3. AN INTELLIGENT SDRE AND ISMC COMBINED SCHEME WITH
3.3. Simulation Results
3.3.2. Employment of the Combined Scheme
In [1], Acarman has demonstrated the efficiency of SDRE scheme in vehicle control. In this chapter, we further improve the SDRE robust performance by incorporating it with the ISMC scheme when there are model uncertainties and/or external disturbances. To employ the combined scheme, we define e1 = [e1, e2, e3, e4, e5, e6, e7]T = [Vx, Vy, γ, δ, λf − λ∗f, λrr− λ∗rr, λrl− λ∗rl]T and u = [δS, Tf, Trr, Trl]T, where λ∗α, α = f, rl, rr, are the peak values, to be tracked, of the wheel slip ratio curves. The governing equations then have the form of (3.2), and the control objective becomes to organize an appropriate controller that effectively brings the error state e1 to the origin. Next, we factorize the nonlinear drift term f1(e1) into a linear structure with SDC matrices. Because we require regulation performance for λf, λrr and λrl, the factorization of f1(e1) exhibit a bias term b(e1), as described by Eq. (3.4). Details of an expression for A1(e1), b(e1) and G1(e1) given in Eq. (3.6) are presented in Appendix.
In this example we assume that the disturbance d = [0, 0, 0, 0, 0, 0, 0.5 sin 20t]T and the output of the brake torque at the rear-left wheel experiences a 40% degradation in magnitude, i.e., [∆G]ij = 0 for all i, j except for [∆G]44 = −0.4, where [·]ij denotes the (i, j)-entry of a matrix. The degradation might result from the abnormal operation of the inverter, braking system and/or wheel motor [41]. The vehicle parameters are assumed from [12] to be m = 403.87kg, Jv = 178.54kgm2, l1 = 1.39m, l2 = 0.61m, l3 = 0.575m, Cf = 3885N/rad, Crr = 4050N/rad, Crl = 4050N/rad, Rw = 0.21m and Jw = 0.567kgm2. The time constant is set to be τ = 30. The road is assumed to be dry with c1 = 1.2801, c2 = 23.99, c3 = 0.52, c4 = 0.02 and λ∗α = 0.15 for all of the three wheels [14]. The other parameters and initial state are κ = 0.6, ρg = 23, ρd(e1) = 0.5, k1 = 10−3, Ds = [04×3 I4×4], R = diag[10−1, 1, 1, 1], Q = diag[10−5, 10−2, 10−2, 10−2, 106, 106, 106] and e1(0) = [30, 0, 0, 10−4, −10−1, −10−1, −10−1]T, where the unit of velocity is meters per second. Note that, we have promoted the weightings on the three slip ratios to make λα → λ∗α as soon as possible and to maximize the antilock braking torque. To alleviate chatter, the control u1 given by (3.12) is replaced with −ρ(DsG1(eǫ 1))Tσ when
k(DsG1(e1))T σk ≤ ǫ, and ǫ is selected to be 5 × 10−3. Verification of Assumption 3.4:
Finally, we need to verify Assumption 3.4 so that the SDRE scheme can be success-fully implemented. Because Q is selected to be a nonsingular matrix, (A(e), Q12(e)) is observable. Due to the special structure of A(e) and B(e) given by (3.6), (A(e), B(e)) is stabilizable if (A1(e1), G1(e1)) is controllable. To investigate the controllability of (A1(e1), G1(e1)), we introduce a matrix M ∈ IR7×7 as follows: the first four columns of M are G1(e1), while the last three columns of M are taken from the last three columns of A1(e1)G1(e1). It is found that M = 03×4 M12
M21 ∗
!
and M21= diag[1τ,JRw
we1,JRw
we1,JRw
we1].
Clearly, M21 is nonsingular if e1 6= 0 (i.e., before the vehicle is fully stopped). Since M is a block triangular matrix, we have that (A1(e1), G1(e1)) is controllable if det(M12) 6= 0.
By direct calculation, we have det(M12) = −2E1JE32E3R3wl3sδsat
wJvm2e31 , where E1, E2 and E3 are three nonzero scalars given in Appendix A. It follows that (A1(e1), G1(e1)) is controllable if sin(δ) 6= 0 (or δ 6= 0). For δ = 0, we replace the fifth column of M by the first column of A21(e1)G1(e1). With this new M and the fact that e5 = λf− λ∗f, e6 = λrr− λ∗rr and e7 = λrl−λ∗rl, it is found that det(M12) = 2Em32JEv3el341JRw33w ·h(1 − λf)(1 + RJ2w
w)E12+ (1 − λrr)E1E2+ (1 − λrl)E1E From (3.24), we observe that 0 ≤ λα ≤ 1 during braking and λα = 1 only when
the wheel is locked. Thus, det(M12) 6= 0 unless all the three wheels are locked (i.e., λf = λrr = λrl= 1). These results verify Assumption 3.4 for the period before the vehicle is fully stopped.
3.3.3 Simulation Results
Numerical results are summarized in Figs. 3.1-3.4. Among these, we consider the following three cases: the first use the SDRE scheme for the nominal system (labeled SDRE0), while the other two adopt the SDRE scheme (labeled SDRE1) and the com-bined scheme (labeled SDRE+ISMC) for the uncertain system (experiencing actuator’s degradation and external disturbance in the actuator). It is observed from Fig. 1(a) that the longitudinal velocity converges to zero for all of the three cases. However, during the control period, the lateral velocity, the angular speed and the steering angle for SDRE1 given in Figs. 3.1(b)-3.1(d) are much larger than the other two cases, which might result
in undesirable instability. Due to the use of a saturation-type function instead of the sign-type function given in (3.12), the state trajectory of the SDRE+ISMC system deviates slightly from that of SDRE0, and it is found from simulation that the smaller the boundary layer width ǫ is, the closer the two trajectories are. From Fig. 3.3, all the sliding variables are found to be within the boundary layer. These results agree with the theoretical results.
It is also observed from Fig. 3.2(c) that the performances of λrl → λ∗rl are not achieved for SDRE1, which results in a worse braking performance, and the oscillation of e7 (i.e., λrl) for SDRE1 comes from the persistent oscillation of d. These results imply that the SDRE+ISMC scheme is more robust than the SDRE1 scheme. Finally, Fig. 3.4 shows the control efforts. Among these, the control curves of Trl for the SDRE1 and SDRE+ISMC schemes have taken the degradation effect into account, i.e., those two curves describe the actual degradation torque output which is 60% of the magnitude of the designed control value. It is seen from Fig. 3.4(d) that the braking torque Trlfor the SDRE+ISMC systems is automatically adjusted to a level such that its degradation curve approximates that of the SDRE0 scheme, to ensure that its state responses are close to those of the SDRE0 system. The oscillations of Trl for the SDRE1 and SDRE+ISMC systems resulted from the compensation for the persistent disturbance excitation. The braking torque Trl for the SDRE+ISMC system is found to be a little larger than that of the SDRE1 system, since the combined scheme provides an additional control u1 for regulation when the system state deviates from the sliding manifold. In contrast, the required steering wheel angle δS for the SDRE+ISMC system is much smaller than that of the SDRE1 system. By direct calculation, the quadratic performance of the three cases has the following relation:
JSDRE1 = 9.2539 × 105 < JSDRE0 = 9.2265 × 105 < JSDRE+ISMC= 9.0261 × 105.
Though the above simulations demonstrate the effectiveness of the combined scheme, the controls are decoupled from each other since the matrix Ds is selected to be Ds = I.
To demonstrate the interactive relation of these actions, in the following we consider Ds
has the following form:
Under the choice of Ds, the sliding variables σ2, σ3, and σ4 will couple to each other. As a result, the required control effort of Trl also be provided by Tf and Trr to compensate the degradation effect. A same scenario for state response can also be observed form Figs.
3.6(a)-(d). However, from Figs. 3.7(a)-(c), the slip ratios of these three wheel are slight variation because the couple effect of sliding variable. Fig. 3.8 shows the couple effect for DsBσ2, DsBσ3, and DsBσ4. Fig 3.9 shows the control efforts. Among these, the control curves of Tf, Trr, Trl are slight different to pervious simulation result.
Although the combined scheme requires a little more control effort than SDRE1, it enables a much smaller yaw rate and lateral velocity than the SDRE1 scheme. Thus, it is concluded from this example that the combined scheme is more robust and safer in braking control than the SDRE scheme alone.
0 1 2 3
Fig. 3.2. Time history of the first four system states.
0 0.5 1 1.5 2 2.5 3
Fig. 3.3. Time history of the last three system states.
0 0.5 1 1.5 2 2.5 3
Fig. 3.4. Time history of the four sliding variables for SDRE+ISMC scheme.
0 1 2 3
Fig. 3.5. Time history of the four control inputs.
0 1 2 3
Fig. 3.6. Time history of the first four system states.
0 0.5 1 1.5 2 2.5 3
Fig. 3.7. Time history of the last three system states.
0 0.5 1 1.5 2 2.5 3
Fig. 3.8. Time history of the four sliding variables for SDRE+ISMC scheme.
0 1 2 3
Fig. 3.9. Time history of the four control inputs.
Appendix 3.A
ai3= 1 − ei− λ∗α
CHAPTER FOUR
A STUDY OF RELIABLE YAW MOMENT CON-TROL FOR ELECTRIC VEHICLE
In this chapter, the SMC reliable control scheme is applied to retain steerability and lead the vehicle to a neutral steer behavior. This chapter is organized as follows. Section 4.1 states the vehicle dynamical model and the problem. It is followed by the design of SMC reliable controllers for ensuring EVs’ safety when actuators’ fault happens. Section 4.4 demonstrates the simulation results.
4.1 Problem Formulation
Consider the following 7 DOF nonlinear vehicle model [16]:
Fig. 4.1. Four wheeled vehicle model.
˙ωi = Rwfxi − Ti
Jw
(4.2) and Mzi = lxi(fxicδi− fyisδi)
+lyi(fxisδi+ fyicδi) (4.3)
for i = 1, · · · , 4. Here, c and s denote respectively the cosine and the sine functions (e.g., cβ = cos β and sδi = sin δi), m, Jv, Jw and Rw denote respectively the vehicle mass, vehicle inertia, wheel inertia and wheel radius, V , β and γ are respectively the vehicle velocity, body side slip angle and yaw rate, ωi, δi and Ti for i = 1, · · · , 4 denote respectively the wheel angular speed, steering wheel angle and brake/drive torque at the ith wheel, fxi and fyi denote the longitudinal and the lateral forces at the ith wheel between the tire patch and the road, respectively, and lxi and lyi denote respectively the lateral and longitudinal distances between the center of gravity (CG) to the ith wheel.
In this chapter, the subscripts i = 1, · · · , 4 are employed to represent respectively the front-right, the front-left, the rear-right and the rear-left wheels, as seen from the driver.
Besides, we assume that δ1 = δ2 = δ, δ3 = δ4 = 0, lx1 = lx3 = −lt, lx2 = lx4 = lt, ly1 = ly2 = lf and ly3 = ly4 = −lr [16]. The longitudinal and the lateral forces are expressed according to the Magic Formula [37] as follows:
fxi(λi) = D sinnC tan−1[Bλi− E (Bλi − tan−1(Bλi)io (4.4) and fyi(αi) = D sinnC tan−1[Bαi− E (Bαi − tan−1(Bαi)io (4.5)
for i = 1, · · · , 4, where the coefficients B, C, D and E denote respectively the stiffness, the shape, the peak, and the curvature factors, λi and αi are respectively the wheel slip ratio and the tire slip angle defined by [16] as
λi = (V − Rwωi)
max{V, Rwωi}, i = 1, · · · , 4 (4.6) and αi =
− tan−1lfγ+V sβV cβ + δ, i = 1, 2
tan−1lrγ−V sβV cβ , i = 3, 4. (4.7) Note that, λi < 0 if the ith wheel is in acceleration, and λi > 0 if it is in deceleration.
In order to have a satisfactory steerability and guide the vehicle to a neutral steer behavior, a reference yaw rate γd determined from the assumption of constant forward
speed, while negotiating a steady state cornering maneuver has been proposed as (4.8)-(4.9) below [4]:
γd = γss
1 + τ pδ (4.8)
and γss = 2V CfCr(lf + lr)
2CfCr(lf + lr)2− mV2(lfCf − lrCr) (4.9) where τ , p, Cf and Cr denote the time constant, the Laplace variable, the front and the rear wheel cornering stiffness, respectively. When the vehicle is operated under normal (non-faulty) condition, the desired yaw rate can be easily tracked by creating a yaw moment from the difference of tire driving torques and/or braking forces between the right and left sides. However, if one of the wheel torque controllers fails to operate (which might result from the inverter failure, the brake system failure, or the wheel motor failure [41]), the vehicle will start to spin and an appropriate reliable controller is needed to assure driving safety and provide better performances. Thus, the objective of this chapter is to organize a suitably control to realize the performance γ → γd when a vehicle experiences wheel actuator failure.
4.2 Reliable Controller Design
4.2.1 Output tracking formulation
Let x = [V, β, γ, ω1, ω2, ω3, ω4], u = [T1, T2, T3, T4] and y = γ. Then Eqs. (4.1)-(4.2) together with the selected output γ constitute a multi-input and single-output (MISO) nonlinear affine system. It is found that the MISO system has relative degree 2, i.e., one has to differentiate y(t) twice to have u(t) explicitly appearing. By direct calculation we have
¨
γ = φ (x) + G (x) u + d (4.10)
where φ(x) ∈ IR, G(x) = [g1(x), · · · , g4(x)] ∈ IR4, d denotes possible uncertainties and/or measurement noises, and the detailed expressions of φ(x) and gi(x) are given as below:
Define
̟1(x) = Bx − EhBx − tan−1(Bx)i (4.11)
̟2(x) = C tan−1[̟1(x)] (4.12)
̟3(x) = BCD cos(̟2(x))[(1 − E)(1 + B2x2) + E]
In this chapter we consider the active reliable output tracking issues for System (4.1)-(4.2), that is, we assume that the actuators’ fault has been successfully detected and diagnosed by an FDD mechanism. The fault may be time varying and include degradation, amplification and outage [28]. Before the occurrence of faults, the engineers may take any kind of control strategy to fulfill their desired system performance. When the fault is detected and diagnosed, the control law is guided to switch to an active reliable law for ensuring system performance. Thus, after the fault is detected, we may divide the actuators into two groups H and F , within which we assume that all of the actuators in H are healthy, while those in F experience faults. This implies that Eq. (4.10) can be rewritten as
¨
γ = φ (x) + GH(x) uH+ GF(x) uF + d (4.19)
where u = [uTH uTF]T and G(x) = [GH(x) GF(x)]. In the rest of the this paper, we assume that uH ∈ IRk, uF ∈ IR4−k and k ≥ 1, since the assumption of rank(GH(x)) = 1 is
necessary for the existence of the equivalent control in SMC design [9]. In addition, we assume that the control inputs in the set of F are diagnosed as
uF = ˆuF+ ∆uF (4.20)
where ˆuF and ∆uF denote the estimated control value and estimated error, respectively.
The estimated error ∆uF is treated as an additional uncertainty that should be compen-sated. Define the output error
e = γ − γd. (4.21)
The control objective is then to force e → 0 through SMC reliable design. Since Eq.
(4.19) is a second-order system, we may assume the sliding surface in the form of
σ = ˙e + k2e (4.22)
where k2 is a positive constant. Clearly, if the system state remains on the sliding surface, then the desired performance of e → 0 can be exponentially achieved with a convergence rate depending on the choice of k2 [9]. From Eqs. (4.19)-(4.22) we have
˙σ = φ(x) + GHuH+ GF(ˆuF + ∆uF) + d − ¨γd+ k2˙e. (4.23) To guarantee the reaching performance, we impose the next assumption:
Assumption 4.1 : There exists a nonnegative functionρ(x, t) such that |GF∆uF + d| ≤ ρ (x, t).
Following the SMC design procedure [9], we choose
uH = −G+H(x) · [φ(x) + GF(x) ˆuF − ¨γd+ k2˙e
+ (ρ(x, t) + η) sgn (σ)] (4.24)
where G+H(x) = GTH(x)hGH(x) GTH(x)i−1 and η is a positive constant which affects the convergence speed of the system state to the sliding surface. Note that, uH, given by (4.24), involves the information of diagnosis. Form (4.23), (4.24) and Assumption 1 we have
σ ˙σ = σ · [GF∆uF + d − (ρ(x, t) + η) sgn (σ)]
≤ −η|σ|. (4.25)
Thus, the system state will reach the sliding surface in a finite amount of time with reaching speed depending on the magnitude of η, and remain there hereafter [9]. After reaching to the sliding surface σ = 0, the tracking error e, from Eq. (4.22), satisfies
˙e + ke = 0. It implies that the tracking error is exponentially convergent to zero and the tracking performance is achieved. In addition to the yaw rate tracking, another factor that affects the vehicle driving stability and cannot be ignored is the magnitude of the body side slip angle β. It is in general expected to have β as small as possible for ensuring vehicle stability; however, when γ ≈ γd, | ˙V | ≈ 0, |β| ≪ 1, ltγd/V ≪ 1 and the tire forces at the two sides are the same, the side slip angle is related to the yaw rate and the steering angle in the following relation [40]:
β ≈ −˙
which is clearly a stable system for β with inputs γdand δ. As a result, the side slip angle is determined from the yaw rate and the steering angle, and it can be made small if γdand δ are appropriately chosen. An analysis of lateral stability and bifurcation phenomena with respect to the variations in the front wheel steering angle has been presented in [29].
Besides, one may also deal with the tracking of yaw rate and side slip angle simultaneously by choosing the weightings on the error of yaw rate and on the error of side slip angle [11]. In this chapter, we only consider the yaw rate tracking matter, i.e., the weighting on the error of side slip angle is chosen to be zero.
4.2.3 FDD mechanism
To detect the actuators’ fault for active reliable task, in this chapter we assume that all the state variables are available for measurement or estimation. In fact, this assumption is feasible for EVs [4]. We adopt the observer and the associated residual signals ri from [22] as (4.27 and (4.28) below:
˙ζi = Rwfxi− Ti
Jw
+ ai(ωi− ζi) (4.27)
and ri = ωi− ζi (4.28)
where i = 1, · · · , 4 and ai > 0 for all i. It was shown in [22] that the actuators’ fault can be detected and diagnosed with diagnosed error converging to zero at an exponential rate ai.
4.3 Simulation Results
In this chapter, we adopt the vehicle’s parameters from [38] as follows: m = 1300kg, Jv = 2000kgm2, Jw = 0.6kgm2, lf = 1.25m, lr = 1.25m, lt= 0.8m and Rw = 0.3m. Under these settings, the normal loads for the four wheels are calculated to be Fzi = mglr/2l = 3188N for i = 1, 2 and Fzi = mglf/2l = 3188N for i = 3, 4. Let the cornering stiffness Cf = Cr = 8741N/rad. The coefficients of the Magic Formula can then be obtained as follows: B = 0.1664, C = 1.65, D = 3579.4 and E = 0.6645 for longitudinal forces; and B = 0.2302, C = 1.3, D = 3152.9 and E = −0.0412 for lateral forces [37]. The control and the observer parameters are selected as τ = 10−3, k2 = 1, η = 1 and ai = 1 for i = 1, · · · , 4.
To alleviate chatter, sgn(σ) is replaced with the saturation function sat(σ/ǫ) and ǫ = 10−1. The steering command for lane change is chosen to be δ = 0.05 sin(1.5708(t − 1)) when 1 ≤ t ≤ 5, and δ = 0 elsewhere [4]. Finally, the road is assumed to be dry, and the initial state and the disturbance are taken as x(0) = [30, 0, 0.5, 102, 102, 102, 102] and d = sin(20t) + 0.5 sin(30t) + 0.1 sin(50t).
To demonstrate the reliable performances, in the following, we will consider three faulty cases: the first concerns only one actuator fault, two actuator faults and three actuators fault. The alarm will be fired if any one of the residual signals exceeds 1 (i.e.,
|ri| > 1). Numerical results are summarized in Figs. 4.2-4.13. Among these, we adopt the following two schemes: the first uses the proposed reliable SMC scheme (labeled RSMC), while the other adopts the conventional (non-reliable) SMC design (labeled SMC). Before alarm, both the two schemes adopt their conventional non-reliable SMC design, i.e., con-trol in the form of (4.24) with all actuators being healthy. Whenever there is an alarm, the associate active reliable controllers are activated according to the FDD information.
4.3.1 One actuator fault
Here, we assume that the rear-left brake actuator fails at t = 2.5, i.e., T4 = 0 after t = 2.5, H = {T1, T2, T3} and F = {T4} after the fault happens. It is observed from Figs. 1(c) and 1(d) that the output tracking error for RSMC is much smaller than that of SMC after fault happens. The longitudinal velocity, as seen from Fig. 4.2(a), of RSMC decreases from 30m/sec to 24.48m/sec, while that of SMC decreases from 30m/sec to 16.3m/sec. This means that the non-reliable design leads to a larger deceleration than the RSMC; however, a large deceleration usually results in a loss of ride comfort [4], which is usually undesired. From Fig. 4.2(b), the magnitude of the side slip angle of RSMC is also significantly reduced after fault happens, compared with that of SMC. With the selected parameters and steering command, it is found that max{ltγd/V }|V =30 = 0.016 and Eq. (4.26) under V ≡ 30 becomes ˙β = −0.90β−γd+0.90δ, which yields the magnitude of β has an upper bound maxt≥0|β| = 0.3678 at around t = 2.60. It is observed from Fig. 4.2(b) that the magnitude of β for RSMC is actually within this bound. Besides, β of SMC is noticed from Fig. 4.2(b) to be convergent to zero slower than that of RSMC, because the output tracking error of SMC in Fig. 4.2(d) converges to zero slower than that of RSMC. Figures 4.3(a)-(d) show the angular speed of the four wheels. All of the four wheels are seen to be decelerated during the change lane period, and those of SMC having angular speed much decreased than those of RSMC. In addition, it is observed from Fig. 4.4(a) that the actuator fault is successfully detected by the observer at around t = 2.506. When the fault is detected, the RSMC scheme switches its controller to the reliable one and the residual signal r4 is seen to quickly decrease to zero. This can also be seen from the alarm signals shown in Fig. 4.4(b), where the alarm value 1 denotes the fault of the fourth actuator. Figures 4.4(c) and 4.4(d) give the history of the four control torques for RSMC and SMC, respectively. It is noted that after the detection of the outage of actuator T4, the other three actuators of RSMC contribute much more control effort than those of SMC to realize the required yaw moment for yaw rate tracking, which are also reflected in Figs. 4.2(b) and 4.2(d) where the output tracking performances of RSMC are much better than those of SMC. After the change lane mission (i.e., t ≥ 5),
the controls of RSMC are seen from Fig. 4.4(c) to maintain near zero level, while those of SMC do not reduce to zero until the output tracking error converges to zero, as seen from Fig. 4.4(d). All the peaks of the healthy control curves and the abrupt change of γ − γd
at t = 5 of the SMC scheme come from the abrupt change of the steering wheel angle.
Finally, Fig. 4 displays the time history of the sliding variables of RSMC and SMC. It is seen that, after the reaching phase time period, the two sliding variables remain inside the boundary layer before the fault happens. After the fault, the sliding variable of RSMC runs out of the boundary layer shortly and then gets back to the boundary layer due to the activation of reliable controller; however, the non-reliable scheme is not able to force its sliding variable stay inside the boundary layer even after the end of change lane mission t ≥ 5. The two abrupt changes of σ of SMC scheme at t = 3 and t = 5 come from the occurrence of fault and control peak of SMC (see Fig. 4.4(d)), respectively. The
Finally, Fig. 4 displays the time history of the sliding variables of RSMC and SMC. It is seen that, after the reaching phase time period, the two sliding variables remain inside the boundary layer before the fault happens. After the fault, the sliding variable of RSMC runs out of the boundary layer shortly and then gets back to the boundary layer due to the activation of reliable controller; however, the non-reliable scheme is not able to force its sliding variable stay inside the boundary layer even after the end of change lane mission t ≥ 5. The two abrupt changes of σ of SMC scheme at t = 3 and t = 5 come from the occurrence of fault and control peak of SMC (see Fig. 4.4(d)), respectively. The