) \dot V_{x} + \max \left(\frac{{I_{w} V_{xr}} |\ddot V_{xr}|}{grR|\mu_{xr}^{\prime}| {(1+\lambda_{r})^{2}}} \right) = \xi_{aM} \dot V_{x} + \zeta_{aM}. \end{align} (5.12) In the deceleration case   \begin{align} |T| \leqslant \left( \frac{mr}{R} + \frac{I_{w}(1-\lambda_{m})}{rR} \right) |\dot V_{x}| + \max \left( \frac{{I_{w} V_{xr}} |\ddot V_{xr}|}{grR\mu_{xr}^{\prime}} \right) = \xi_{dM} \dot V_{x} + \zeta_{dM}. \end{align} (5.13) 5.3. Discussion and bounds fulfilment 5.3.1. Generic bound The bound given in Equations (5.6) and (5.7) is rather generic, since it contains expressions in λ, yielding expressions in Vx (see, e.g. Equation (3.2)). Thus, it cannot be used very simply. 5.3.2. Simplistic bound The simplistic bound of Equations (5.8) and (5.9) is far too pessimistic. Indeed, e.g. for the trajectory given in Fig. 4, the above bound in the acceleration case is 9696.828 N, when the real maximum on T is 1.948 N. It is thus unusable. 5.3.3. A simple but realistic bound The simple bound given in Equations (5.12) and (5.13) yields a maximum of 1.962 N which is a much better bound than the previous one, wrt the real maximum of 1.948 N. Remark 5.1 Note that the bounding functions (5.12) and (5.13) are valid for any type of reference trajectory and not only the one given in (4.5). Fig. 12. View largeDownload slide Bounds (5.12) and (5.13) on T. Fig. 12. View largeDownload slide Bounds (5.12) and (5.13) on T. Fig. 13. View largeDownload slide Error between T and the bounds (5.12) and (5.13). Fig. 13. View largeDownload slide Error between T and the bounds (5.12) and (5.13). Recall the form of the bounds given in (5.3)   \begin{align*} \dot V_{xm} &= -\frac{\Delta_{V\!d}}{\Delta_{td}}, \quad \dot V_{xM} = \frac{\Delta_{V\!u}}{\Delta_{tu}} \\ \Delta_{V\!u} &= V_{hu} - V_{lu}, \quad \Delta_{V\!d} = V_{hd} - V_{ld}, \quad \Delta_{tu} = t_{eu} - t_{bu}, \quad \Delta_{td} = t_{ed} - t_{bd} \end{align*}and suppose ΔVu and ΔVd are being given by practical considerations (e.g. speed limits). From the bounds obtained in (5.12) and (5.13), we then have In the acceleration case   \begin{align} |T| \leqslant \xi_{aM} \dot V_{x} + \zeta_{aM} \leqslant \xi_{aM} \dot V_{xM} + \zeta_{aM} = \xi_{aM} \frac{\Delta_{V\!u}}{\Delta_{tu}} + \zeta_{aM}. \end{align} (5.14) In the deceleration case   \begin{align} |T| \leqslant \xi_{dM} \dot V_{x} + \zeta_{dM} \leqslant - \xi_{dM} \dot V_{xm} + \zeta_{dM} = \xi_{dM} \frac{\Delta_{V\!d}}{\Delta_{td}} + \zeta_{dM}. \end{align} (5.15) Then, to ensure some prescribed bound on the torque   \begin{align} |T| \leqslant T_{Ma} \ \text{ on acceleration, and }\ |T| \leqslant T_{Md} \ \textrm{ on deceleration} \end{align} (5.16)it is sufficient to impose the following bounds on Δtu, Δtd:   $$ \Delta_{tu}> \frac{\xi_{dM} \Delta_{V\!d}}{T_{Md} - \zeta_{dM}}, \quad \Delta_{ta} > \frac{\xi_{aM} \Delta_{V\!a}}{T_{Ma} - \zeta_{aM}}. $$ In Fig. 12, we have the bounds (5.12) and (5.13) in dashed line ($$\xi _{aM} \dot V_{x} + \zeta _{aM}$$ and $$\xi _{dM} \dot V_{x} + \zeta _{dM}$$) and the torque T in solid line, and in Fig. 13 is depicted the error between the previous two. Note that the maximum error is 1.398.10−2, which is 0.18% of Tr’s maximum. 6. Conclusion We have elaborated a simple yet efficient scheme for tracking a reference speed of a longitudinal vehicle model with torque constraint. The flatness character of the model enabled to embed the constraint fulfilment in the trajectory design. We considered a special class of functions for the class output, namely combinations of $$\log (\cosh (t))$$ type functions. More general classes of functions will be considered in the future, together with some other types of constraints. References Bakker, E., Nyborg, L. & Pacejka, H. B. ( 1987) Tyre modelling for use in vehicle dynamics studies. SAE Technical Paper . SAE International. Bekcheva, M., Greco, L., Mounier, H. & Quadrat, A. ( 2015) Euler-bernoulli beam flatness based control with constraints. Proc. of IEEE 9th International Workshop on Multidimensional (nD) Systems (IEEE nDS 2015) . Vila Real, Portugal. Bekcheva, M., Mounier, H. & Greco, L. ( 2017) Control of differentially at linear delay systems with constraints. Proc. of 20th IFAC World Congress , Toulouse, France. Chamseddine, A., Theilliol, D., Zhang, Y., Join, C. & Rabbath, C.-A. ( 2013) Active fault-tolerant control system design with trajectory re-planning against actuator faults and saturation: application to a quadrotor unmanned aerial vehicle. Int. J. Adapt. Control Signal Process , 29, 1-- 23. doi: https://doi.org/10.1002/acs.2451. Google Scholar CrossRef Search ADS   Ellis, J. R. ( 1969) Vehicle Dynamics . London, United Kingdom: Business Books. Faiz, N., Agrawal, S. K. & Murray, R. M. ( 2001) Trajectory planning of differentially at systems with dynamics and inequalities. J. Guid. Control Dyn. , 24, 219-- 227. doi: https://doi.org/10.2514/2.4732. Google Scholar CrossRef Search ADS   Fliess, M., Levine, J., Martin, P. & Rouchon, P. ( 1995) Flatness and defect of nonlinear systems: introductory theory and examples. Int. J. Control , 61, 1327-- 1361. Google Scholar CrossRef Search ADS   Flores, M. E. & Milam, M. B. ( 2006) Trajectory generation for differentially at systems via NURBS basis functions with obstacle avoidance. 2006 American Control Conference . doi: https://doi.org/10.1109/acc.2006.1657645. Gillespie, T. D. ( 1992) Fundamentals of Vehicle Dynamics . Premiere Series Bks. Warrendale, Pennsylvania, USA: Society of Automotive Engineers. Hsu, L.-Y. & Chen, T.-L. ( 2013) An optimal wheel torque distribution controller for automated vehicle trajectory following. IEEE Trans. Vehicular Technol. , 62, 2430-- 2440. doi: https://doi.org/10.1109/tvt.2013.2246593. Google Scholar CrossRef Search ADS   Hsu, L.-Y., Weng, K. & Chen, T.-L. ( 2010) A constrained wheel torque controller for lane following system using control distribution. Proceedings of the 2010 American Control Conference . doi: https://doi.org/10.1109/acc.2010.5530720. Kahveci, N. E. & Ioannou, P. A. ( 2010) Cruise control with adaptation and wheel torque constraints for improved fuel economy. 2010 IEEE Intelligent Vehicles Symposium . doi: https://doi.org/10.1109/ivs.2010.5547986. Keck, A., Knierim, K. L. & Sawodny, O. ( 2015) SAMMY - an algorithm for efficient computation of a smooth path for reference trajectory generation. 2015 6th International Conference on Automation, Robotics and Applications (ICARA) . doi: https://doi.org/10.1109/icara.2015.7081133. Kiencke, U. & Daiss, A. ( 1994) Estimation of tyre friction for enhaced ABS- systems. Proceedings of the International Symposium on Advanced Vehicle Control . Tsukuba-shi, Japan, pp. 515-- 520. Kiencke, U. & Nielsen, L. ( 2000) Automotive Control Systems: For Engine, Driveline and Vehicle, 1st edn . Secaucus, NJ, USA: Springer, Inc. Li, S., Li, K., Rajamani, R. & Wang, J. ( 2011) Model predictive multi-objective vehicular adaptive cruise control. IEEE Trans. Control Syst. 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( 2011) Optimal drive of electric vehicles using an inversion-based trajectory generation approach. IFAC Proc. Vol. , 44, 14519-- 14526. doi: https://doi.org/10.3182/20110828–6-it-1002.01986. Google Scholar CrossRef Search ADS   Rajamani, R. ( 2011) Vehicle Dynamics and Control . Mechanical Engineering Series. US: Springer. Ross, M. & Fahroo, F. ( 2004) Pseudospectral methods for optimal motion planning of differentially at systems. IEEE Trans. Automat. Contr. , 49, 1410-- 1413. doi: https://doi.org/10.1109/tac.2004.832972. Google Scholar CrossRef Search ADS   Ruppel, T., Knierim, K. L. & Sawodny, O. ( 2011) Analytical multi-point trajectory generation for differentially flat systems with output constraints. IFAC Proc. Vol. , 44, 950-- 955. doi: https://doi.org/10.3182/20110828–6-it-1002.00527. Tarbouriech, S. & Turner, M. ( 2009) Anti-windup design: an overview of some recent advances and open problems. 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( 2016) Optimal trajectory generation for car-type mobile robot using spline interpolation. FAC-PapersOnLine , 49, 601-- 606. doi: https://doi.org/10.1016/j.ifacol.2016.03.121. Google Scholar CrossRef Search ADS   © The Author(s) 2018. Published by Oxford University Press on behalf of the Institute of Mathematics and its Applications. All rights reserved. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png IMA Journal of Mathematical Control and Information Oxford University Press

Flatness-based longitudinal vehicle control with embedded torque constraint

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Abstract

Abstract This paper aims at establishing a simple yet efficient solution to the problem of trajectory tracking with input constraint of a non-linear longitudinal vehicle model. We make use of differential flatness by embedding the constraint into the reference trajectory design. 1. Introduction The aim of this paper is to come up with simple yet efficient control techniques for vehicle longitudinal speed control with constraint on its torque. More precisely, we consider a longitudinal non-linear model including a simple adherence/friction law (see, e.g. Ellis, 1969; Gillespie, 1992; Kiencke & Nielsen, 2000; Mitschke & Wallentowitz, 2004; Rajamani, 2011). For this model, we consider the problem of tracking a reference speed trajectory with constraint on the input torque. Traditional treatment of such a problem includes model predictive control (Li et al., 2011) and use of optimization techniques (Hsu & Chen, 2013; Hsu et al., 2010). Some other works use adaptive anti-windup techniques (Kahveci & Ioannou, 2010; Tarbouriech & Turner, 2009), saturated inputs (Valmorbida et al., 2013), to name a few. The constraint is embedded in the flat output trajectory design. Thus, the closed-loop tracking controller naturally satisfies the required constraint, without the recourse to costly optimization procedure. A key advantage of the advocated technique is that the physical meaning is kept throughout the whole process, a feature often lost in MPC (Model Predictive Control) or other optimization-based techniques. More precisely, a dynamical system with m inputs is differentially flat (Fliess et al., 1995) if there exists a so-called flat output ω with m components ω = (ω1, … , ωm) such that: first, these components are functions of the system’s variables (endogenous character); second, the ωis are differentially independent, i.e. they do not satisfy a differential equation involving themselves only (independent character); third, all the system’s variables can be expressed as non-linear functions of the ωis and of a finite number of their derivatives (parametrization property). Thus, when a system variable is subject to a constraint, the latter is directly translated into a flat output constraint, thanks to the parametrization property. The tracking problem with constraints is thus elaborated in two steps: first, design a flat output reference trajectory ωr satisfying all the required constraints; second, design a closed-loop feedback control law ensuring the tracking of ωr with stability. The constraints satisfaction is ensured by design, since it is embedded in the reference trajectory elaboration process. The involved constraints can be given on any system variable, since all of the system is parametrized by the flat output. The constraint is enforced on the reference variables and is ensured practically on the actual variables since the tracking error is meant to tend to zero in general exponentially. Ensuring the constraints on the flat output is simplified by specializing the flat output reference trajectory to specific classes of functions with convenient properties, such as closedness wrt differentiation or being a solution of a differential equation. To the best of the author’s knowledge, almost all the current works on differentially flat systems with constraints are managed through optimization procedures (Chamseddine et al., 2013; Faiz et al., 2001; Flores & Milam, 2006; Keck et al., 2015; Petit & Sciarretta, 2011; Ross & Fahroo, 2004; Tsuei & Milam, 2016; Walambe et al., 2016). In the study by Löwis & Rudolph (2003), no optimization technique is used, but the flat output trajectory is not known in advance; thus, the trajectory is built step by step, by concatenating pieces. The only work partially related to our approach is the study by Ruppel et al. (2011), where the constraints appear solely on derivatives of the flat output, which is specialized to piecewise polynomial functions. Preliminary results related to the present one have been presented for linear systems with delays in the study by Bekcheva et al. (2017) and for an Euler–Bernoulli beam in the study by Bekcheva et al. (2015). Other works related to the present theme include differential flatness-based techniques for longitudinal and lateral vehicle dynamics (Menhour et al., 2014). The paper is organized as follows. In the next section, the model is recalled. In Section 3, the flatness of the model is established, and a closed-loop feedback tracking controller is given in Section 4. The torque constraint management is dealt with in Section 5. 2. Longitudinal model The equations of the vehicle dynamics can be written as follows (see, e.g. Ellis, 1969; Gillespie, 1992; Kiencke & Nielsen, 2000; Mitschke & Wallentowitz, 2004; Rajamani, 2011):   \begin{align} m\dot{V}_{x} = F_{x} \qquad\quad \end{align} (2.1a)  \begin{align} \qquad\ I_{w}\dot{\omega} = RT-r F_{x} - F_{o} \end{align} (2.1b)with the following slip ratio and forces:   \begin{align} F_{x} = \mu_{x} (\lambda) F_{z}, \quad \lambda = \frac{V_{x}-r{\omega}}{\max(V_{x}, r{\omega})} \end{align} (2.2a)  \begin{align} F_{z} = mg\qquad\qquad\qquad\qquad\qquad\ \ \end{align} (2.2b)  \begin{align} F_{o} = - F_{a} - F_{s} \qquad\qquad\qquad\qquad\! \end{align} (2.2c)  \begin{align} \qquad\qquad\quad\ R_{x} = mg C_{r}, \quad F_{a} = \frac{\rho\, C_{a}A{V_{x}^{2}}}{2}, \quad F_{s} = m g \sin \alpha. \end{align} (2.2d) The notations for the models (2.1a) and (2.1b) are as follows: Vx is the longitudinal speed of the vehicle, m its mass, Fx the longitudinal tyre force, Iw the inertia moment of the wheel, ω the angular wheel speed, R the damping coefficient of the driveline, T the engine torque, r the effective tyre radius, Fo the other forces exerted on the car body. The expressions of the forces are given in Equations (2.2a)–(2.2d), with the following notations: μx is the adherence function, Fz the normal force on the tyre, λ the slip ratio, g the gravity constant, Rx the rolling resistance force, Fa the longitudinal aerodynamic drag force, ρ is the air volumic mass, A is frontal area of the vehicle, Ca is the drag coefficient, Fs the force due to the road slope and α the road slope angle. A possible model for μx(λ) introduced by Kiencke & Daiss, (1994) and depicted in Fig. 1 is given by the function  \begin{align} \mu_{x} (\lambda) = \frac{a \lambda}{b + c |\lambda| + \lambda^{2}}. \end{align} (2.3) One easily obtains that the maximum μ* of such a curve occurs at λ* with   $$ \lambda^{\ast} = \sqrt{b}, \quad \mu^{\ast} = \frac{a}{c + 2 \sqrt{b}}. $$ Conversely, the constants a, b, c can be expressed as functions of μ*, λ* and μ1  $$ a = \frac{\mu^{\ast} \mu_{1} \left(1-\lambda^{\ast}\right)^{2}}{\mu^{\ast} - \mu_{1}}, \quad b ={\lambda^{\ast}}^{2}, \quad c = \frac{\mu_{1} \left(1+{\lambda^{\ast}}^{2}\right) - 2 \mu^{\ast} \lambda^{\ast}}{\mu^{\ast} - \mu_{1}} $$where μ1 = μ(1) is the value of the function μ at λ = 1 (i.e. at wheel lock). Note that a and b are strictly positive constants. Fig. 1. View largeDownload slide Adherence function μ(λ). Fig. 1. View largeDownload slide Adherence function μ(λ). Remark 2.1 Another, quite popular, model is the Pacejka one (Bakker et al., 1987; Pacejka, 2006). We have not used the latter for simplicity reasons, but a similar, although more complex, analysis could be made with Pacejka’s model. The measured outputs are traditionally the wheel speed (e.g. through ABS (Anti Blocking System) encoders). We shall here suppose that the speed Vx of the vehicle’s centre of gravity is either measured or reconstructed via an observer or an estimator (see, e.g. the study by Villagra et al., 2008, a previous work of some author of the present paper). 3. Differential flatness of the model The model (2.1) is trivially flat, with flat output Vx. Indeed,   \begin{align} \dot{V}_{x} = g \mu_{x} (\lambda). \end{align} (3.1) Then,   \begin{align} \lambda = \mu^{-1} \left( \frac{\dot{V}_{x}}{g} \right)\hskip-2pt. \end{align} (3.2) Now one has to distinguish two acceleration and deceleration cases (implied by the form of λ in (2.2a)). Acceleration case, where $$r\omega \geqslant V_{x}$$  $$ \lambda = \frac{V_{x}}{r\omega} - 1 = \mu^{-1} \left( \frac{\dot V_{x}}{g}\right)\hskip-2pt. $$ Hence   \begin{align} \omega = \frac{V_{x}}{r \left[ 1 + \mu^{-1} \left( \frac{\dot V_{x}}{g}\right)\right]}. \end{align} (3.3) And thus   $$ \ddot{V}_{x} = g \mu_{x}^{\prime} (\lambda) \dot \lambda = g \mu_{x}^{\prime} (\lambda)\frac{1}{r\omega^{2}} \left( {\omega} \dot V_{x} - \dot{\omega} V_{x}\right)\hskip-2pt. $$ Deceleration case, where $$r\omega \leqslant V_{x}$$  $$ \lambda = 1 - \frac{r\omega}{V_{x}} = \mu^{-1} \left( \frac{\dot V_{x}}{g}\right)\hskip-2pt. $$ Hence   \begin{align} \omega = \frac{V_{x}}{r}\, \left[ 1 - \mu^{-1} \left( \frac{\dot V_{x}}{g}\right)\right]\hskip-2pt. \end{align} (3.4) And thus   $$ \ddot{V}_{x} = g \mu_{x}^{\prime} (\lambda) \dot \lambda = g \mu_{x}^{\prime} (\lambda)\frac{r}{{V_{x}^{2}}} \left({\omega} \dot V_{x} - \dot{\omega} V_{x}\right)\hskip-2pt. $$ Thus, one has the following dynamics in Vx:   \begin{align} \ddot V_{x} = \frac{g\mu_{x}^{\prime}}{\max \left(r\omega^{2}, \frac{{V_{x}^{2}}}{r}\right)} \left[ \omega \dot V_{x} + \frac{V_{x}}{I_{w}}\, \left( mr\dot V_{x} + F_{o} + RT \right) \right] \end{align} (3.5)and the control input T is then obtained as   \begin{align} T = \frac{1}{R}\, \left[ \left( mr + \frac{I_{w} \omega}{V_{x}} \right) \dot V_{x} + F_{o} - \frac{{I_{w}}\max \left(r^{2}\omega^{2}, {V_{x}^{2}}\right)}{gr{V_{x}}\mu_{x}^{\prime}}\, \ddot V_{x} \right]\hskip-2pt. \end{align} (3.6) Remark 3.1 The reader could have the (quite normal) feeling that the laws (3.3) and (3.4) yield a discontinuity when the vehicle switches from acceleration to deceleration (leading to a chattering-like phenomenon). First note that this can only occur at extremely low slip, i.e. when rω − Vx ≪ 1, where the μ() curve is in the linear zone (and thus the μ−1 also); thus   $$ \mu^{-1} \left( \frac{\dot V_{x}}{g}\right) \approx \beta \frac{\dot V_{x}}{g}. $$ Moreover, when the vehicle switches from acceleration to deceleration (or vice versa), one has $$|\dot V_{x} | \ll 1$$. Thus, in (3.3), one has   $$ \frac{1}{1 + \mu^{-1} \left( \frac{\dot V_{x}}{g}\right)} = 1 - \mu^{-1} \left( \frac{\dot V_{x}}{g}\right) + o \left( \left( \frac{\dot V_{x}}{g}\right)^{\!\!2} \right)\hskip-2pt. $$ Thus, the expression of ω is   $$ \omega = \frac{V_{x}}{r \left[ 1 + \mu^{-1} \left( \frac{\dot V_{x}}{g}\right)\right]} = \frac{V_{x}}{r} \left[ 1 - \mu^{-1} \left( \frac{\dot V_{x}}{g}\right) + o \left( \left( \frac{\dot V_{x}}{g}\right)^{\!\!2} \right) \right] $$whose term in $$o\left((V_{x}/g)^{2}\right)$$ is exactly the one of (3.4). Thus, in case of acceleration–deceleration switching, the expression of ω is continuous and differentiable. 4. Trajectory tracking 4.1. Trajectory tracking control law Recalling the flat output dynamics (3.5) and setting the right member equal to a new input v, one obtains the linearizing feedback   $$ \omega \dot V_{x} + \frac{V_{x}}{I_{w}}\, \left( mr\dot V_{x} + F_{o} + RT \right) = \frac{\max \left(r^{2}\omega^{2}, {V_{x}^{2}}\right)}{gr\mu_{x}^{\prime}}\, v $$transforming the flat output dynamics (3.5) to   $$ \ddot V_{x} = v. $$ Setting the new input v to   $$ v = \ddot V_{xr} - K_{p} e_{V_{x}} - K_{d} \dot e_{V_{x}}, \quad e_{V_{x}} = V_{x} - V_{xr} $$with Kp, Kd > 0 yields an exponentially stable error dynamics. The original input is then obtained as   \begin{align} T = \frac{1}{R}\, \left[ \left( mr + \frac{I_{w} \omega}{V_{x}} \right) \dot V_{x} + F_{o} - \frac{{I_{w}}\max \left(r^{2}\omega^{2}, {V_{x}^{2}}\right)}{gr{V_{x}}\mu_{x}^{\prime}}\, v \right] \end{align} (4.1)  \begin{align} v = \ddot V_{xr} - K_{p} e_{V_{x}} - K_{d} \dot e_{V_{x}}.\qquad\qquad\qquad\qquad\qquad\qquad\! \end{align} (4.2) Remark 4.1 Note that, in (4.2), one could have used equally a second order sliding mode or a model free control law, for instance, in order to gain in robustness. 4.1.1. Open- and closed-loop tracking Let Vxr be a reference trajectory for the flat output Vx. Denoting by Tr, the following open-loop control law, one has by direct substitution from (3.6):   \begin{align} T_{r} = \frac{1}{R}\, \left[ \left( mr + \frac{I_{w} \omega_{r}}{V_{xr}} \right) \dot V_{xr} + F_{o} - \frac{{I_{w}}\max \left(r^{2}{\omega_{r}^{2}}, V_{xr}^{2}\right)}{gr{V_{x}}\mu_{xr}^{\prime}}\, \ddot V_{xr} \right] \end{align} (4.3) Thus, the Equations (4.1) and (4.2) can be rewritten as   \begin{align} T = T_{r} - \frac{1}{R}\, \left( K_{p} e_{V_{x}} + K_{d} \dot e_{V_{x}} \right)\hskip-2pt. \end{align} (4.4) We thus see that, if the error $$e_{V_{x}}$$ and its derivative $$\dot e_{V_{x}}$$ remain small (which is the case when the tracking performance is good), the closed-loop torque T remains close to the open loop one Tr. 4.2. Trajectory tracking scenario 4.2.1. Trajectory form We shall choose a trajectory Vxr(t) of the following form:   \begin{align} V_{xr} (t) = \Omega_{\,{\boldsymbol{p}}_{u},\ {\boldsymbol{p}}_{d}} (t) = \Theta_{\,{\boldsymbol{p}}_{u}} (t) - \Theta_{\,{\boldsymbol{p}}_{d}} (t)\qquad\qquad\qquad\qquad\qquad\qquad\ \end{align} (4.5)  \begin{align} \Theta_{\,{\boldsymbol{p\ast}}} (t) &= \frac{V_{h \ast} - V_{l\ast}}{2 \left(t_{e\ast} - t_{b\ast}\right)}\, \left(\textrm{logCh}_{\sigma_{\ast}} \!(t-t_{b\ast}) + \textrm{logCh}_{-\sigma_{\ast}} \!(t-t_{e\ast}) \right) + \frac{V_{h\ast} - V_{l\ast}}{2}\\ \textrm{logCh}_{\sigma} (t) &= \frac{1}{\sigma}\, \log \left(\cosh (\sigma t)\right) \nonumber \\ \nonumber{\boldsymbol{p}}_{\ast} &\in \left\{{\boldsymbol{p}}_{u}, {\boldsymbol{p}}_{d} \right\}, \quad{\boldsymbol{p}}_{u} = (t_{bu}, t_{eu}, V_{lu}, V_{hu}, \sigma_{u}),\ \ {\boldsymbol{p}}_{d} = (t_{bd}, t_{ed}, V_{ld}, V_{hd}, \sigma_{d}) \end{align} (4.6) The forms of $$ \Omega_{\,{\boldsymbol{p}}_{u},\ {\boldsymbol{p}}_{d}}$$ and $$ \Theta_{\,\boldsymbol{p}}$$ are depicted in Figs 2 and 3. The speeds Vl* and Vh* are the beginning and reached speeds, respectively; tb* and te* are the beginning and ending times of speed change. The real σ* is a stiffness parameter: the higher σ*, the closer $$\textrm{logCh}_{\sigma _{\ast }}(t)$$ is from |t|. Remark 4.2 One could have chosen a $$\tanh $$-like trajectory for Vxr. The chosen form (which amounts to a combination of primitives of $$\tanh $$) is a smooth (in fact entire) approximation of a trajectory yielding a piecewise constant acceleration. The difference te*− tb* is related to the acceleration, while the stiffness σ is related to the jerk. A $$\tanh $$-like trajectory would furnish only a single design parameter (the stiffness). Fig. 2. View largeDownload slide An example of $$ V_{xr}(t)=\Omega_{\,{\boldsymbol{p}}_{u},\ {\boldsymbol{p}}_{d}}(t) $$ trajectory. Fig. 2. View largeDownload slide An example of $$ V_{xr}(t)=\Omega_{\,{\boldsymbol{p}}_{u},\ {\boldsymbol{p}}_{d}}(t) $$ trajectory. Fig. 3. View largeDownload slide An example of $$ \Theta_{\,\boldsymbol{p}}(t)$$ function. Fig. 3. View largeDownload slide An example of $$ \Theta_{\,\boldsymbol{p}}(t)$$ function. The associated trajectory parameters are: σ = 0.5, tbu = 20s, teu = 35s, tbd = 70s, ted = 85s. 4.2.2. A physical constraint Using Equation (2.1a), we have   $$ \dot V_{x} = g \mu_{x} (\lambda). $$ Since the μx curve is imposed by the tyre/ground physics, we should ensure that $$ \dot{V}_{x}$$ does not exceed the maximum (resp. minimum) of gμx. In other words, the chosen trajectory will be such that the physical constraint   \begin{align} \left|\dot V_{xr}\right| \leqslant g \max\limits_{\lambda \in [-1,1]} (\mu_{x} (\lambda)) \end{align} (4.7)is met, where   $$ \max\limits_{\lambda \in [-1,1]}(\mu_{x} (\lambda)) = \mu_{x} (\lambda^{\ast}) = \mu^{\ast} $$is given by (see Equation (2.3) and below)   $$ \mu^{\ast} = \frac{a}{c + 2 \sqrt{b}}, \quad \textrm{with} \quad \lambda^{\ast} = \sqrt{b}. $$ We shall consider the following   \begin{align} \max\limits_{t \in{\mathbb{R}}} \left|\dot V_{xr} (t)\right| = g\left(\mu^{\ast} - \epsilon_{\mu_{x}}\right) \:{\triangleq}\: g \mu_{M} \end{align} (4.8)where $$\epsilon _{\mu _{x}}$$ is such that $$\epsilon _{\mu _{x}} / \mu ^{\ast } \ll 1$$. This corresponds to   \begin{align} \lambda_{M} = \mu^{-1} (\mu_{M}) = \lambda^{\ast} - \epsilon_{\lambda} \end{align} (4.9)where ϵλ is such that ϵλ/λ*≪ 1. 4.2.3. Trajectory tracking The trajectory tracking of $$ V_{xr}=\Omega_{\,{\boldsymbol{p}}_{u},\ {\boldsymbol{p}}_{d}} $$ is depicted in Figs 4 and 5. Fig. 4. View largeDownload slide Speed trajectory tracking; in blueVx and in Vxr, standard trajectory. Fig. 4. View largeDownload slide Speed trajectory tracking; in blueVx and in Vxr, standard trajectory. Fig. 5. View largeDownload slide Speed trajectory tracking error Vx − Vxr, standard trajectory. Fig. 5. View largeDownload slide Speed trajectory tracking error Vx − Vxr, standard trajectory. The chosen parameters are the following: initial conditions Vx0 = 5 m/s, ω0 = 16.67 rad/s, starting speed Vlu = Vld = 5 m/s, reached speed Vhu = Vhd = 15 m/s. We see on Figs 4 and 5 that the trajectory tracking is achieved with a very good precision, since the maximum error Vx − Vxr in Fig. 5 is 2.055.10−5. The slip ratio λ and the adherence function μ(λ) are plotted in Figs 6 and 7. Remark that this slip ratio λ remains very small (the maximum of λ is 4.613.10−4). The parameters of the function μ(λ) are a = 3.661, b = 0.022, c = 5.153. Fig. 6. View largeDownload slide Slip ratio λ, standard trajectory. Fig. 6. View largeDownload slide Slip ratio λ, standard trajectory. Fig. 7. View largeDownload slide Adherence μx, standard trajectory. Fig. 7. View largeDownload slide Adherence μx, standard trajectory. The control law T and the error T − Tr are depicted in Figs 8 and 9. The chosen feedback gains are Kp = 200, Kd = 10. Finally, the closed-loop torque T is very close to the open-loop torque Tr, as can be seen on Fig. 9: the maximum error (in absolute value) T − Tr is − 1.4.10−6. Fig. 8. View largeDownload slide Closed-loop control T, standard trajectory. Fig. 8. View largeDownload slide Closed-loop control T, standard trajectory. Fig. 9. View largeDownload slide Control error T − Tr, standard trajectory. Fig. 9. View largeDownload slide Control error T − Tr, standard trajectory. 5. Torque constraint management Since the constraints will be expressed in terms of the flat output Vx and its derivatives, we have to compute analytically the first derivatives of Vx. 5.1. Trajectory first derivatives The derivatives of $$ \Omega$$ are the following:   \begin{align} \dot V_{xr} =&\, \dot \Omega_{\,{\boldsymbol{p}}_{u},\ {\boldsymbol{p}}_{d}} (t) = \frac{V_{hu} - V_{lu}}{2 (t_{eu} - t_{bu})}\, \left( \tanh \left( \sigma_{u} (t-t_{bu}) \right) + \tanh \left( -\sigma_{u} (t-t_{eu}) \right) \right)\qquad\ \nonumber\\ &-\frac{V_{hd} - V_{ld}}{2 (t_{ed} - t_{bd})}\, \left( \tanh \left( \sigma_{d} (t-t_{bd}) \right) + \tanh \left( -\sigma_{d} (t-t_{ed}) \right) \right) \end{align} (5.1)  \begin{align} \ddot V_{xr} =&\, \ddot{\Omega}_{\,{\boldsymbol{p}}_{u},\ {\boldsymbol{p}}_{d}} (t) = \frac{\sigma_{u} (V_{hu} - V_{lu})}{2 (t_{eu} - t_{bu})}\, \left( \tanh^{2} \left( -\sigma_{u} (t-t_{eu}) \right) - \tanh^{2} \left( \sigma_{u} (t-t_{bu}) \right) \right) \nonumber\\ &-\frac{\sigma_{d} (V_{hd} - V_{ld})}{2 (t_{ed} - t_{bd})}\, \left( \tanh^{2} \left( \sigma_{d} (t-t_{ed}) \right) - \tanh^{2} \left( -\sigma_{d} (t-t_{bd}) \right) \right) \end{align} (5.2)For the example depicted in Fig. 2, we get the derivatives in Figs 10 and 11. The maximum and minimum of $$\dot{\Omega}_{\,{\boldsymbol{p}}_{u},\ {\boldsymbol{p}}_{d}}$$ and $$\ddot{\Omega}_{\,{\boldsymbol{p}}_{u},\ {\boldsymbol{p}}_{d}}$$ are   \begin{align} \max \left(\dot{\Omega}_{\,{\boldsymbol{p}}_{u},\ {\boldsymbol{p}}_{d}} (t)\right) = \frac{V_{hu} - V_{lu}}{2 (t_{eu} - t_{bu})}, \quad \min \left(\dot{\Omega}_{\,{\boldsymbol{p}}_{u},\ {\boldsymbol{p}}_{d}} (t)\right) = -\frac{V_{hd} - V_{ld}}{2 (t_{ed} - t_{bd})} \end{align} (5.3)  \begin{align} \max \left(\ddot{\Omega}_{\,{\boldsymbol{p}}_{u},\ {\boldsymbol{p}}_{d}} (t)\right) = \max \left(\frac{\sigma_{u} (V_{hu} - V_{lu})}{2 (t_{eu} - t_{bu})}, \frac{\sigma_{d} (V_{hd} - V_{ld})}{2 (t_{ed} - t_{bd})}\right)\qquad\qquad\ \end{align} (5.4)  \begin{align} \min \left(\ddot{\Omega}_{\,{\boldsymbol{p}}_{u},\ {\boldsymbol{p}}_{d}} (t)\right) = -\max \left(\frac{\sigma_{u} (V_{hu} - V_{lu})}{2 (t_{eu} - t_{bu})}, \frac{\sigma_{d} (V_{hd} - V_{ld})}{2 (t_{ed} - t_{bd})}\right).\qquad\ \ \ \end{align} (5.5) Fig. 10. View largeDownload slide The first derivative $$\dot V_{xr} = \dot{\Omega}_{\,{\boldsymbol{p}}_{u},\ {\boldsymbol{p}}_{d}}$$ (acceleration). Fig. 10. View largeDownload slide The first derivative $$\dot V_{xr} = \dot{\Omega}_{\,{\boldsymbol{p}}_{u},\ {\boldsymbol{p}}_{d}}$$ (acceleration). Fig. 11. View largeDownload slide The first derivative $$\ddot V_{xr} = \ddot{\Omega}_{\,{\boldsymbol{p}}_{u},\ {\boldsymbol{p}}_{d}}$$ (jerk). Fig. 11. View largeDownload slide The first derivative $$\ddot V_{xr} = \ddot{\Omega}_{\,{\boldsymbol{p}}_{u},\ {\boldsymbol{p}}_{d}}$$ (jerk). We then set   \begin{align*} \Delta_{V\!u} &= V_{hu} - V_{lu}, \quad \Delta_{V\!d} = V_{hd} - V_{ld}, \quad \Delta_{tu} = t_{eu} - t_{bu}, \quad \Delta_{td} = t_{ed} - t_{bd} \\[-2pt] \dot V_{xm} &= -\frac{\Delta_{V\!d}}{\Delta_{td}}, \quad \dot V_{xM} = \frac{\Delta_{V\!u}}{\Delta_{tu}}, \quad \ddot V_{xm} = -\max \left(\frac{\sigma_{u} \Delta_{V\!u}}{\Delta_{tu}}, \frac{\sigma_{d} \Delta_{V\!d}}{\Delta_{td}}\right), \quad \ddot V_{xM} = - V_{xm}. \end{align*} 5.2. Torque expression and simple bounds We shall give in this subsection various bounds, postponing a discussion about them to Subsection 5.3. 5.2.1. Torque expression amenable to be bounded Recall the expression obtained for the trajectory tracking feedback law in Equation (4.1).   $$ T = \frac{1}{R}\, \left[ \left( mr + \frac{I_{w} \omega}{V_{x}} \right) \dot V_{x} + F_{o} - \frac{{I_{w}}\max \left(r^{2}\omega^{2}, {V_{x}^{2}}\right)}{gr{V_{x}}\mu_{x}^{\prime}}\, \ddot V_{x} \right] $$ Then, we have In the acceleration case, where $$r\omega \geqslant V_{x}$$, $$\lambda \leqslant 0$$  $$ \frac{\omega}{V_{x}} = \frac{1}{r(1+\lambda)}. $$ In the deceleration case, where $$r\omega \leqslant V_{x}$$, $$\lambda \geqslant 0$$  $$ \frac{\omega}{V_{x}} = \frac{1-\lambda}{r}. $$ Thus, the expression for the torque is In the acceleration case   $$ T = \frac{1}{R}\, \left[ \left( mr + \frac{I_{w}}{r(1+\lambda)} \right) \dot V_{x} + F_{o} + \frac{{I_{w}}}{gr\mu_{x}^{\prime} {(1+\lambda)^{2}}}\, {V_{x}} \ddot V_{x} \right]\!. $$ In the deceleration case   $$ T = \frac{1}{R}\, \left[ \left( mr + \frac{I_{w}(1-\lambda)}{r} \right) \dot V_{x} + F_{o} - \frac{{I_{w}}}{gr\mu_{x}^{\prime}}\, {V_{x}} \ddot V_{x} \right]\!. $$ 5.2.2. Generic bound We have the following bounds for |T|: In the acceleration case   \begin{align} |T| \leqslant \frac{1}{R}\, \left[ \left( mr + \frac{I_{w}}{r(1+\lambda)} \right) \dot V_{x} + |F_{o}| + \frac{{I_{w}}}{gr|\mu_{x}^{\prime}| {(1+\lambda)^{2}}}\, {V_{x}} |\ddot V_{x}| \right]\!. \end{align} (5.6) In the deceleration case   \begin{align} |T| \leqslant \frac{1}{R}\, \left[ \left( mr + \frac{I_{w}(1-\lambda)}{r} \right) |\dot V_{x}| + |F_{o}| + \frac{{I_{w}}}{gr\mu_{x}^{\prime}}\, {V_{x}} |\ddot V_{x}| \right]\!. \end{align} (5.7) 5.2.3. A simplistic bound A simplistic bound is given by considering minimum (in denominators) and maximum (in numerators) values for the various expressions in the bounding formulas (5.6) and (5.7). In the acceleration case   \begin{align} |T| \leqslant \frac{1}{R}\, \left[ \left( mr + \frac{I_{w}}{r(1+\lambda_{m})} \right) \dot V_{x\hskip.4pt M} + \frac{{I_{w}}}{gr|\mu_{x\hskip.4ptm}^{\prime}| {(1+\lambda_{m})^{2}}}\, {V_{x\hskip.4ptM}} |\ddot V_{x\hskip.4ptM}| \right]. \end{align} (5.8) In the deceleration case   \begin{align} |T| \leqslant \frac{1}{R}\, \left[ \left( mr + \frac{I_{w}(1-\lambda_{m})}{r} \right) |\dot V_{x\hskip.4pt M}| + \frac{{I_{w}}}{gr\mu_{x\hskip.4pt m}^{\prime}}\, {V_{x\hskip.4pt M}} |\ddot V_{x\hskip.4pt M}| \right] \end{align} (5.9) with the following notations (see, in particular, Equation (4.9))   \begin{align} \dot V_{x\hskip.4pt M} = \max \left(\frac{\Delta_{V\hskip.4pt u}}{\Delta_{t\hskip.4pt u}}, \frac{\Delta_{V\!\hskip.4pt d}}{\Delta_{t\hskip.4pt d}} \right), \quad \ddot V_{xM} = \max \left(\frac{\sigma_{u} \Delta_{V\!u}}{\Delta_{tu}}, \frac{\sigma_{d} \Delta_{V\!d}}{\Delta_{td}}\right) \end{align} (5.10)  \begin{align} \quad \lambda_{m} = -\lambda_{M} = \mu^{-1} (\mu_{M}) = -\lambda^{\ast} + \epsilon_{\lambda}, \quad \mu^{\prime}_{m} = \mu^{\prime} (\lambda^{\ast} - \epsilon_{\mu^{\prime}}). \end{align} (5.11) 5.2.4. A simple but realistic bound We shall then consider the following more realistic bounding function: In the acceleration case   \begin{align} |T| \leqslant \left( \frac{mr}{R} + \frac{I_{w}}{rR(1+\lambda_{m})} \right) \dot V_{x} + \max \left(\frac{{I_{w} V_{xr}} |\ddot V_{xr}|}{grR|\mu_{xr}^{\prime}| {(1+\lambda_{r})^{2}}} \right) = \xi_{aM} \dot V_{x} + \zeta_{aM}. \end{align} (5.12) In the deceleration case   \begin{align} |T| \leqslant \left( \frac{mr}{R} + \frac{I_{w}(1-\lambda_{m})}{rR} \right) |\dot V_{x}| + \max \left( \frac{{I_{w} V_{xr}} |\ddot V_{xr}|}{grR\mu_{xr}^{\prime}} \right) = \xi_{dM} \dot V_{x} + \zeta_{dM}. \end{align} (5.13) 5.3. Discussion and bounds fulfilment 5.3.1. Generic bound The bound given in Equations (5.6) and (5.7) is rather generic, since it contains expressions in λ, yielding expressions in Vx (see, e.g. Equation (3.2)). Thus, it cannot be used very simply. 5.3.2. Simplistic bound The simplistic bound of Equations (5.8) and (5.9) is far too pessimistic. Indeed, e.g. for the trajectory given in Fig. 4, the above bound in the acceleration case is 9696.828 N, when the real maximum on T is 1.948 N. It is thus unusable. 5.3.3. A simple but realistic bound The simple bound given in Equations (5.12) and (5.13) yields a maximum of 1.962 N which is a much better bound than the previous one, wrt the real maximum of 1.948 N. Remark 5.1 Note that the bounding functions (5.12) and (5.13) are valid for any type of reference trajectory and not only the one given in (4.5). Fig. 12. View largeDownload slide Bounds (5.12) and (5.13) on T. Fig. 12. View largeDownload slide Bounds (5.12) and (5.13) on T. Fig. 13. View largeDownload slide Error between T and the bounds (5.12) and (5.13). Fig. 13. View largeDownload slide Error between T and the bounds (5.12) and (5.13). Recall the form of the bounds given in (5.3)   \begin{align*} \dot V_{xm} &= -\frac{\Delta_{V\!d}}{\Delta_{td}}, \quad \dot V_{xM} = \frac{\Delta_{V\!u}}{\Delta_{tu}} \\ \Delta_{V\!u} &= V_{hu} - V_{lu}, \quad \Delta_{V\!d} = V_{hd} - V_{ld}, \quad \Delta_{tu} = t_{eu} - t_{bu}, \quad \Delta_{td} = t_{ed} - t_{bd} \end{align*}and suppose ΔVu and ΔVd are being given by practical considerations (e.g. speed limits). From the bounds obtained in (5.12) and (5.13), we then have In the acceleration case   \begin{align} |T| \leqslant \xi_{aM} \dot V_{x} + \zeta_{aM} \leqslant \xi_{aM} \dot V_{xM} + \zeta_{aM} = \xi_{aM} \frac{\Delta_{V\!u}}{\Delta_{tu}} + \zeta_{aM}. \end{align} (5.14) In the deceleration case   \begin{align} |T| \leqslant \xi_{dM} \dot V_{x} + \zeta_{dM} \leqslant - \xi_{dM} \dot V_{xm} + \zeta_{dM} = \xi_{dM} \frac{\Delta_{V\!d}}{\Delta_{td}} + \zeta_{dM}. \end{align} (5.15) Then, to ensure some prescribed bound on the torque   \begin{align} |T| \leqslant T_{Ma} \ \text{ on acceleration, and }\ |T| \leqslant T_{Md} \ \textrm{ on deceleration} \end{align} (5.16)it is sufficient to impose the following bounds on Δtu, Δtd:   $$ \Delta_{tu}> \frac{\xi_{dM} \Delta_{V\!d}}{T_{Md} - \zeta_{dM}}, \quad \Delta_{ta} > \frac{\xi_{aM} \Delta_{V\!a}}{T_{Ma} - \zeta_{aM}}. $$ In Fig. 12, we have the bounds (5.12) and (5.13) in dashed line ($$\xi _{aM} \dot V_{x} + \zeta _{aM}$$ and $$\xi _{dM} \dot V_{x} + \zeta _{dM}$$) and the torque T in solid line, and in Fig. 13 is depicted the error between the previous two. 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IMA Journal of Mathematical Control and InformationOxford University Press

Published: Feb 8, 2018

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