Estimation of Equivalent Input Disturbance ... - Semantic Scholar
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IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 56, NO. 6, NOVEMBER 2007
Estimation of Equivalent Input Disturbance Improves Vehicular Steering Control Jin-Hua She, Member, IEEE, Xin Xin, Member, IEEE, and Yasuhiro Ohyama, Member, IEEE
Abstract—This paper describes a simple effective approach to vehicular steering control based on the estimation of an equivalent input disturbance. A control system configuration with a disturbance estimator and a design method based on linear matrix inequalities are explained. The simulation results demonstrate the validity of the method. Index Terms—Disturbance estimation, disturbance rejection, H∞ control, linear matrix inequality (LMI), vehicle control, vehicle dynamics, vehicle steering.
I. I NTRODUCTION
Fig. 1.
A
UTOMATIC steering control is a key element of intelligent transportation systems (see, for example, [1]–[8]). It involves two techniques: lane keeping, in which the steering control system tracks the center of the current lane, and lane changing, in which it steers in order to track a reference input for a given lateral motion. Since disturbances caused by wind and road conditions seriously affect control results, a good disturbance-rejection performance is essential and has been the focus of many studies. For example, Lin et al. [6] used a steady-state Kalman filter to estimate the lateral velocity of a vehicle and the magnitudes of external disturbances acting on it, Güvenç et al. [7] used a disturbance observer to suppress external disturbances, and Yamamoto et al. [8] devised a disturbance-estimation algorithm. However, each method has drawbacks: Lin et al. [6] only considered step-type disturbances, it is difficult to tune the filter parameters in [7] because the filter design must guarantee both the causality of the estimator and the stability of the whole control system, and [8] requires the differentiation of measured outputs. This paper addresses steering control for a straight road. We employ an equivalent-input-disturbance (EID) estimator [9] that estimates an equivalent disturbance on the control input channel to reject disturbances caused by wind, yaw disturbance torque, etc., thus providing a good steering performance. This method has three advantages: First, no a priori information about disturbances is required. The only restriction is that the Manuscript received October 10, 2005; revised July 12, 2006 and April 12, 2007. This work was supported in part by a Grant-in-Aid for Scientific Research (C), Japan Society for the Promotion of Science, under Grant 18560259. The review of this paper was coordinated by Dr. M. Abul Masrur. J.-H. She and Y. Ohyama are with the School of Bionics, Tokyo University of Technology, Tokyo 192-0982, Japan (e-mail: [email protected]; [email protected]). X. Xin is with the Faculty of Computer Science and System Engineering, Okayama Prefectural University, Soja 719-1197, Japan (e-mail: [email protected]). Digital Object Identifier 10.1109/TVT.2007.904543
Bicycle model of the vehicle (top view).
output produced by the disturbances must be bounded, but this condition is generally satisfied. Second, no differentiation of measured outputs is needed. Third, the design of the control system is very simple; the configuration can be viewed as a conventional control system enhanced by the plugging-in of a disturbance estimator. Simulation results have demonstrated the validity of this method. II. M ODEL OF V EHICLE This paper assumes that the vehicle has front-wheel steering, that there are no sudden accelerations or decelerations, and that the front wheels turn only through a small angle. Therefore, the velocity is constant, and the rolling motion can be ignored. Thus, the vehicle can be thought of as traveling on a flat surface, which is affected only by lateral and yawing motions. A further assumption is that the sideslip angle and the yaw rate are very small. Since a complicated model is not necessary to simulate the lateral motion of highway vehicles (see [10]), this paper employed the simple two-degree-of-freedom bicycle model [11] (Fig. 1). The parameters are x(y) ψ δ β v m Iz d τd lf (lr )
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x(y)-position of the center of gravity (CG) of a vehicle (in meters); yaw angle with respect to the x-axis (in radians); angle of the front wheel (in radians); sideslip angle (in radians); longitudinal velocity (in meters per second); total mass of the vehicle (in kilograms); yaw moment of inertia (in kilograms square meter); equivalent lateral disturbance force (in newtons); equivalent yaw disturbance torque (in newtons meter); distance from CG to the front (rear) axle (in meters);
SHE et al.: ESTIMATION OF EQUIVALENT INPUT DISTURBANCE IMPROVES VEHICULAR STEERING CONTROL
Fig. 2.
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Configuration of the steering control system.
ld
distance from CG to the position at which d has an effect (in meters); Cf (Cr ) cornering stiffness of the front (rear) tires (in newtons per radian). The dynamic equations of the lateral motion are the following: d2 y C +C m dt2 = − f v r dy dt + (Cf + Cr )ψ Cf lf −Cr lr dψ − v dt + Cf δ + d Cf lf −Cr lr dy d2 ψ Iz dt2 = − v dt + (Cf lf − Cr lr )ψ Cf lf2 +Cr lr2 dψ − v dt + Cf lf δ + τd τd = dld . The state-space expression is dξ dt = Aξ + Bδ + Bd d y = Cξ
(2)
where ξ = [y 0 0 A= 0 0 Bd
a1 a3 b1
dy dt
ψ
1 a1 v
0 a3 v
dψ T dt ]
0 −a1 0 −a3
0 a2 v
1 a4 v
,
(1)
0 b B = 1 0 b2
T 0 1 bd1 0 = C= , 0 0 bd2 ld 0 Cf + Cr Cf lf − Cr lr , a2 = − =− m m Cf lf2 + Cr lr2 Cf lf − Cr lr =− , a4 = − Iz Iz Cf Cf lf 1 ld , b2 = = , bd1 = , bd2 = . m Iz m Iz
A simple check of controllability and observability shows that, for this model, (A, B) is controllable for a nonzero speed and that (C, A) is observable except at two low speeds (e.g., for the nominal parameters used in Section IV, the speeds are 14.26 and −14.26 km/h). Therefore, in this paper, we consider the case in which the plant is controllable and observable. However, if the plant is not controllable or not observable, then we only need to consider its controllable and observable subsystem because we just want to produce an effective control input to achieve satisfactory steering control; the unobservable subsystem does not influence the output, and the effect of the uncontrollable subsystem on the output can be regarded as a disturbance. Moreover, a simple verification shows that this plant has no zeros on the imaginary axis, regardless of the parameters. III. D ESIGN OF C ONTROL S YSTEM An EID is a disturbance on the control input channel that produces the same effect on the controlled output as actual disturbances do [9]. In fault diagnosis, what we want to know are the exact place and kind of disturbance (fault). In contrast, in vehicular steering control, we use the steering angle δ to reject the disturbances, which makes it more reasonable to estimate an EID on the control input channel than to estimate the disturbances themselves. Since an EID always exists for a controllable and observable plant with no zeros on the imaginary axis [9], [12], we focused on its estimation. We employed the control system configuration in [9] and constructed the steering control system in Fig. 2, where B + :=
BT . BTB
(3)
Note that controllability guarantees that B T B = 0. This configuration can be viewed as a conventional control system enhanced by the plugging-in of a disturbance estimator. Since
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only the steering control for a straight road is considered here, the lane-change command can be characterized as a step signal. Therefore, a step-type servo system was selected. Remark 1: Note that a full-order state observer is employed in Fig. 2. A reduced-order state observer is also conceivable. However, when a disturbance exists, the available states of the real plant may be different from those of the plant with an EID. If we ignore this and use a reduced-order state observer, the difference between the states will produce an error in the state estimation, thereby degrading the tracking precision. In order to obtain an EID with a high precision, it is important to guarantee that the output of the observer converges to that of the real plant. This necessitates the use of a full-order state observer. The rationale behind the construction of this disturbance estimator is explained in the following section. A. Estimation and Rejection of Disturbances A full-order state observer is used to estimate the state of the vehicle dξˆ ˆ = Aξˆ + Bu + LC(ξ − ξ). (4) dt On the other hand, the following is true for the state of the plant ¯ with an EID ξ¯ and the actual value of the EID d:
dξˆ = Aξˆ + Bδ + B d¯ + B∆d. dt
(11)
In the aforementioned equation, assuming the estimate of the EID to be dˆ = d¯ + ∆d
(12)
dξˆ ˆ = Aξˆ + B(δ + d). dt
(13)
gives
Therefore, if we want the state of the plant with an EID ξ¯ to ˆ then the difference between be equal to that of the observer ξ, ¯ ˆ ξ and ξ can be described as the error in estimating the EID. Equations (4) and (13) yield ˆ B(dˆ + δ − u) = LC(ξ − ξ).
(14)
ˆ then a least-squares solution is If we solve (14) for d, ˆ + u − δ. dˆ = B + LC(ξ − ξ)
(15)
dˆ is filtered by F (s) (Fig. 2), which selects the angular frequency band for disturbance estimation, and the filtered disturbance estimate d˜ is ˆ d˜ = L−1 F (s)D(s)
dξ¯ ¯ = Aξ¯ + Bδ + B d. dt
(5)
∆ξ = ξˆ − ξ¯
(6)
ˆ ˆ and L[·] and L−1 [·] denote the Laplace where D(s) = L[d(t)], and inverse Laplace transforms, respectively. Combining d˜ with the original control law yields
(7)
˜ δ = u − d.
Letting
and substituting that into (5) yields dξˆ = Aξˆ + Bδ + B d¯ + dt
d∆ξ − A∆ξ . dt
As shown in the following discussion, there exists a control input ∆d that precisely produces the state ∆ξ. That is, the following holds: d∆ξ − A∆ξ = B∆d. dt
(8)
To prove the aforementioned statement, we construct a ∆d directly from ∆ξ. Since (A, B) is controllable, a state transformation (∆ξ = T ∆ξc ) with the following controllable canonical form [13]: d∆ξc = Ac ∆ξc + Bc ∆d dt ∆ξ = ∆ξc3 ∆ξc4 ]T c2 c [ ∆ξc1 ∆ξ 0 1 0 0 0 1 0 (9) 0 Ac = T −1 AT = 0 0 0 1 −α0 −α1 −α2 −α3 Bc = T −1 B = [ 0 0 0 1 ]T gives us ∆d ∆d =
Substituting (8) into (7) yields
d∆ξc4 + α0 ∆ξc1 + α1 ∆ξc2 + α2 ∆ξc3 + α3 ∆ξc4 . dt (10)
(16)
For this control law, the EID added onto the input channel of ˜ Therefore, the modified control law the plant reduces to d¯ − d. improves the disturbance-rejection performance. This method has the following three main features. 1) The configuration of the control system is very simple. 2) The disturbance-rejection performance can be tuned by adjusting the low-pass filter F (s) and the observer gain L. 3) The stability of the closed-loop system can be divided into two parts: the feedback gains KP and KR , and L and F (s). The only difference between this system and a conventional control system is the incorporation of a disturbance estimator, and the only design parameters that are related to the estimator are the low-pass filter and observer, which are discussed in the succeeding discussions. B. Design of Observer For a conventional control system, the closed-loop poles are those of the full state feedback plus those of the observer (separation theorem) [14]. To show that the separation theorem
SHE et al.: ESTIMATION OF EQUIVALENT INPUT DISTURBANCE IMPROVES VEHICULAR STEERING CONTROL
Fig. 3.
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Control system without exogenous inputs.
holds for our system configuration, let the exogenous inputs (r and d) be equal to zero. The vehicle is then described by dξ dt = Aξ + Bδ (17) y = Cξ. Equations (4), (6), (16), and (17) yield d∆ξ ˜ = (A − LC)∆ξ + B d. dt
(18)
On the other hand, (15) is equivalent to ˜ dˆ = −B + LC∆ξ + d.
(19)
Based from (17)–(19) and Fig. 2, we can redraw the block diagram in Fig. 2 as Fig. 3. The system can be decomposed into a lower part consisting of the state feedback and an upper part consisting of the state observer and low-pass filter. Since the state in the upper part is not controllable from the lower part and since the state in the lower part is not observable from the upper part, we can independently design the poles of these two parts, i.e., the closed-loop poles are those of the state feedback plus those of the observer and low-pass filter. Since the former can be designed by existing methods, the rest of this paper focuses on the latter. The transfer function from d˜ to dˆ is GL (s) = 1 − B + LC [sI − (A − LC)]−1 B
(20)
F (s) and L must not destroy the stability of the control system. That is, they must guarantee the stability of the upper part in Fig. 3. From the small-gain theorem [15], it follows that the stability of the upper part is guaranteed if GL F ∞ < 1.
then we obtain Fig. 4 from Fig. 3. A state-space realization of the generalized plant Gg (s) = G(s)F (s) = Dg + Cg (sI − Ag )−1 Bg
(22)
(25)
is
dxf dt d∆ξ dt
z = yL =
(21)
If we choose the angular frequency band for disturbance rejection to be Ωr = {ω : ω ≤ ωr }
then, to reject disturbances, it is best to choose F (jω) ≈ 1 in the range Ωr . For a given F (s), condition (21) and Fig. 3 necessitate an L such that |GL (jω)| < 1/|F (jω)|. In the succeeding discussion, we show that the design of L can be converted into an H∞ static output feedback synthesis problem (see Appendix A for the definition of the H∞ control problem). If we break the control loop in the upper part in Fig. 3 at the input of the filter F (s), introduce two new signals (z and w), which are the signals before and after the break point, respectively, and assume the state space representations of F (s) and G(s) to be dx f dt = Af xf + Bf w F (s) : (23) yf = Cf xf d∆ξ dt = A∆ξ + Byf − uL G(s) : z = yf − B + uL (24) yL = C∆ξ
=
Af BCf Cf 0 Ag Cg1 Cg2 Ag Cg
xf ∆ξ 0 0 −B + w uL C 0 0 xf Bg1 Bg2 ∆ξ Dg11 Dg12 w Dg21 0 uL xf ∆ξ Bg (26) w Dg uL 0 A
Bf 0
0 −I
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TABLE I NOMINAL PARAMETERS OF VEHICLE AND SIMULATION CONDITIONS
Step 2: Choose Ωr and a corresponding F (s). Step 3: Design L by solving the minimization problem for the trace of P X + W Y under the constraints of the LMIs for the H∞ control problem. IV. S IMULATIONS Simulations were carried out for a straight road using the vehicle parameters in [11] (Table I). First, we chose
Fig. 4. Formulation of the design of L using H∞ control framework.
where I is an identity matrix of size dim(A), and dim(A) is the size of the square matrix A. The design problem is thus converted into one that is designing a control law uL = LyL
Q = diag{1, 1, 10, 10, 10}, R = 10
(30)
and obtained KP and KR by optimizing the performance index ∞ J= [ ξeT
(27)
xT Re ] Q
ξe + Rδe2 dt xRe
(31)
0
such that the closed-loop transfer function from w to z, which is Gzw (s) (= GL (s)F (s)), satisfies Gzw ∞ < 1.
where ξe (t) = ξ(t) − ξ(∞)
(28)
Note that the H∞ static output feedback synthesis problem is not convex and is difficult to solve. Suppose that this problem is solvable for (26). Then, there exists a controller of order less than dim(Af ) + dim(A) [15], [16], and it can be obtained, for example, by using a solution (X, Y ) of three linear matrix inequalities (LMIs) (see Appendix B), where X and Y are positive matrices. Based on the design method for reducedorder H∞ output control [19] (see Appendix B), there exists a controller such that the bound on the order is Bg dim(Af ) + dim(A) − rank (29) + rank Dg12 . Dg12 In fact, such a controller can be obtained by minimizing the trace of P X + W Y for any given real positive definite matrices P and W under the constraints of the three LMIs. Furthermore, numerical investigations have shown that a controller of order less than the bound (29) is obtainable. By exploiting the special structure of Gg (s) in (26), we T T ] = dim(A) and rank Dg12 = 1. Thus, have rank [BgT Dg12 the order of L(s) is dim(Af ) + 1. Clearly, L(s) is not static. Therefore, we employ a minimization procedure (see Lemma 2 in Appendix B) to obtain a static controller or, in other words, the observer gain L. The aforementioned discussion is summarized in the following design algorithm for a steering control system. Design algorithm: Step 1: Design KP and KR for a conventional step-type servo system by an existing method (e.g., the optimal control method).
xRe (t) = xR (t) − xR (∞) δe (t) = δ(t) − δ(∞) for the step-type reference input r(t). The resulting KP and KR are KP = [ −1.0760 −0.3117 −5.4296 −0.7896 ] (32) KR = 1.0000. Then, we determined Ωr by setting ωr to 30 rad/s and F (s) to 1 F (s) = T s+1 , T = 0.0333 s (33) Af Bf −1/T 1/T = . Cf 0 1 0 From the last section, the order of L(s) obtained by solving the H∞ synthesis problem (26)–(28) is two. By using the new design algorithm and by solving the H∞ synthesis problem (26)–(28) with P = W = I, we obtained the static observer gain L = [ 37.307
1038.0
26.941
570.49 ]T .
(34)
The resulting transfer function is Gzw (s) =
n3 s3 + n2 s2 + n1 s + n0 s4 + d3 s3 + d2 s2 + d1 s + d0
d3 = 43.044,
d2 = 1270.1,
d1 = 5936.2
d0 = 61603,
n3 = 43.044,
n2 = 255.23
n1 = 2857.7,
n0 = 2400.3.
(35)
SHE et al.: ESTIMATION OF EQUIVALENT INPUT DISTURBANCE IMPROVES VEHICULAR STEERING CONTROL
Fig. 5.
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Bode plot of Gzw (s).
The Bode plot of Gzw (s) (Fig. 5) satisfies (28), and the control system is stable. For comparison, a conventional control system (state observer and state feedback) was also designed. The state feedback gain was chosen to be that in (32). The resulting poles of the closed-loop system with state feedback are −22.7799, −3.2265 ± j4.0487, and −1.3428 ± j1.1754. Generally speaking, the poles of the observer should be to the left of those poles on the complex plane [13]. Trial and error yielded the values −25.0, −30.0, −31.0, and −35.0. These yield the observer gain Lcv = [ 115.26
4785.6
1116.1
6840.9 ]T .
(36)
It is shown in the succeeding discussion that, although a comparison of the two systems revealed that they have almost the same response characteristics for a step reference input, the conventional control system had much poorer robustness.
Fig. 6. Simulation results for lane-changing control of vehicle with nominal parameters [conventional method: (32) and (36); new method: (32)–(34)].
A. Simulation Results for Lane-Changing Control Fig. 6 shows the lateral response (dotted line) and steering angle of a vehicle, with the nominal parameters listed in Table I under zero initial conditions for the conventional control method [(32) and (36)]. The reference input for the lane change was r = 4 × 1(t − 1)
m.
(37) Fig. 7. Disturbance.
The following disturbances (Fig. 7) were added: −3000 × [0.2 + sin 0.1π(t − 7) +2 sin 0.2π(t − 7) + sin π(t − 7) d= +0.4 sin 5π(t − 7)] N, 0,
t≥7 t < 7.
(38)
Fig. 6 also shows the response obtained by our method [(32)–(34)] (solid line). Incorporating a disturbance estimator dramatically reduced the peak-to-peak tracking error from 0.3669 to 0.0300 m, and the steady-state tracking error is only 8% of that of the conventional control system. Stability tests showed that both the conventional control system and the one obtained by our method were stable for
parameters in the following ranges: m = 1500 × (100 ± 20)% kg Iz = 3000 × (100 ± 20)% kg · m2 Cf = 50000 × (100 ± 50)% N/rad Cr = 70000 × (100 ± 50)% N/rad. Fig. 8 shows the results for the worst case m = 1800 kg, Iz = 3600 kg · m2 Cf = 25000 N/rad, Cr = 35000 N/rad
(39)
(40)
and the disturbances in (38) under zero initial conditions. Even though the conventional control system was stable, it did not maintain control of the car because the steering angle was
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Fig. 8. Simulation results for lane-changing control of vehicle with the parameters in (40).
restricted, and the disturbances forced the steering wheel all the way to the left or right (0.6981 rad = 40◦ ). However, with our method, the car followed the reference lane command with a steady-state peak-to-peak tracking error of 0.0825 m. Note that, in this case, the estimates of the equivalent disturbances d˜ are quite different from those for the nominal case because the differences between the parameters and the nominal values are considered to be a kind of disturbance (see graph of d˜ for 1 ≤ t ≤ 6 s) and that our method also estimates and restrains their influence. Therefore, even though the parameters were different from the nominal values, the disturbances were suppressed, and the car tracked the reference command with satisfactory precision. B. Simulation Results for Lane-Keeping Control For lane-keeping control (r = 0), the incorporation of a disturbance estimator also markedly reduces the tracking error. As an example, Fig. 9 shows some results for a straight road and a vehicle, with the parameters in (40) under zero initial conditions, when the disturbances in (38) were added. Also in this case, control broke down for the conventional control system. However, our method suppressed the disturbances and kept the car in the specified lane with satisfactory precision (steady-state peak-to-peak tracking error is 0.0825 m). Remark 2: The simulation results show that plugging an EID estimator into a conventional control system (state ob-
Fig. 9. Simulation results for lane-keeping control of a vehicle with the parameters in (40).
server and state feedback) improves the disturbance-rejection performance. In fact, the analysis in [17] shows that it adds a new degree of freedom to the system, and the output caused by the disturbances is related to the term 1 − F (s). Therefore, the disturbance-rejection performance can greatly be improved by choosing F (jω) ≈ 1 in the angular frequency band for disturbance rejection. V. C ONCLUSION This paper presents a new method of steering control for a vehicle with front-wheel steering. First, a method that estimates an EID was described. Then, a steering control configuration with an EID estimator was presented. Unlike existing methods, this one is very simple. Furthermore, the design of the observer, which is related to disturbance estimation, can be separated from the design of the state feedback. A design algorithm was presented, and the simulation results demonstrated the validity of the method. This paper considered steering control only for a straight road, but a general formulation must take the radius of curvature (ROC) of the road into account. The problem can then be divided into two cases, based on whether preview information on the road ROC is available. If it is available, a feedforward control scheme [1] can be employed, but if it is not available,
SHE et al.: ESTIMATION OF EQUIVALENT INPUT DISTURBANCE IMPROVES VEHICULAR STEERING CONTROL
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Fig. 10. H∞ control.
the control problem can be formulated as a disturbancerejection problem [2]. Therefore, even when the road ROC cannot be used, our method still provides an estimate of the equivalent disturbance due to road ROC, and thus, a satisfactory control can be achieved.
Fig. 11. H∞ norm of Gzw (s).
and for a multiple-input multiple-output system, it is Gzw ∞ = sup σ ¯ (Gzw (jω))
(45)
0≤ω≤∞
A PPENDIX A H∞ C ONTROL As explained in [15], in the design of the control system in Fig. 10, Gg (s) is a generalized plant containing the physical plant and some weighting functions and is linear timeinvariant; the signal w(t) ∈ Rr contains all external inputs, including references, disturbances, and sensor noise; the output z(t) ∈ Rm is a controlled error signal; uL (t) ∈ Rp is the control input; and yL (t) ∈ Rq is the observation output. The stabilizable and detectable realization of Gg (s) is
z yL
w = Gg (s) uL =
Gg11 (s) Gg21 (s)
where σ ¯ (A) is the maximum singular value of A. On the other hand, if we define
w(t)2 =
z(t)2 =
∞
0
∞
wT (t)w(t)dt
z T (t)z(t)dt
0
1/2
1/2
then the H∞ norm of the stable system Gsw (s) is Gg12 (s) Gg22 (s)
w uL
(42)
where xg ∈ Rn is the state vector. Note that the closedloop transfer function from w(t) to z(t), which is Gzw (s), is a linear fractional transform of Gg (s) and L(s) and is given by Gzw (s) = Gg11 (s) + Gg12 (s)L(s)[I − Gg22 (s) L(s)]−1 Gg21 (s). The H∞ control problem is to design an H∞ controller L(s), which makes the H∞ norm of Gzw (s) less than one, i.e., Gzw ∞ < 1.
0≤ω≤∞
A PPENDIX B F ULL - AND R EDUCED -O RDER O UTPUT H∞ F EEDBACK C ONTROLLERS All the results in this Appendix are from Iwasaki and Skelton [16], Gahinet and Apkarian [18], and Xin [19]. Here, the set of all p × q constant real matrices is denoted as Rp×q . A⊥ is a matrix satisfying N(A⊥ ) = R(A) and A⊥ A⊥T > 0, where N(A) and R(A) are the null and range spaces of matrix A, respectively. Let a state-space realization of the controller L(s) be L(s) =
(43)
For a single-input single-output system [w(t) ∈ R and z(t) ∈ R], the H∞ norm of the system is the largest gain in its Bode plot (Fig. 11) Gzw ∞ = sup |Gzw (jω)|
(46)
That is, it is the maximum ratio of the output to the input energy.
and the state-space realization is dx dtg = Ag xg + Bg1 w + Bg2 uL z = Cg1 xg + Dg11 w + Dg12 uL yL = Cg2 xg + Dg21 w
z(t)2 . w(t) w(t)=0 2
Gzw ∞ = sup
(41)
(44)
AL CL
BL DL
.
(47)
From [16] and [18], H∞ control problem for the plant (42) is solvable if and only if there exists P∞ > 0 such that
Acl P∞ + P∞ AT cl Ccl P∞ T Bcl
T P∞ Ccl −I T Dcl
Bcl Dcl < 0 −I
(48)
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where Acl Ccl
Bcl Dcl Aˆ = ˆ C1
Aˆ Cˆ1 Cˆ2
=
ˆ1 ˆ2 B B ˆ ˆ ˆ 11 + D ˆ 12 L [ C2 D ˆ1 ˆ2 B B ˆ 11 ˆ 12 D D ˆ ˆT D21 L
ˆ 21 ] D
(49)
If MD = ∅ and (X∞ , Y∞ ) ∈ MD are given, then P∞ satisfies (48) X∞ P12 P∞ = (58) T P12 I where I is an identity matrix of size rank (X∞ − Y∞−1 ), and P12 has a full column rank and satisfies T = X∞ − Y∞−1 . P12 P12
Ag 0 Cg1
0 0 0
Bg1 0 Dg11
Bg2 0 Dg12
InL 0
Cg2 0
0 Inc
Dg21 0
T DL T CL
BLT AT L
0
(50)
where nL = dim(AL ). Rewriting (48)–(50) yields the following LMI: ˆL ˆ Cˆ + (B ˆL ˆ C) ˆ T+Ω0
MC (Y ) < 0 MB (X) < 0, X I ≥0 I Y
Dg21 ] + rank Dg21 .
Moreover, for any given real positive definite matrices P and W , let (Xm , Ym ) be a solution of the minimization problem min trace(P X + W Y )
ˆ by solving (51). We use the following lemma. Next, we obtain L Lemma 1 [16], [18]: The H∞ control problem for the plant (42) is solvable if and only if Md = ∅, where
X = X T > 0,
Bg2 + rank Dg12 Dg12
T + Bg1 Bg1 T Dg11 Bg1
T Y Ag + AT g Y + Cg1 Cg1 QY = T T Bg1 Y + Dg11 Cg1
T XCg1
(55)
T + Bg1 Dg11 T Dg11 Dg11 − I
(56) T Y Bg1 + Cg1 Dg11 . T Dg11 Dg11 − I (57)
[1] H. Peng and M. Tomizuka, “Preview control for vehicle lateral guidance in highway automation,” Trans. ASME, J. Dyn. Syst. Meas. Control, vol. 115, no. 4, pp. 679–686, 1993. [2] R. T. O’Brien, P. A. Iglesias, and T. J. Urban, “Vehicle lateral control for automated highway systems,” IEEE Trans. Control Syst. Technol., vol. 4, no. 3, pp. 266–273, May 1996. [3] R. H. Byrne, C. T. Abdallah, and P. Dorato, “Experimental results in robust lateral control of highway vehicles,” IEEE Control Syst. Mag., vol. 18, no. 2, pp. 70–76, Apr. 1998. [4] C. S. Kim, J. I. Bae, S. P. Hong, M. H. Lee, and F. Harashima, “H∞ steering control for the unmanned vehicle system,” in Proc. 27th Annu. Conf. IEEE IECON, 2001, pp. 2139–2143. [5] C. Hatipoglu, Ü. Özguner, and K. Redmill, “Automated lane change controller design,” IEEE Trans. Intell. Transp. Syst., vol. 4, no. 1, pp. 13–22, Mar. 2003.
SHE et al.: ESTIMATION OF EQUIVALENT INPUT DISTURBANCE IMPROVES VEHICULAR STEERING CONTROL
[6] C.-F. Lin, A. G. Ulsoy, and D. J. LeBlanc, “Vehicle dynamics and external disturbance estimation for vehicle path prediction,” IEEE Trans. Control Syst. Technol., vol. 8, no. 3, pp. 508–518, May 2000. [7] B. A. Güvenç, T. Bünte, D. Odenthal, and L. Günenç, “Robust two degree of freedom vehicle steering controller design,” in Proc. Amer. Control Conf., 2001, pp. 13–18. [8] M. Yamamoto, Y. Kagawa, and A. Okuno, “Robust control for automated lane keeping against lateral disturbance,” in Proc. IEEE/IEEJ/JSAI Int. Conf. Intell. Transp. Syst., 1999, pp. 240–245. [9] J.-H. She, M. Fang, Y. Ohyama, H. Hashimoto, and M. Wu, “Improving disturbance rejection performance based on an equivalent-inputdisturbance approach,” IEEE Trans. Ind. Electron, vol. 54, no. 6. to be published. [10] R. T. O’Brien, T. J. Urban, and P. A. Iglesias, “Lane change maneuver via modern steering control methods,” in Proc. ITS Amer. Conf., 1995, pp. 1–9. [11] H. Kumamoto, I. Sakamoto, K. Tenmoku, and H. Shimoura, “Vehicle steering control by reduced-dimension sliding mode theory,” Trans. Soc. Instrum. Contr. Eng. (SICE), vol. 34, no. 5, pp. 393–399, 1998, (in Japanese). [12] L. R. Hunt, G. Meyer, and R. Su, “Noncausal inverses for linear systems,” IEEE Trans. Autom. Control, vol. 41, no. 4, pp. 608–611, Apr. 1996. [13] T. Kailath, Linear Systems. Englewood Cliffs, NJ: Prentice-Hall, 1980. [14] B. D. O. Anderson and J. B. Moore, Optimal Control—Linear Quadratic Methods. Englewood Cliffs, NJ: Prentice-Hall, 1989. [15] K. Zhou, J. C. Doyle, and K. Glover, Robust and Optimal Control. Upper Saddle River, NJ: Prentice-Hall, 1996. [16] T. Iwasaki and R. E. Skelton, “All controllers for the H∞ control problem: LMI existence conditions and state space formulas,” Automatica, vol. 30, no. 8, pp. 1307–1317, 1994. [17] J.-H. She, X. Xin, and T. Yamaura, “Analysis and design of control system with equivalent-input-disturbance estimation,” in Proc. IEEE Int. Conf. Control Appl., 2006, pp. 1463–1469. [18] P. Gahinet and P. Apkarian, “A linear matrix inequality approach to H∞ control,” Int. J. Robust Nonlinear Control, vol. 4, no. 4, pp. 421–448, 1994. [19] X. Xin, “Reduced-order controllers for the H∞ control problem with unstable invariant zeros,” Automatica, vol. 40, no. 2, pp. 319–326, Feb. 2004.
Jin-Hua She (A’94–M’99) received the B.S. degree in engineering from Central South University, Changsha, China, in 1983 and the M.S. and Ph.D. degrees in engineering from Tokyo Institute of Technology, Tokyo, Japan, in 1990 and 1993, respectively. In 1993, he joined the Department of Mechatronics, School of Engineering, Tokyo University of Technology, and in April 2004, he transferred to the School of Bionics, Tokyo University of Technology, where he is currently an Associate Professor. His current research interests include the application of control theory, repetitive control, expert control, Internet-based engineering education, and robotics. Dr. She is a member of the Society of Instrument and Control Engineers and the Institute of Electrical Engineers of Japan. He, together with M. Wu and M. Nakano, received the Paper Prize in Control Engineering Practice from the International Federation of Automatic Control (IFAC) in 1999.
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Xin Xin (S’94–A’95–M’02) received the B.S. degree from the University of Science and Technology of China, Hefei, China, in 1987, the Ph.D. degree from Southeast University, Nanjing, China, in 1993, and the Ph.D. degree in engineering from Tokyo Institute of Technology, Tokyo, Japan, in 2000. From 1991 to 1993, he worked on his Ph.D. dissertation at Osaka University, Osaka, Japan, as a coadvised student of China and Japan, with a Japanese Government Scholarship. From 1993 to 1995, he was a Postdoctoral Researcher with Southeast University and then became an Associate Professor. From 1996 to 1997, he did research on advanced industrial technology for the New Energy and Industrial Technology Development Organization, Japan. From April 1997 to March 2000, he was a Research Associate with the Department of Control and Systems Engineering, Tokyo Institute of Technology. Since April 2000, he has been an Associate Professor with the Faculty of Computer Science and System Engineering, Okayama Prefectural University, Soja, Japan. He, together with C. B. Feng and Y. P. Tian, coauthored Robust Control System Design (in Chinese) (Nanjing: Southeast University Press, 1995) and has written over 80 journal and international conference papers. His current research interests include robotics, dynamics and control of nonlinear and complex systems, and robust control. Dr. Xin received a paper award at the 3rd Annual Conference on Control Systems of the Society of Instrument and Control Engineers of Japan in 2004.
Yasuhiro Ohyama (M’96) received the B.S., M.S., and Ph.D. degrees in engineering from Tokyo Institute of Technology, Tokyo, Japan, in 1980, 1982, and 1985, respectively. From 1985 to 1991, he was a Director of the Advanced Control Laboratory Inc., Tokyo, where he worked on developing controllers for industrial robots and on CAD systems for control design. He is currently a Professor with the School of Bionics, Tokyo University of Technology, where he researches the applications of control theory, robotics, and engineering education. Dr. Ohyama is a member of the Society of Instrument and Control Engineers and the Institute of Electrical Engineers of Japan.
IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 56, NO. 6, NOVEMBER 2007
Estimation of Equivalent Input Disturbance Improves Vehicular Steering Control Jin-Hua She, Member, IEEE, Xin Xin, Member, IEEE, and Yasuhiro Ohyama, Member, IEEE
Abstract—This paper describes a simple effective approach to vehicular steering control based on the estimation of an equivalent input disturbance. A control system configuration with a disturbance estimator and a design method based on linear matrix inequalities are explained. The simulation results demonstrate the validity of the method. Index Terms—Disturbance estimation, disturbance rejection, H∞ control, linear matrix inequality (LMI), vehicle control, vehicle dynamics, vehicle steering.
I. I NTRODUCTION
Fig. 1.
A
UTOMATIC steering control is a key element of intelligent transportation systems (see, for example, [1]–[8]). It involves two techniques: lane keeping, in which the steering control system tracks the center of the current lane, and lane changing, in which it steers in order to track a reference input for a given lateral motion. Since disturbances caused by wind and road conditions seriously affect control results, a good disturbance-rejection performance is essential and has been the focus of many studies. For example, Lin et al. [6] used a steady-state Kalman filter to estimate the lateral velocity of a vehicle and the magnitudes of external disturbances acting on it, Güvenç et al. [7] used a disturbance observer to suppress external disturbances, and Yamamoto et al. [8] devised a disturbance-estimation algorithm. However, each method has drawbacks: Lin et al. [6] only considered step-type disturbances, it is difficult to tune the filter parameters in [7] because the filter design must guarantee both the causality of the estimator and the stability of the whole control system, and [8] requires the differentiation of measured outputs. This paper addresses steering control for a straight road. We employ an equivalent-input-disturbance (EID) estimator [9] that estimates an equivalent disturbance on the control input channel to reject disturbances caused by wind, yaw disturbance torque, etc., thus providing a good steering performance. This method has three advantages: First, no a priori information about disturbances is required. The only restriction is that the Manuscript received October 10, 2005; revised July 12, 2006 and April 12, 2007. This work was supported in part by a Grant-in-Aid for Scientific Research (C), Japan Society for the Promotion of Science, under Grant 18560259. The review of this paper was coordinated by Dr. M. Abul Masrur. J.-H. She and Y. Ohyama are with the School of Bionics, Tokyo University of Technology, Tokyo 192-0982, Japan (e-mail: [email protected]; [email protected]). X. Xin is with the Faculty of Computer Science and System Engineering, Okayama Prefectural University, Soja 719-1197, Japan (e-mail: [email protected]). Digital Object Identifier 10.1109/TVT.2007.904543
Bicycle model of the vehicle (top view).
output produced by the disturbances must be bounded, but this condition is generally satisfied. Second, no differentiation of measured outputs is needed. Third, the design of the control system is very simple; the configuration can be viewed as a conventional control system enhanced by the plugging-in of a disturbance estimator. Simulation results have demonstrated the validity of this method. II. M ODEL OF V EHICLE This paper assumes that the vehicle has front-wheel steering, that there are no sudden accelerations or decelerations, and that the front wheels turn only through a small angle. Therefore, the velocity is constant, and the rolling motion can be ignored. Thus, the vehicle can be thought of as traveling on a flat surface, which is affected only by lateral and yawing motions. A further assumption is that the sideslip angle and the yaw rate are very small. Since a complicated model is not necessary to simulate the lateral motion of highway vehicles (see [10]), this paper employed the simple two-degree-of-freedom bicycle model [11] (Fig. 1). The parameters are x(y) ψ δ β v m Iz d τd lf (lr )
0018-9545/$25.00 © 2007 IEEE
x(y)-position of the center of gravity (CG) of a vehicle (in meters); yaw angle with respect to the x-axis (in radians); angle of the front wheel (in radians); sideslip angle (in radians); longitudinal velocity (in meters per second); total mass of the vehicle (in kilograms); yaw moment of inertia (in kilograms square meter); equivalent lateral disturbance force (in newtons); equivalent yaw disturbance torque (in newtons meter); distance from CG to the front (rear) axle (in meters);
SHE et al.: ESTIMATION OF EQUIVALENT INPUT DISTURBANCE IMPROVES VEHICULAR STEERING CONTROL
Fig. 2.
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Configuration of the steering control system.
ld
distance from CG to the position at which d has an effect (in meters); Cf (Cr ) cornering stiffness of the front (rear) tires (in newtons per radian). The dynamic equations of the lateral motion are the following: d2 y C +C m dt2 = − f v r dy dt + (Cf + Cr )ψ Cf lf −Cr lr dψ − v dt + Cf δ + d Cf lf −Cr lr dy d2 ψ Iz dt2 = − v dt + (Cf lf − Cr lr )ψ Cf lf2 +Cr lr2 dψ − v dt + Cf lf δ + τd τd = dld . The state-space expression is dξ dt = Aξ + Bδ + Bd d y = Cξ
(2)
where ξ = [y 0 0 A= 0 0 Bd
a1 a3 b1
dy dt
ψ
1 a1 v
0 a3 v
dψ T dt ]
0 −a1 0 −a3
0 a2 v
1 a4 v
,
(1)
0 b B = 1 0 b2
T 0 1 bd1 0 = C= , 0 0 bd2 ld 0 Cf + Cr Cf lf − Cr lr , a2 = − =− m m Cf lf2 + Cr lr2 Cf lf − Cr lr =− , a4 = − Iz Iz Cf Cf lf 1 ld , b2 = = , bd1 = , bd2 = . m Iz m Iz
A simple check of controllability and observability shows that, for this model, (A, B) is controllable for a nonzero speed and that (C, A) is observable except at two low speeds (e.g., for the nominal parameters used in Section IV, the speeds are 14.26 and −14.26 km/h). Therefore, in this paper, we consider the case in which the plant is controllable and observable. However, if the plant is not controllable or not observable, then we only need to consider its controllable and observable subsystem because we just want to produce an effective control input to achieve satisfactory steering control; the unobservable subsystem does not influence the output, and the effect of the uncontrollable subsystem on the output can be regarded as a disturbance. Moreover, a simple verification shows that this plant has no zeros on the imaginary axis, regardless of the parameters. III. D ESIGN OF C ONTROL S YSTEM An EID is a disturbance on the control input channel that produces the same effect on the controlled output as actual disturbances do [9]. In fault diagnosis, what we want to know are the exact place and kind of disturbance (fault). In contrast, in vehicular steering control, we use the steering angle δ to reject the disturbances, which makes it more reasonable to estimate an EID on the control input channel than to estimate the disturbances themselves. Since an EID always exists for a controllable and observable plant with no zeros on the imaginary axis [9], [12], we focused on its estimation. We employed the control system configuration in [9] and constructed the steering control system in Fig. 2, where B + :=
BT . BTB
(3)
Note that controllability guarantees that B T B = 0. This configuration can be viewed as a conventional control system enhanced by the plugging-in of a disturbance estimator. Since
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only the steering control for a straight road is considered here, the lane-change command can be characterized as a step signal. Therefore, a step-type servo system was selected. Remark 1: Note that a full-order state observer is employed in Fig. 2. A reduced-order state observer is also conceivable. However, when a disturbance exists, the available states of the real plant may be different from those of the plant with an EID. If we ignore this and use a reduced-order state observer, the difference between the states will produce an error in the state estimation, thereby degrading the tracking precision. In order to obtain an EID with a high precision, it is important to guarantee that the output of the observer converges to that of the real plant. This necessitates the use of a full-order state observer. The rationale behind the construction of this disturbance estimator is explained in the following section. A. Estimation and Rejection of Disturbances A full-order state observer is used to estimate the state of the vehicle dξˆ ˆ = Aξˆ + Bu + LC(ξ − ξ). (4) dt On the other hand, the following is true for the state of the plant ¯ with an EID ξ¯ and the actual value of the EID d:
dξˆ = Aξˆ + Bδ + B d¯ + B∆d. dt
(11)
In the aforementioned equation, assuming the estimate of the EID to be dˆ = d¯ + ∆d
(12)
dξˆ ˆ = Aξˆ + B(δ + d). dt
(13)
gives
Therefore, if we want the state of the plant with an EID ξ¯ to ˆ then the difference between be equal to that of the observer ξ, ¯ ˆ ξ and ξ can be described as the error in estimating the EID. Equations (4) and (13) yield ˆ B(dˆ + δ − u) = LC(ξ − ξ).
(14)
ˆ then a least-squares solution is If we solve (14) for d, ˆ + u − δ. dˆ = B + LC(ξ − ξ)
(15)
dˆ is filtered by F (s) (Fig. 2), which selects the angular frequency band for disturbance estimation, and the filtered disturbance estimate d˜ is ˆ d˜ = L−1 F (s)D(s)
dξ¯ ¯ = Aξ¯ + Bδ + B d. dt
(5)
∆ξ = ξˆ − ξ¯
(6)
ˆ ˆ and L[·] and L−1 [·] denote the Laplace where D(s) = L[d(t)], and inverse Laplace transforms, respectively. Combining d˜ with the original control law yields
(7)
˜ δ = u − d.
Letting
and substituting that into (5) yields dξˆ = Aξˆ + Bδ + B d¯ + dt
d∆ξ − A∆ξ . dt
As shown in the following discussion, there exists a control input ∆d that precisely produces the state ∆ξ. That is, the following holds: d∆ξ − A∆ξ = B∆d. dt
(8)
To prove the aforementioned statement, we construct a ∆d directly from ∆ξ. Since (A, B) is controllable, a state transformation (∆ξ = T ∆ξc ) with the following controllable canonical form [13]: d∆ξc = Ac ∆ξc + Bc ∆d dt ∆ξ = ∆ξc3 ∆ξc4 ]T c2 c [ ∆ξc1 ∆ξ 0 1 0 0 0 1 0 (9) 0 Ac = T −1 AT = 0 0 0 1 −α0 −α1 −α2 −α3 Bc = T −1 B = [ 0 0 0 1 ]T gives us ∆d ∆d =
Substituting (8) into (7) yields
d∆ξc4 + α0 ∆ξc1 + α1 ∆ξc2 + α2 ∆ξc3 + α3 ∆ξc4 . dt (10)
(16)
For this control law, the EID added onto the input channel of ˜ Therefore, the modified control law the plant reduces to d¯ − d. improves the disturbance-rejection performance. This method has the following three main features. 1) The configuration of the control system is very simple. 2) The disturbance-rejection performance can be tuned by adjusting the low-pass filter F (s) and the observer gain L. 3) The stability of the closed-loop system can be divided into two parts: the feedback gains KP and KR , and L and F (s). The only difference between this system and a conventional control system is the incorporation of a disturbance estimator, and the only design parameters that are related to the estimator are the low-pass filter and observer, which are discussed in the succeeding discussions. B. Design of Observer For a conventional control system, the closed-loop poles are those of the full state feedback plus those of the observer (separation theorem) [14]. To show that the separation theorem
SHE et al.: ESTIMATION OF EQUIVALENT INPUT DISTURBANCE IMPROVES VEHICULAR STEERING CONTROL
Fig. 3.
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Control system without exogenous inputs.
holds for our system configuration, let the exogenous inputs (r and d) be equal to zero. The vehicle is then described by dξ dt = Aξ + Bδ (17) y = Cξ. Equations (4), (6), (16), and (17) yield d∆ξ ˜ = (A − LC)∆ξ + B d. dt
(18)
On the other hand, (15) is equivalent to ˜ dˆ = −B + LC∆ξ + d.
(19)
Based from (17)–(19) and Fig. 2, we can redraw the block diagram in Fig. 2 as Fig. 3. The system can be decomposed into a lower part consisting of the state feedback and an upper part consisting of the state observer and low-pass filter. Since the state in the upper part is not controllable from the lower part and since the state in the lower part is not observable from the upper part, we can independently design the poles of these two parts, i.e., the closed-loop poles are those of the state feedback plus those of the observer and low-pass filter. Since the former can be designed by existing methods, the rest of this paper focuses on the latter. The transfer function from d˜ to dˆ is GL (s) = 1 − B + LC [sI − (A − LC)]−1 B
(20)
F (s) and L must not destroy the stability of the control system. That is, they must guarantee the stability of the upper part in Fig. 3. From the small-gain theorem [15], it follows that the stability of the upper part is guaranteed if GL F ∞ < 1.
then we obtain Fig. 4 from Fig. 3. A state-space realization of the generalized plant Gg (s) = G(s)F (s) = Dg + Cg (sI − Ag )−1 Bg
(22)
(25)
is
dxf dt d∆ξ dt
z = yL =
(21)
If we choose the angular frequency band for disturbance rejection to be Ωr = {ω : ω ≤ ωr }
then, to reject disturbances, it is best to choose F (jω) ≈ 1 in the range Ωr . For a given F (s), condition (21) and Fig. 3 necessitate an L such that |GL (jω)| < 1/|F (jω)|. In the succeeding discussion, we show that the design of L can be converted into an H∞ static output feedback synthesis problem (see Appendix A for the definition of the H∞ control problem). If we break the control loop in the upper part in Fig. 3 at the input of the filter F (s), introduce two new signals (z and w), which are the signals before and after the break point, respectively, and assume the state space representations of F (s) and G(s) to be dx f dt = Af xf + Bf w F (s) : (23) yf = Cf xf d∆ξ dt = A∆ξ + Byf − uL G(s) : z = yf − B + uL (24) yL = C∆ξ
=
Af BCf Cf 0 Ag Cg1 Cg2 Ag Cg
xf ∆ξ 0 0 −B + w uL C 0 0 xf Bg1 Bg2 ∆ξ Dg11 Dg12 w Dg21 0 uL xf ∆ξ Bg (26) w Dg uL 0 A
Bf 0
0 −I
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TABLE I NOMINAL PARAMETERS OF VEHICLE AND SIMULATION CONDITIONS
Step 2: Choose Ωr and a corresponding F (s). Step 3: Design L by solving the minimization problem for the trace of P X + W Y under the constraints of the LMIs for the H∞ control problem. IV. S IMULATIONS Simulations were carried out for a straight road using the vehicle parameters in [11] (Table I). First, we chose
Fig. 4. Formulation of the design of L using H∞ control framework.
where I is an identity matrix of size dim(A), and dim(A) is the size of the square matrix A. The design problem is thus converted into one that is designing a control law uL = LyL
Q = diag{1, 1, 10, 10, 10}, R = 10
(30)
and obtained KP and KR by optimizing the performance index ∞ J= [ ξeT
(27)
xT Re ] Q
ξe + Rδe2 dt xRe
(31)
0
such that the closed-loop transfer function from w to z, which is Gzw (s) (= GL (s)F (s)), satisfies Gzw ∞ < 1.
where ξe (t) = ξ(t) − ξ(∞)
(28)
Note that the H∞ static output feedback synthesis problem is not convex and is difficult to solve. Suppose that this problem is solvable for (26). Then, there exists a controller of order less than dim(Af ) + dim(A) [15], [16], and it can be obtained, for example, by using a solution (X, Y ) of three linear matrix inequalities (LMIs) (see Appendix B), where X and Y are positive matrices. Based on the design method for reducedorder H∞ output control [19] (see Appendix B), there exists a controller such that the bound on the order is Bg dim(Af ) + dim(A) − rank (29) + rank Dg12 . Dg12 In fact, such a controller can be obtained by minimizing the trace of P X + W Y for any given real positive definite matrices P and W under the constraints of the three LMIs. Furthermore, numerical investigations have shown that a controller of order less than the bound (29) is obtainable. By exploiting the special structure of Gg (s) in (26), we T T ] = dim(A) and rank Dg12 = 1. Thus, have rank [BgT Dg12 the order of L(s) is dim(Af ) + 1. Clearly, L(s) is not static. Therefore, we employ a minimization procedure (see Lemma 2 in Appendix B) to obtain a static controller or, in other words, the observer gain L. The aforementioned discussion is summarized in the following design algorithm for a steering control system. Design algorithm: Step 1: Design KP and KR for a conventional step-type servo system by an existing method (e.g., the optimal control method).
xRe (t) = xR (t) − xR (∞) δe (t) = δ(t) − δ(∞) for the step-type reference input r(t). The resulting KP and KR are KP = [ −1.0760 −0.3117 −5.4296 −0.7896 ] (32) KR = 1.0000. Then, we determined Ωr by setting ωr to 30 rad/s and F (s) to 1 F (s) = T s+1 , T = 0.0333 s (33) Af Bf −1/T 1/T = . Cf 0 1 0 From the last section, the order of L(s) obtained by solving the H∞ synthesis problem (26)–(28) is two. By using the new design algorithm and by solving the H∞ synthesis problem (26)–(28) with P = W = I, we obtained the static observer gain L = [ 37.307
1038.0
26.941
570.49 ]T .
(34)
The resulting transfer function is Gzw (s) =
n3 s3 + n2 s2 + n1 s + n0 s4 + d3 s3 + d2 s2 + d1 s + d0
d3 = 43.044,
d2 = 1270.1,
d1 = 5936.2
d0 = 61603,
n3 = 43.044,
n2 = 255.23
n1 = 2857.7,
n0 = 2400.3.
(35)
SHE et al.: ESTIMATION OF EQUIVALENT INPUT DISTURBANCE IMPROVES VEHICULAR STEERING CONTROL
Fig. 5.
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Bode plot of Gzw (s).
The Bode plot of Gzw (s) (Fig. 5) satisfies (28), and the control system is stable. For comparison, a conventional control system (state observer and state feedback) was also designed. The state feedback gain was chosen to be that in (32). The resulting poles of the closed-loop system with state feedback are −22.7799, −3.2265 ± j4.0487, and −1.3428 ± j1.1754. Generally speaking, the poles of the observer should be to the left of those poles on the complex plane [13]. Trial and error yielded the values −25.0, −30.0, −31.0, and −35.0. These yield the observer gain Lcv = [ 115.26
4785.6
1116.1
6840.9 ]T .
(36)
It is shown in the succeeding discussion that, although a comparison of the two systems revealed that they have almost the same response characteristics for a step reference input, the conventional control system had much poorer robustness.
Fig. 6. Simulation results for lane-changing control of vehicle with nominal parameters [conventional method: (32) and (36); new method: (32)–(34)].
A. Simulation Results for Lane-Changing Control Fig. 6 shows the lateral response (dotted line) and steering angle of a vehicle, with the nominal parameters listed in Table I under zero initial conditions for the conventional control method [(32) and (36)]. The reference input for the lane change was r = 4 × 1(t − 1)
m.
(37) Fig. 7. Disturbance.
The following disturbances (Fig. 7) were added: −3000 × [0.2 + sin 0.1π(t − 7) +2 sin 0.2π(t − 7) + sin π(t − 7) d= +0.4 sin 5π(t − 7)] N, 0,
t≥7 t < 7.
(38)
Fig. 6 also shows the response obtained by our method [(32)–(34)] (solid line). Incorporating a disturbance estimator dramatically reduced the peak-to-peak tracking error from 0.3669 to 0.0300 m, and the steady-state tracking error is only 8% of that of the conventional control system. Stability tests showed that both the conventional control system and the one obtained by our method were stable for
parameters in the following ranges: m = 1500 × (100 ± 20)% kg Iz = 3000 × (100 ± 20)% kg · m2 Cf = 50000 × (100 ± 50)% N/rad Cr = 70000 × (100 ± 50)% N/rad. Fig. 8 shows the results for the worst case m = 1800 kg, Iz = 3600 kg · m2 Cf = 25000 N/rad, Cr = 35000 N/rad
(39)
(40)
and the disturbances in (38) under zero initial conditions. Even though the conventional control system was stable, it did not maintain control of the car because the steering angle was
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Fig. 8. Simulation results for lane-changing control of vehicle with the parameters in (40).
restricted, and the disturbances forced the steering wheel all the way to the left or right (0.6981 rad = 40◦ ). However, with our method, the car followed the reference lane command with a steady-state peak-to-peak tracking error of 0.0825 m. Note that, in this case, the estimates of the equivalent disturbances d˜ are quite different from those for the nominal case because the differences between the parameters and the nominal values are considered to be a kind of disturbance (see graph of d˜ for 1 ≤ t ≤ 6 s) and that our method also estimates and restrains their influence. Therefore, even though the parameters were different from the nominal values, the disturbances were suppressed, and the car tracked the reference command with satisfactory precision. B. Simulation Results for Lane-Keeping Control For lane-keeping control (r = 0), the incorporation of a disturbance estimator also markedly reduces the tracking error. As an example, Fig. 9 shows some results for a straight road and a vehicle, with the parameters in (40) under zero initial conditions, when the disturbances in (38) were added. Also in this case, control broke down for the conventional control system. However, our method suppressed the disturbances and kept the car in the specified lane with satisfactory precision (steady-state peak-to-peak tracking error is 0.0825 m). Remark 2: The simulation results show that plugging an EID estimator into a conventional control system (state ob-
Fig. 9. Simulation results for lane-keeping control of a vehicle with the parameters in (40).
server and state feedback) improves the disturbance-rejection performance. In fact, the analysis in [17] shows that it adds a new degree of freedom to the system, and the output caused by the disturbances is related to the term 1 − F (s). Therefore, the disturbance-rejection performance can greatly be improved by choosing F (jω) ≈ 1 in the angular frequency band for disturbance rejection. V. C ONCLUSION This paper presents a new method of steering control for a vehicle with front-wheel steering. First, a method that estimates an EID was described. Then, a steering control configuration with an EID estimator was presented. Unlike existing methods, this one is very simple. Furthermore, the design of the observer, which is related to disturbance estimation, can be separated from the design of the state feedback. A design algorithm was presented, and the simulation results demonstrated the validity of the method. This paper considered steering control only for a straight road, but a general formulation must take the radius of curvature (ROC) of the road into account. The problem can then be divided into two cases, based on whether preview information on the road ROC is available. If it is available, a feedforward control scheme [1] can be employed, but if it is not available,
SHE et al.: ESTIMATION OF EQUIVALENT INPUT DISTURBANCE IMPROVES VEHICULAR STEERING CONTROL
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Fig. 10. H∞ control.
the control problem can be formulated as a disturbancerejection problem [2]. Therefore, even when the road ROC cannot be used, our method still provides an estimate of the equivalent disturbance due to road ROC, and thus, a satisfactory control can be achieved.
Fig. 11. H∞ norm of Gzw (s).
and for a multiple-input multiple-output system, it is Gzw ∞ = sup σ ¯ (Gzw (jω))
(45)
0≤ω≤∞
A PPENDIX A H∞ C ONTROL As explained in [15], in the design of the control system in Fig. 10, Gg (s) is a generalized plant containing the physical plant and some weighting functions and is linear timeinvariant; the signal w(t) ∈ Rr contains all external inputs, including references, disturbances, and sensor noise; the output z(t) ∈ Rm is a controlled error signal; uL (t) ∈ Rp is the control input; and yL (t) ∈ Rq is the observation output. The stabilizable and detectable realization of Gg (s) is
z yL
w = Gg (s) uL =
Gg11 (s) Gg21 (s)
where σ ¯ (A) is the maximum singular value of A. On the other hand, if we define
w(t)2 =
z(t)2 =
∞
0
∞
wT (t)w(t)dt
z T (t)z(t)dt
0
1/2
1/2
then the H∞ norm of the stable system Gsw (s) is Gg12 (s) Gg22 (s)
w uL
(42)
where xg ∈ Rn is the state vector. Note that the closedloop transfer function from w(t) to z(t), which is Gzw (s), is a linear fractional transform of Gg (s) and L(s) and is given by Gzw (s) = Gg11 (s) + Gg12 (s)L(s)[I − Gg22 (s) L(s)]−1 Gg21 (s). The H∞ control problem is to design an H∞ controller L(s), which makes the H∞ norm of Gzw (s) less than one, i.e., Gzw ∞ < 1.
0≤ω≤∞
A PPENDIX B F ULL - AND R EDUCED -O RDER O UTPUT H∞ F EEDBACK C ONTROLLERS All the results in this Appendix are from Iwasaki and Skelton [16], Gahinet and Apkarian [18], and Xin [19]. Here, the set of all p × q constant real matrices is denoted as Rp×q . A⊥ is a matrix satisfying N(A⊥ ) = R(A) and A⊥ A⊥T > 0, where N(A) and R(A) are the null and range spaces of matrix A, respectively. Let a state-space realization of the controller L(s) be L(s) =
(43)
For a single-input single-output system [w(t) ∈ R and z(t) ∈ R], the H∞ norm of the system is the largest gain in its Bode plot (Fig. 11) Gzw ∞ = sup |Gzw (jω)|
(46)
That is, it is the maximum ratio of the output to the input energy.
and the state-space realization is dx dtg = Ag xg + Bg1 w + Bg2 uL z = Cg1 xg + Dg11 w + Dg12 uL yL = Cg2 xg + Dg21 w
z(t)2 . w(t) w(t)=0 2
Gzw ∞ = sup
(41)
(44)
AL CL
BL DL
.
(47)
From [16] and [18], H∞ control problem for the plant (42) is solvable if and only if there exists P∞ > 0 such that
Acl P∞ + P∞ AT cl Ccl P∞ T Bcl
T P∞ Ccl −I T Dcl
Bcl Dcl < 0 −I
(48)
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IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 56, NO. 6, NOVEMBER 2007
where Acl Ccl
Bcl Dcl Aˆ = ˆ C1
Aˆ Cˆ1 Cˆ2
=
ˆ1 ˆ2 B B ˆ ˆ ˆ 11 + D ˆ 12 L [ C2 D ˆ1 ˆ2 B B ˆ 11 ˆ 12 D D ˆ ˆT D21 L
ˆ 21 ] D
(49)
If MD = ∅ and (X∞ , Y∞ ) ∈ MD are given, then P∞ satisfies (48) X∞ P12 P∞ = (58) T P12 I where I is an identity matrix of size rank (X∞ − Y∞−1 ), and P12 has a full column rank and satisfies T = X∞ − Y∞−1 . P12 P12
Ag 0 Cg1
0 0 0
Bg1 0 Dg11
Bg2 0 Dg12
InL 0
Cg2 0
0 Inc
Dg21 0
T DL T CL
BLT AT L
0
(50)
where nL = dim(AL ). Rewriting (48)–(50) yields the following LMI: ˆL ˆ Cˆ + (B ˆL ˆ C) ˆ T+Ω0
MC (Y ) < 0 MB (X) < 0, X I ≥0 I Y
Dg21 ] + rank Dg21 .
Moreover, for any given real positive definite matrices P and W , let (Xm , Ym ) be a solution of the minimization problem min trace(P X + W Y )
ˆ by solving (51). We use the following lemma. Next, we obtain L Lemma 1 [16], [18]: The H∞ control problem for the plant (42) is solvable if and only if Md = ∅, where
X = X T > 0,
Bg2 + rank Dg12 Dg12
T + Bg1 Bg1 T Dg11 Bg1
T Y Ag + AT g Y + Cg1 Cg1 QY = T T Bg1 Y + Dg11 Cg1
T XCg1
(55)
T + Bg1 Dg11 T Dg11 Dg11 − I
(56) T Y Bg1 + Cg1 Dg11 . T Dg11 Dg11 − I (57)
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[6] C.-F. Lin, A. G. Ulsoy, and D. J. LeBlanc, “Vehicle dynamics and external disturbance estimation for vehicle path prediction,” IEEE Trans. Control Syst. Technol., vol. 8, no. 3, pp. 508–518, May 2000. [7] B. A. Güvenç, T. Bünte, D. Odenthal, and L. Günenç, “Robust two degree of freedom vehicle steering controller design,” in Proc. Amer. Control Conf., 2001, pp. 13–18. [8] M. Yamamoto, Y. Kagawa, and A. Okuno, “Robust control for automated lane keeping against lateral disturbance,” in Proc. IEEE/IEEJ/JSAI Int. Conf. Intell. Transp. Syst., 1999, pp. 240–245. [9] J.-H. She, M. Fang, Y. Ohyama, H. Hashimoto, and M. Wu, “Improving disturbance rejection performance based on an equivalent-inputdisturbance approach,” IEEE Trans. Ind. Electron, vol. 54, no. 6. to be published. [10] R. T. O’Brien, T. J. Urban, and P. A. Iglesias, “Lane change maneuver via modern steering control methods,” in Proc. ITS Amer. Conf., 1995, pp. 1–9. [11] H. Kumamoto, I. Sakamoto, K. Tenmoku, and H. Shimoura, “Vehicle steering control by reduced-dimension sliding mode theory,” Trans. Soc. Instrum. Contr. Eng. (SICE), vol. 34, no. 5, pp. 393–399, 1998, (in Japanese). [12] L. R. Hunt, G. Meyer, and R. Su, “Noncausal inverses for linear systems,” IEEE Trans. Autom. Control, vol. 41, no. 4, pp. 608–611, Apr. 1996. [13] T. Kailath, Linear Systems. Englewood Cliffs, NJ: Prentice-Hall, 1980. [14] B. D. O. Anderson and J. B. Moore, Optimal Control—Linear Quadratic Methods. Englewood Cliffs, NJ: Prentice-Hall, 1989. [15] K. Zhou, J. C. Doyle, and K. Glover, Robust and Optimal Control. Upper Saddle River, NJ: Prentice-Hall, 1996. [16] T. Iwasaki and R. E. Skelton, “All controllers for the H∞ control problem: LMI existence conditions and state space formulas,” Automatica, vol. 30, no. 8, pp. 1307–1317, 1994. [17] J.-H. She, X. Xin, and T. Yamaura, “Analysis and design of control system with equivalent-input-disturbance estimation,” in Proc. IEEE Int. Conf. Control Appl., 2006, pp. 1463–1469. [18] P. Gahinet and P. Apkarian, “A linear matrix inequality approach to H∞ control,” Int. J. Robust Nonlinear Control, vol. 4, no. 4, pp. 421–448, 1994. [19] X. Xin, “Reduced-order controllers for the H∞ control problem with unstable invariant zeros,” Automatica, vol. 40, no. 2, pp. 319–326, Feb. 2004.
Jin-Hua She (A’94–M’99) received the B.S. degree in engineering from Central South University, Changsha, China, in 1983 and the M.S. and Ph.D. degrees in engineering from Tokyo Institute of Technology, Tokyo, Japan, in 1990 and 1993, respectively. In 1993, he joined the Department of Mechatronics, School of Engineering, Tokyo University of Technology, and in April 2004, he transferred to the School of Bionics, Tokyo University of Technology, where he is currently an Associate Professor. His current research interests include the application of control theory, repetitive control, expert control, Internet-based engineering education, and robotics. Dr. She is a member of the Society of Instrument and Control Engineers and the Institute of Electrical Engineers of Japan. He, together with M. Wu and M. Nakano, received the Paper Prize in Control Engineering Practice from the International Federation of Automatic Control (IFAC) in 1999.
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Xin Xin (S’94–A’95–M’02) received the B.S. degree from the University of Science and Technology of China, Hefei, China, in 1987, the Ph.D. degree from Southeast University, Nanjing, China, in 1993, and the Ph.D. degree in engineering from Tokyo Institute of Technology, Tokyo, Japan, in 2000. From 1991 to 1993, he worked on his Ph.D. dissertation at Osaka University, Osaka, Japan, as a coadvised student of China and Japan, with a Japanese Government Scholarship. From 1993 to 1995, he was a Postdoctoral Researcher with Southeast University and then became an Associate Professor. From 1996 to 1997, he did research on advanced industrial technology for the New Energy and Industrial Technology Development Organization, Japan. From April 1997 to March 2000, he was a Research Associate with the Department of Control and Systems Engineering, Tokyo Institute of Technology. Since April 2000, he has been an Associate Professor with the Faculty of Computer Science and System Engineering, Okayama Prefectural University, Soja, Japan. He, together with C. B. Feng and Y. P. Tian, coauthored Robust Control System Design (in Chinese) (Nanjing: Southeast University Press, 1995) and has written over 80 journal and international conference papers. His current research interests include robotics, dynamics and control of nonlinear and complex systems, and robust control. Dr. Xin received a paper award at the 3rd Annual Conference on Control Systems of the Society of Instrument and Control Engineers of Japan in 2004.
Yasuhiro Ohyama (M’96) received the B.S., M.S., and Ph.D. degrees in engineering from Tokyo Institute of Technology, Tokyo, Japan, in 1980, 1982, and 1985, respectively. From 1985 to 1991, he was a Director of the Advanced Control Laboratory Inc., Tokyo, where he worked on developing controllers for industrial robots and on CAD systems for control design. He is currently a Professor with the School of Bionics, Tokyo University of Technology, where he researches the applications of control theory, robotics, and engineering education. Dr. Ohyama is a member of the Society of Instrument and Control Engineers and the Institute of Electrical Engineers of Japan.
