2. PRELIMINARIES
2.1. State Dependent Riccati Equation
In this section, we recall the concepts of State Dependent Riccati Equation (SDRE) [1], [5]-[7], [31], [32].
Consider the following class of nonlinear control systems
˙x = f(x) + B(x)u (2.1)
where x ∈ IRnand u ∈ IRmdenote the system and control inputs, respectively, f(x) ∈ IRn, B(x) ∈ IRn×m and f(0) = 0. In addition, we consider the following performance index
J =
Z ∞ 0
hxTQ(x)x + uTR(x)uidt (2.2)
where QT(x) = Q(x) ≥ 0 and RT(x) = R(x) > 0. The procedure of SDRE is summarized as follows:
• Symbolically factorize f(x) into the form of f(x) = A(x)x, where A(x) ∈ IRn×n.
• Check the stabilizability of (A(x), B(x)) and the observability of (A(x), C(x)) sym-bolically to ensure the solvability of the following SDRE:
AT(x)P (x) + P (x)A(x) − P (x)B(x)R−1(x)BT(x)P (x) + Q(x) = 0. (2.3)
where C(x) ∈ IRp×n has full rank and satisfies Q(x) = CT(x)C(x).
• Solve the SDRE for P (x) to produce the SDRE controller u = −R−1(x)BT(x)P (x)x.
Note that the SDRE scheme to the stabilization of nonlinear control systems need to symbolically factorize the drift term in the form of f(x) = A(x)x, and then using this A(x)
to check system’s stabilizability and observability symbolically at every state for ensuring the solvability of an associated state-dependent Riccati equations. In doing so, the SDRE algorithm fully captures the nonlinearities of the system, bringing the nonlinear system to a (non-unique) linear structure having state-dependent coefficient (SDC) matrices, and minimizing a nonlinear performance index having a quadratic-like structure. Moreover, the nonuniqueness of the factorization creates extra degrees of freedom, which can be used to enhance controller performance.
2.2 Sliding Mode Control
The implementation of the Sliding Mode Control (SMC) consists of two main phases.
First, we should construct the sliding surface such that the system states restricted to the sliding surface will produce the desired behavior. Second, we construct switched feedback gain which derive the plant state trajectory to the sliding surface in finite time (i.e. σT ˙σ ≤ −η ksk for some η ¿0) and restrict the state to sliding surface (i.e. s = 0 and ˙s = 0). Suppose at t0, the state trajectory of the plant intercepts the sliding surface and a sliding mode exists for all t > t0. The existence of a sliding mode implies (1) ˙s = 0, and (2)s = 0 for all t > t0. The system’s motions on the sliding surface can be given an interesting geometric interpretation, as an “averag” of the systems’ dynamics on both sides of the surface. The system while in sliding mode can be written as
˙s = 0. (2.4)
By solving the above equation formally for the control input, we obtain an expression for u called the equivalent control, ueq which can be interpreted as the continuous control law that would maintain ˙s = 0 if the dynamics were exactly known. For instance, for a system of the form
¨
x = f + u, x ∈ IR. (2.5)
In order to be converged to a desired trajectory x(t)≡xd(t), we define a sliding surface s = 0.
s = (d
dt+ λ)˜x = ˙˜x + λ˜x, (2.6)
here, define the tracking error ˜x = x − xd. We then have
˙s = ¨x − ¨xd+ λ ˙˜x = f + u − ¨xd+ λ ˙˜x (2.7)
and the system dynamics while in sliding mode is, of course,
ueq = −f + ¨xd− λ ˙˜x. (2.8)
Controller design is the second phase of the SMC design procedure. Here the goal is to determine switched feedback gains which derive the plant state trajectory to the sliding surface and maintain a sliding mode condition. The presumption is that the sliding surface has been designed. Among several approach (e.g. the diagonalization method and hierarchical control method), augmenting the equivalent control is one popular approach.
This structure of control of system (2.5) is
u = ueq+ ure (2.9)
where ureis the discontinuous or the switched part of Eq.(2.9). Consider the system (2.5), we have ueq = −f + ¨xd− λ ˙˜x. In order to satisfy sliding condition Eq.(2.12), we add to ueq a term discontinuous across the surface s = 0, and let
u = ueq+ ure
= ueq− ksgn(s) (2.10)
where sgn denotes the sign function.
sgn(s) =
1 if s > 0 0 if s = 0
−1 if s < 0
(2.11)
By choosing k to be a positive scalar, 1
2 d
dts2 = ˙s · s = −ksgn(s) · s = −k|s|. (2.12) Therefore, the sliding variable s will keep at zero. Practically, Eq. (2.12) states that the squared “distance” to the surface, as measured by s2, decrease along system trajectory.
Thus, it constrains trajectories to points towards the surface S(t), as illustrated in Fig.
2.1. In particular, once on the surface, the system trajectories remain on the the surface.
In other words, satisfying sliding condition Eq. (2.12), makes the surface an invariant set.
Furthermore, as we shall see, Eq. (2.12) also implies that some disturbances or dynamics uncertainties can be tolerated while still keeping the surface an invariant set.
2.3 Integral Sliding Mode Control
In this section, we review the concepts of integral sliding mode control scheme. First, consider the following nonlinear matched uncertainties system :
˙x(t) = f(x) + G(x) {u + ∆dm} (2.13)
where x ∈ IRn is a vector of states, u ∈ IRm is a vector of control inputs. f(x) and G(x) are known nonlinear functions. kdmk ≤ γm, γ > 0 which represent the matched uncertainties. In the ISMC approach, a law of the form
u(x, t) = u0 + u1 (2.14)
is proposed. The nominal control u0(x) is responsible for the performance of the nominal system; u1(x) is a discontinuous control action that rejects the perturbations by ensuring the sliding motion, design as
u1 =
where ρ > γm. and sliding surface define as
σ(x, t) = Ds
in (2.16) can be though as a trajectory of the system in the absence of perturbations and in the presence of the nominal control u0, that is, as a nominal trajectory for a given initial condition x(t0). With this remark in mind, σ(x) can be considered a penalizing factor of the difference between the actual and the nominal trajectories, projected along
G (hence, the name projection matrix, not to be confused with a projection operator).
Notice that at t = t0, s(x, t) = 0, so the system always starts at the sliding manifold.
To determine the motion equations at the sliding manifold we use the equivalent control method. The derivative of s along time is
˙s = Ds· [ ˙x − (f (x) + G(x)u0)]
= Ds· [(f(x) + G(x)u + G(x)dm) − (f (x) + G(x)u0)]
= DsG(x) · (u + dm− u0). (2.18)
Therefore, the equivalent control ueq = u0 − dm by solving the equation ˙σ = 0. By substituting ueq for u1 in (2.13), we obtain the sliding dynamics (motion eqations on the sliding manifold) is
˙xeq= f(x) + G(x) · u0. (2.19)
It is found from (2.19) that the matched type uncertainties can be completely rejected, and the sliding dynamics and the nominal system dynamics are exactly the same. To prove u1 can maintain the sliding mode, we choose a Lyapunov function as V = 1/2σTσ.
Differentiating V with respect time using with respect to time using (2.14), (2.15) and (2.18) yields
V˙ = σT ˙σ
= σTDsG(x) · (u + dm− u0)
= σTDsG(x) ·
−ρ · [DsG (x)]T s
[DsG (x)]T s + dm
≤ −ρ (DsG(x))Tσ + kdmk · (DsG(x))Tσ
≤ (−ρ + γm) · (DsG(x))Tσ
< 0. (2.20)
since DG(x) is of full rank and σ(x(t0), t0) = 0, then the controller (2.14) guarantees that the sliding mode σ = 0 can be maintained, ∀t ∈ [t0, ∞).
2.4 Burckhardt Tire Friction Model
The friction behavior of the wheel can be approximated with parametric characteristics, as shown in Fig. 2.2. The friction, or adhesion coefficient µ is defined as the ratio of the
frictional force in the wheel plane Ff ric and the wheel ground contact force Fz: Ff ric
Fz
. (2.21)
The calculation of friction forces can be carried out using the method of Burckhard [16]:
µ (λ) = c1·1 − e−c2·λ− c3λ· e−c4·λ·vcog·1 − c5FZ2 (2.22) λ = V − Rwω
max(V, Rwω) (2.23)
where c1, c2, and c3 are given for various road surfaces in table 2.1. c4 lies in the rang 0.02s/m to 0.04s/m which influence of a higher drive velocity. c5 is the influence of a higher wheel load. c4 and c5 have a maximum value of 1, i.e. they lead to a reduction of the friction coefficient. In this thesis, we neglect the influence of a higher wheel load and adopt as tire friction model of vehicle brake control.
Table 2.1: Parameters for friction coefficient characteristics (Burckhardt Tire Friction Model)
c1 c2 c3
Asphalt, dry 1.2801 23.99 0.52 Asphalt, wet 0.857 33.822 0.347 Snow 0.1946 94.129 0.0646
0 0.2 0.4 0.6 0.8 1 0
0.2 0.4 0.6 0.8 1 1.2 1.4
λ
friction coefficient µ
asphalt, dry
snow
asphalt, wet
Fig. 2.1. Friction coefficient (Burckhardt Tire Friction Model)
2.5 Magic Formula for Longitudinal and Lateral Force
In general, there are many descriptions of the model for longitudinal and lateral force.
In recent years, an well-known empirical method , the so-called Magic Formula, is devel-oped for characterizing tire behavior and used in vehicle handling simulation. The Magic Formula can be used to fit experimental tire data for characterizing the relationships between the cornering face and tire slip angle or braking effort and skid. Thus, in this thesis, we adopt the Magic Formula form [37] to represent the nonlinear cornering forces.
It is expressed by
y(x) = D sin n C tan−1 h Bx − EBx − tan−1Bx i o (2.24) where B is called the stiffness factor, C the shape factor, D the peak factor, and E the curvature factor, respectively. Where y(x) represent longitudinal force fx and lateral force fy. When y(x) represent longitudinal force fx, then x denotes slip ratio at wheel.When y(x) represent lateral force, then x denotes tire slip angle at wheel.One example of the parameter values for Magic Formula type longitudinal and cornering force are adopted from [37] as given in Table 2.2. Fig. 2.3 depicts the relationship between longitudinal force and skid. Fig. 2.4 depicts the relationship between cornering force and tire side slip
angle. In this thesis, we adopt Magic Formula as tire model of yaw moment control.
Table 2.2: Magic Formula coefficient (Fz = 3118.3N)
B C D E
longitudinal force 0.1664 1.65 3579.4 0.6645 lateral force 0.23 1.3 3152.9 -0.4216
0 0.2 0.4 0.6 0.8 1
0 500 1000 1500 2000 2500 3000 3500 4000
slip ratio using Magic Formula.
longitudinal force F x (N)
Fig. 2.2. Longitudinal force using Magic Formula
0 5 10 15 20 0
500 1000 1500 2000 2500 3000 3500
tire side slip angle α (deg.) cornering force F y (N)
Fig. 2.3. Cornering force using Magic Formula
CHAPTER THREE
AN INTELLIGENT SDRE AND ISMC COMBINED SCHEME WITH APPLICATION TO VEHICLE BRAKE CONTROL
In this chapter, combination of the SDRE and ISMC scheme is studied and applied to vehicle brake system when actuators degradation. The organized of this chapter as follow.
Section 3.2 states the problem and the main goal of this paper. Section 3.3 then describes the design of a combination of the SDRE and ISMC schemes. Section 3.4 discusses the application of the analytic results to the antilock brake control of a three-wheel vehicle system.
3.1 Problem Formulation
Consider a class of nonlinear systems described by the following equation:
˙
x = f(x) + G(x) [(I + ∆G)u + d] (3.1)
where x ∈ IRn and u ∈ IRm denote the system states and the control inputs respectively,
∆G and d describe possible matched type uncertainties, f(x) ∈ IRn and G(x) ∈ IRn×m. In this chapter, we assume that G(x) 6= 0 for all x 6= 0, and make the following three assumptions:
Assumption 3.1 : The origin of the nominal system ˙x = f(x) + G(x)u of System (3.1) is asymptotically stabilizable.
Assumption 3.2 : I + ∆G ≥ κI for some κ > 0 in the sense of ξT(I + ∆G)ξ ≥ κ · kξk2 for all ξ 6= 0.
Assumption 3.3 : There exist a positive constant ρg and a nonnegative function ρd(x) such that k∆Gk ≤ κ · ρg and kdk ≤ κ · ρd(x).
When ∆G is a diagonal matrix and a controller has been organized, Assumption 3.2 implies that each actuator experiences no change in the output direction and only experiences degradation and/or amplification. In electrical vehicles, this degradation and/or amplification are associated with the loss of control effectiveness of steering and/or wheel torque controllers [41]. The objective of this chapter is to organize an appropriate controller to realize the regulation performance x → xd, where xd is a nonzero constant vector.
3.2 Controller Design
Due to the many advantages of the SDRE and the ISMC approaches mentioned in the Introduction section, we combine the two schemes for the controller design in this chapter.
First, the SDRE design is used for the nominal system. Then, we adopt the ISMC strategy to compensate for the error when the system state deviates from the nominal system’s trajectory.
3.2.1 SDRE Design for Nominal System
To employ the SDRE approach, the regulation problem needs to be converted into the stabilization problem. For this purpose, we introduce the error state e1 = x − xd. System (3.1) then becomes
˙e1 = f1(e1) + G1(e1) [(I + ∆G)u + d] (3.2)
where f1(e1) := f(e1+ xd) and G1(e1) := G(e1+ xd). Using this error state dynamics, the regulation problem becomes the stabilization problem, i.e., x → xd if and only if e1 → 0.
To implement the SDRE design, we assume that the quadratic performance index is given by (3.3) below:
J =
Z ∞ 0
heT1Q(e1)e1+ uTR(e1)uidt (3.3)
where QT(e1) = Q(e1) ≥ 0 and RT(e1) = R(e1) > 0. The SDRE scheme requires the condition f1(0) = 0 to factorize the drift term f1(e1) in Eq. (3.2) into the form of A1(e1)e1. However, there might be a state-dependent bias term, say b(e1), that makes f1(0) 6= 0,
i.e.,
f1(e1) = A1(e1)e1+ b(e1) (3.4) with b(0) 6= 0. To solve this problem, we adopt a strategy from [7] to express b(e1) as b(e1) =hb(ez1)iz and augment the system with a stable auxiliary state z as (3.5) below:
˙z = −k1z (3.5)
where k1 is a small positive constant that makes z change slowly, and z(0) 6= 0 so that
b(e1)
z is smooth during the control period. With these settings, the augmented system formed by (3.5) and the nominal system of (3.2) has following form:
˙e = A(e)e + B(e)u (3.6)
where A(e) = A1(e1) b(ez1) 0 −k1
!
, B(e) = G1(e1) 0
!
and e = [eT1, z]T. To successfully implement the SDRE scheme, the following assumption for the existence of a unique positive definite matrix solution of an associated algebraic Riccati equation (ARE) is needed [17]:
Assumption 3.4 : The pair nA(e), Q1/2(e)o is pointwise observable and {A(e), B(e)}
is pointwise stabilizable.
According to the SDRE design procedure, the SDRE controller then has the following form [7]
u0 = −R−1(e)BT(e)P (e)e (3.7)
where P (e) is the unique positive definite matrix solution of the following ARE:
AT(e)P (e) + P (e)A(e) − P (e)B(e)R−1(e)BT(e)P (e) + Q(e) = 0. (3.8) 3.2.2 ISMC Design for the Compensation of Uncertainties
After selecting the nominal controller, the system might experience uncertainties, in-cluding external disturbances and actuators’ output degradation and/or amplification.
To compensate for these effects, we adopt the ISMC design because of its advantages, including robustness, rapid response and easy implementation.
According to the ISMC design procedure [5], [6], the sliding manifold for System (3.2) is given by the following equation:
σ = Ds
e1(t) − e1(t0) −
Z t t0
f1(e1) + G1(e1)u0 dτ
(3.9)
where Ds∈ IRm×n is choose such that DsG1(e1) has full rank and u0 is given by (3.7). It follows from Eqs. (3.2) and (3.9) that
˙σ = Ds[ ˙e1− f1(e1) − G1(e1)u0]
= DsG1(e1) [(I + ∆G)u + d − u0] . (3.10)
To keep the system state on the sliding manifold, we chose the overall control law to be
u= u0+ u1 (3.11)
where u0 is given by (3.7) and
u1 =
0 if σ = 0;
−ρk[D[DssGG11(e(e11)])]TTσσk otherwise (3.12) with ρ chosen to satisfy ρ > ρgku0k + ρd(e1). Although the coefficient ρ requires the information about u0, it can be easily obtained after the calculation of u0 because the upper bounds ρg and ρd(e1) can be estimated offline. We then have the next result:
Theorem 3.1 : Suppose that Assumptions 3.1-3.4 hold. Then the error states e1 given in the nominal system (3.6) with the SDRE controller and the uncertain system described by (3.2) with the control law given by (3.11)-(3.12) are identical.
proof: From Eqs. (3.10)-(3.12) and Assumptions 3.2-3.3 we have
σT ˙σ = σTDsG1(e1) [(I + ∆G)u1− ∆G · u0+ d]
≤ (−ρκ + k∆Gk · ku0k + kdk) · [DsG1(e1)]Tσ
≤ κ (−ρ + ρgku0k + ρd(e1)) · [DsG1(e1)]Tσ
< 0 (3.13)
whenever [DsG1(e1)]Tσ 6= 0 or σ 6= 0 because DsG1(e1) is a nonsingular matrix. From (3.9), it is clear that σ(e1(t0)) = 0. Thus, from (3.13), we have σ(e1(t)) = 0 for all t ≥ t0,
i.e., the system state remains on the sliding manifold for all t ≥ t0. To determine the sliding dynamics (the equations of motion on the sliding manifold), the equivalent control method is used [6]. The equivalent control is obtained by solving the equation ˙σ = 0 from Eq. (3.10) as follows:
ueq = (I + ∆G)−1(u0 − d). (3.14)
By substituting ueq into (3.2), we have the sliding dynamics ˙e1 = f1(e1)+G1(e1)u0, which is the nominal system under the SDRE controller and the proof is completed.
It is found from the proof of Theorem 3.1 that the matched type uncertainties can be completely rejected, and that the sliding dynamics and the nominal system dynamics are identical. Because the system state under the ISMC scheme starts from the sliding manifold, it follows that the state trajectories of the uncertain system using the combined scheme and the nominal system under SDRE scheme are identical. Therefore, with this ISMC design the engineer can organize another optimal controller (other than the SDRE design) according to system requirements creating a desired system state trajectory for the state of the uncertain system to follow.
3.3 Simulation Results
3.3.1 Vehicle dynamics
A three wheeled vehicle (TWV) model has been described in [12]. For simplicity, we consider only the yaw plane motion, in which the simplified yaw plane vehicle model can be described as follows: the body frame, which is fixed to vehicle’s center of gravity (CG), is denoted by x and y. The positive x- and y-axes represent the forward direction and right-hand side respectively, as seen by the driver. The vehicle dynamics then have the following form:
Fig. 3.1. Three wheeled vehicle model.
Here, Vx and Vy are the components of the velocity of the CG in the x- and y-directions respectively, γ denotes the angular speed about the vertical axis (z-axis), m denotes the total mass of the TWV, Jv denotes the moment of inertia in the vertical axis of the body frame, l1 and l2 are the longitudinal distances from the CG to the front axle and the rear axle respectively, l3 denotes the lateral distance between CG and the left or right wheel, Fx and Fy represent the external forces acting on the body along the x- and y-axes respectively, Rα and Lα are the longitudinal and the lateral forces at each wheel between the tire patches and the road respectively, where α = f, rl, rr and the subscripts
f , rl and rr denote the front, the rear-left and the rear-right wheels respectively, cα, α = f, rl, rr, denotes the cornering stiffness for the three wheels, δsat is the actual output value of the steering wheel angle δ, defined by δsat = 16π if δ > 16π ; δsat = −16π if δ < −16π; δsat = 16π · sin(δ) if −16π ≤ δ ≤ 16π, and c and s denote the cosine and the sine functions respectively. In addition to the vehicle dynamics, we adopt the tire friction model from Burckhardt [14] in this chapter to simulate the antilock brake system, where Rα at each wheel can be expressed as
Rα = µαNα, α = f, rr, rl. (3.22)
Here, Nf = mglL1, Nrr = mgl2L2, Nrl = mgl2L2, L = l1+ l2, g is the acceleration of gravity and µα, α = f, rr, rl, are the tire-road coefficients of friction defined by
µα =hc11 − e−c2λα− c3λαi· e−c4λαVx (3.23)
where c1, c2, c3 and c4 are four parameters introduced in [14],
λα =
( Vx−Rwωα
Vx during braking;
Vx−Rwωα
rwωα during acceleration (3.24)
for α = f, rr, rl are the slip ratios of the three wheels, and Rw and ωα denote the radius and the angular velocity of the wheel respectively. To study the braking performance, we consider only the slip ratio in braking mode. To achieve an optimal antilock braking performance, the wheel slip ratio is guided to track its peak value for producing maximum negative acceleration [14]. For this purpose, we differentiate λα and using the fact that
˙ωα= (µαrwNα− Tα)/Jw [14], yield
˙λα= V˙x(1 − λα) Vx
+ RwTα− µαRw2Nα
VxJw
, α = f, rr, rl (3.25)
where Jw denotes the inertia moment of wheel and Tα, α = f, rl, rr, are the three brake torques. Finally, the vehicle model is augmented with the following steering dynamics [1]:
τ ˙δ = −δ + δS (3.26)
where τ is time constant and δS is the steering wheel angle generated by the SDRE controller. The overall system is then described by (3.15)-(3.16) and (3.25)-(3.26).
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
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