Trackability Filtering for Underactuated Systems - Semantic Scholar
CONFIDENTIAL. Limited circulation. For review only.
Trackability Filtering for Underactuated Systems Eric E. Sandoz and Petar V. Kokotovi´c and Jo˜ao P. Hespanha
Abstract— Underactuated systems are commonplace and present a challenge in designing tracking controllers. Foremost among these are vehicles, like passenger cars and aircraft. Such systems with n outputs and p < n control inputs are not right invertible and continuous control laws cannot be designed to track arbitrary reference signals YN (t) ∈ Rn . While switching controllers can be designed to make the tracking error converge in the average, many physical actuators cannot produce the discontinuous motion required. We present an Trackability Filter design which produces an augmented reference signal as close as possible to the original reference which the underactuated system can track.
I. I NTRODUCTION As more effort is placed on control of autonomous vehicles there is a growing need for control design methods which are computationally efficient, robust and command feasible control effort by the actuators. Often for cost reasons or simple historical inertia, many systems, vehicles in particular, are underactuated, their physical actuators can only produce bounded continuous motion, and the entire position state of the vehicle is of interest. The last issue precludes the relegation of some states to zero dynamics, as their trajectories cannot be ignored. Model Predictive Control can address all of these issues [3], but at the cost of significant computational complexity. Switching methods often require control actions which are not feasible by many actuators. One could solve inverse dynamics problems to find all possible signals that the system can track, but this is an extensive offline calculation. A simpler approach would be to develop a filter which takes as input a nominal tracking signal and produces an augmented version which the underactuated system can track with zero error. This paper proposes such a Trackability Filter design. We will begin by looking at how we would design a tracking controller if our system was fully actuated. This will then guide us in dealing with the limitations imposed by an underactuated system. For the general nonlinear system ξ˙ = f (ξ, up ) x1 = h(ξ)
(1)
This work supported by Ford Motor Co. and NSF J.P. Hespanha is with Faculty of Electrical and Computer Engineering, University of California, Santa Barbara, USA
[email protected] P.V. Kokotovi´c is with Faculty of Electrical and Computer Engineering, University of California, Santa Barbara, USA [email protected] E.E. Sandoz is a graduate student at University of California, Santa Barbara, USA, [email protected]
with state ξ, input up and output x1 , who’s normal form is z˙ x˙ 1 x˙ 2 x˙ r
= = = .. . =
φ(z, x) x2 x3 Lrf h(T −1 (x)) + Lg Lf h(T −1 (x))up n
(2)
p
Where each xi ∈ R , i = 1, . . . , r and up ∈ R , p < n. The system has uniform vector relative degree r and and it is assumed that Lg Lf h(·) has uniform full column rank p and is Lipschitz continuous on a domain D with Lipschitz rank L. Because Lg Lf h(·) is not square, it has no inverse and the system is not right invertible thus tracking of arbitrary signals YN (t) ∈ C r is not possible. The normal form of system (1) shown in (2) is produced through the diffeomorphism x = T (ξ) where h(ξ) x1 Lf h(ξ) x2 T (ξ) = (3) = .. .. . . Lr−1 h(ξ) f
xr
See [7][6] for details. Only systems with input to state stable (ISS) zero dynamics will be addressed here, as they may be neglected henceforth without changing the control design. When confronted with a system which is not right invertible as is the case when the dimension of up is p < n ¯ 1 ) ∈ Rp . it is common to choose a new output function h(x Then compute a new normal form for a system of dimension rp which is right invertible, leaving an (n − p)r dimensional system of zero dynamics. How does one choose which states ¯ 1 ) and or combination of states to ignore when picking h(x will the zero dynamics be stable? What if one cares about all states in x1 and has performance goals for them? A good example of this problem exists in both ground and aerial vehicles. In ground vehicles the position states consist of X and Y position in a plane and a heading or yaw angle ψ. For straight line maneuvers (i.e. reference signals) it would ¯ = [Y ψ]T , but what about a 90◦ be sufficient to choose h turn? All three variables change during a turn maneuver, and ignoring any one of them may produce unacceptable deviation from a nominal maneuver path. It is commonplace in the active steering/steering control literature [3] to consider only the lateral deviation state Y and heading angle ψ and to ignore X (i.e. pushing X into zero dynamics). This is valid for maneuvers like lane changes where the heading angle is only perturbed slightly, but this approach is not sufficient for large heading angle maneuvers. Similarly for aerial vehicles
Preprint submitted to 2008 American Control Conference. Received September 21, 2007.
CONFIDENTIAL. Limited circulation. For review only. it is commonplace to push the roll and pitch angle dynamics into the zero dynamics, as for most maneuvers these states should be bounded in the neighborhood of zero. But when a loop or a roll maneuver is commanded these states become the ones which must track a reference and others become less important. Our approach is to provide a tool which can smoothly handle all of these cases without moving to a hybrid switching design between a set of controllers for different operational cases. We wish to solve a tracking problem and assume that we have a reference signal YN (t) and its first r time derivatives. Our goal is to choose a control law for up such that x1 (t) → YN (t) as t → ∞. Let us for convenience rename the reference signal and its derivatives. Y1 Y˙ i Y˙ r
:= YN := Yi+1 , i = 1, . . . , r − 1 := Yu
(4)
Which is in the same block chain of integrators form as (2). To solve the tracking problem we form the tracking error coordinates as ei
:= xi − Yi , i = 1, . . . , r
(5)
Then our tracking error system becomes e˙ 1 e˙ 2 e˙ r
= e2 = e3 .. . = Fr (x) + Gr (x)u
(6)
linearization control approach [7] choosing our control law to be u ˆp = Gr (x)−1 (Ke − Fr (x)) e := (eT1 eT2 · · · eTr )T
(9)
Which would produce the following exponentially stable error system e˙ = Ae e
(10)
Which, for the positive definite radially unbounded Lyapunov function Ve = eT Pe e would produce the following Lyapunov function derivative V˙ e
=
2eT Pe Ae e = −eT Qe e < 0, ∀x 6= 0
(11)
Which implies that V (t) converges to zero and since Pe is positive definite, this implies that our tracking error e1 (t) → 0 as t → ∞. Unfortunately our actual control signal is not of dimension n, the matrix Gr (x) is not square and thus its inverse does not exist. We cannot impliment the control law of (9) as a result. However we will use (9) with our modified Trackable Filter system presented in the next section. This paper is orgainized into the following five sections: Section 2 discusses the concept of Trackable Filtering followed by Section 3 which constructively develops the control law for the Trackable Filtering concept. Section 4 presents a simulation example of the design method applied to a standard front wheel steering car model which is a common example of an underactuated system. Future work is presented in Section 5 and concluding remarks are made in Section 6.
where x := (xT1 xT2 · · · xTr )T (7) Fr (x) := Lrf h(T −1 (x)), Gr (x) := Lg Lf h(T −1 (x)) Henceforth, a vector without a numbered subscript like x refers to the stack of all of the n-dimensional sub-state vectors as in (7). a feedback gain matrix Ke = Choose Ke,1 Ke,2 · · · Ke,r , Ke,i ∈ Rn×n such that the matrix 0 In 0 ··· 0 .. 0 0 In . (8) Ae = . .. In Ke,1 Ke,2 Ke,3 · · · Ke,r is Hurwitz and Pe = PeT > 0 is the solution to the Lyapunov equation Pe Ae + ATe Pe = −Qe for some positive definite symmetric matrix Qe . At this point if our system was fully actuated then our control signal would be u ˆp ∈ Rn (as opposed to our actual p control up ∈ R , p < n) and Gr (x) was uniformly invertible over all x in a domain of interest, then following the feedback
II. T RACKABILITY F ILTERING C ONCEPT Systems with n outputs and p < n inputs are not right invertible and continuous control laws cannot be designed to track any arbitrary reference signal Y1 (t) ∈ C r . The best that we can hope for is follow Y1 (t) as closely as possible and maintain a bounded tracking error. Furthermore, we would like to design a controller which allows the control designer the freedom to choose in which states or combination of states the tracking error is injected in and to be able to compute for known reference signals a tracking error tollerance a priori. In [1] a method utilizing nonlinear damping for dealing with this problem is presented but it does not allow as much freedom to control to which states the tracking error goes. Motion our system output will lie on a p-dimensional manifold in n-dimensional space. This is clear from the zero dynamics evolving on their own. Ignoring the ISS zero dynamics of (2) φ(z, x) we construct a new normal form with ¯ the output function h(x) ∈ Rp . We use the same form of the coordinate transformation T (x) as in (3) and the control law (9), with the only difference being dimension. The result is an exponentially stable tracking error system and ¯ z , e, Y ) which possess unknown stability zero dynamics φ(¯
Preprint submitted to 2008 American Control Conference. Received September 21, 2007.
CONFIDENTIAL. Limited circulation. For review only. properties and are driven by the reference signal Y which acts as a disturbance to the zero dynamics. ¯ z , e, Y ) z¯˙ = φ(¯ e¯˙ = A¯e e¯
(12)
Only the r-dimensional error vector output e1 can be con¯ z , 0, Y ). From this trolled. When tracking, e ≡ 0 and z¯˙ = φ(¯ it is clear that the zero dynamics are completely determined by signals outside our control. Thus we can only move our system output in an p-dimensional manifold. What we would like to do is construct a reference signal lying within this manifold which is as close as possible to our nominal reference signal. Such a signal can be tracked with zero error and represents the best that our underactuated system can do. Let YN (t) be the nominal reference signal which we would like to track, but which may not lie within our trackable pdimensional manifold. Then let the trackable reference signal be described as YT
= YN + d1
(13)
Where d1 represents the vector between YN and the trackable manifold, and is the output of an ISS relative degree r filter. d˙1 d˙2 d˙r
overdetermined problem and can easily satisfy the condition in (17). Since our total control vector now is u = [uTp uTd ]T ∈ n+p R and we have an excess of degrees of freedom, there are multiple ways to solve this problem. We want to choose ud carefully as it will act as a disturbance to the ISS trackablity filter. Furthermore, as ud will inevitably be a function of d, a poor choice could render the trackablity filter unstable. Intuitively we would like the plant control signal up to do as much work as it possibly can, as anything not done by up must be done by ud which will only inject error between our nominal reference signal and the one which we can track. Thus minimizing ud is desirable. For clarity let Gr (x)up − ud Gr (x)up − ud
The plant control up can only effect vectors lying in the range space of Gr (x). To make sure up does as much as it can, let us decompose v into the sum of two vectors with one lying in the range of Gr (x). We will now drop functional dependencies where they may clutter equations. Let v T G⊥ r Gr T ⊥ G⊥ r Gr
= d2 = d3 .. . = Kd d + u d
(14)
We begin our design by constructing new error variables. Let = Yi + di = xi − YT,i = xi − Yi − di
(15)
Then our error system becomes
e˙ r
= e2 = e3 .. . = Fr (x) + Gr (x)up − Yu − Kd d − ud
(16)
From our ideal control law in (9), we would like to render Fr (x) + Gr (x)up − Yu − Kd d − ud
=
(19)
0n−p×p
= In−p
Where is orthogonal to Gr and each column is of unit length such that B = [Gr G⊥ r ] forms a basis set of vectors, enabling us to perform the decomposition. We now have Gr u p − u d
= Ke e (17)
as that would make (16) equivalent to the exponentially stable system (10) which guarantees that x1 (t) → YT (t) as t → ∞. While up lacks sufficient degrees of freedom on its own to accomplish this task, ud ∈ Rn thus we have an
= Gr vp + G⊥ r vd
(20)
Moving the Gr vp to the left hand side of the equation we get Gr (up − vp ) − ud
= G⊥ r vd
(21)
Choosing up = vp our problem reduces to
III. C ONSTRUCTIVE T RACKABLE F ILTER D ESIGN
e˙ 1 e˙ 2
= Gr vp + G⊥ r vd
G⊥ r
Where ud ∈ Rn is a control term to be chosen later. The Trackable Filter concept simply produces a filter which generates, from the nominal reference YN (t) a signal YT (t) which lies within the trackable p-dimensional manifold of motion for our plant, and the plant controller up is designed to track YT (t).
YT,i ei
= −Fr (x) + Yu (t) + Kd d + Ke e = v(e, d, t) (18)
ud = −G⊥ r vd
(22) T
We now compute vd and vp . Left multiplying (19) by G⊥ r removes the vp term due to the orthogonality of Gr and G⊥ r T and yields vd = G⊥ v. Following a similar approach to find r vp we left multiply (19) by GTr producing GTr Gr vp = GTr v. By assumption Gr has uniform full column rank, or rank p, the p × p matrix GTr Gr is thus uniformly invertible, and vp = (GTr Gr )−1 GTr v. Our two control laws are summarized below (GTr Gr )−1 GTr v
up
=
ud v
⊥ = −G⊥ r Gr v = −Fr + Yu + Kd d + Ke e
T
(23)
Applying these two control laws to (16) and (14). We get e˙ 1 e˙ 2 e˙ r
= e2 = e3 .. . = Ke e
Preprint submitted to 2008 American Control Conference. Received September 21, 2007.
(24)
CONFIDENTIAL. Limited circulation. For review only. which is exponentially stable, and equivalent to e˙ = Ae e. The trackablity filter system is now d˙1 d˙2 d˙r H
= d2 = d3 .. . = Kd d − H(−Fr + Yu + Kd d + Ke e) =
V˙ ≤ −µe |e|2 − (µd − c1 − c2 L)|d|2 + c2 L|d||e| + |d|(c2 LY¯ + c3 Y¯u ) reassigning constants c4 = µd − c1 − c2 L, c5 = c2 L, and c6 = c2 LY¯ + c3 Y¯u simplifies to
(25)
⊥T G⊥ r Gr
V˙ ≤ −µe |e|2 − c4 |d|2 + c5 |e||d| + c6 |d| κc2
The cross term can be bounded by c5 |e||d| ≤ 25 |e|2 + κ1 |d|2 where κ > 0 is a free parameter to assist the analysis
Which we can rewrite as d˙ = Ad d + Bd ∆ 0 .. Bd = . ∆
Collecting terms, particularly with respect to |d|2 we get
In = −H(−Fr + Yu + Kd d + Ke e)
(26) (27) (28)
Where Pd = PdT > 0 is the positive definite solution to the Lyapunov equation Pd Ad + ATd Pd = −Qd for some positive definite symmetric matrix Qd .
V˙
= −eT Qe e − dT Qd d −2dT Pd Bd H(−Fr + Yu + Kd d + Ke e) (29)
The product Prd = Pd Bd selects the last n columns of Pd , or the rth block column of Pd . We now search for upper bounds on our remaining terms, all constants are positive. Let c1 = 2kPdr HKd k if this tighter bound can be computed. Generally is a function H(x, YT , d) and not a constant, however it is uniformly rank p and norm 1. If the first bound cannot be computed, then use the more conservative bound c1 = 2kPdr kkKd k ≥ 2kPdr HKd k. Next, by assumption Fr (x) is Lipschitz thus kFr (x)k ≤ L|x| on some domain D ⊂ Rrn and we can bound the second term with 2kPdr HFr k ≤ c2 L(|e| + |Y | + |d|) where c2 = 2kPdr k ≥ 2kPdr Hk. Again tighter bounds are possible if structure of H is known for all possible x. Since x = e+Y +d its norm can be upper bounded by |x| ≤ |e|+Y¯ +|d|, where |Y (t)| ≤ Y¯ , ∀t. Let c3 = 2kPrd Ke k. Next we have to consider our disturbance term 2kPdr HYu k ≤ c2 Y¯u where |Yu (t)| ≤ Y¯u . Let µe , µd > 0 be the smallest eigenvalues of Qe , Qd respectively. Now we can upper bound our Lyapunov function derivative. V˙ ≤ −µe |e|2 − µd |d|2 + c1 |d|2 + c2 L|d|(|e| + Y¯ + |d|) + c3 |d|Y¯u
(30)
≤
V˙ V˙
≤
−c7 |e| − c8 |d|2 + c6 |d| κc2 c7 = µe − 5 2 1 c8 = c4 − (31) κ Finally arranging terms to examine the bounded disturbance c6 we have
A. Stability Analysis Our last step is to ensure that the trackability filter is ISS with respect to bounded input Yu (t) ∈ L∞ ∩ C 0 and exponentially decaying input e(t) ∈ L2 ∩ L∞ . Of clear concern is the term −HKd d which is potentially working against the stabilizing term Kd d. The matrix H has rank n − p, thus has p eigenvalues at 0, and its norm is unity kHk = 1. With the Lyapunov function V = eT Pe e + dT Pd d its derivative is
V˙
−c7 |e| − |d|(c8 |d| − c6 ) c6 ≤ 0, when |d| ≥ c8
(32)
Our Lyapunov function will decrease along the trajectories of e(t) and d(t) when |d| > cc68 . The necessary conditions for a stable design on a domain D are summarized below µd
> c1 + c2 L +
1 κ
κc22 L2 2 Rewriting the necessary conditions in terms of the system and design parameters a stable design requires µe
>
µd
> 2kPdr HKd k + 2kPdr HkL +
µe
>
κkPdr Hk2 L2
=
⊥T G⊥ r Gr
H
1 κ (33)
for any κ > 0. In summary when the control law of (23) is employed, the necessary conditions of (33) must be met for a stable design. B. Design Extensions Recall that our choice of ud was non-unique and many different choices can be made. For the example of car steering they system has three outputs and two control inputs. This means that ud needs to produce only one dimension of information. We can choose a different basis set of vectors than Gr and G⊥ r . One could choose to inject tracking error only in the lateral speed direction for example, thereby having up focus exclusively on on the forward and angular velocity directions. Such an approach guarantees that the tracking error is predictably injected only into one of the three states that make up d1 . This gives the designer a good deal of flexibility to control how and where the tracking error goes. The stability analysis doesn’t change, only the contents of the H matrix do.
Preprint submitted to 2008 American Control Conference. Received September 21, 2007.
CONFIDENTIAL. Limited circulation. For review only. Another appealing option is to make the basis B a uniformly orthonormal function of d. Then designing a supervisory control which changes which states or combination of states gets the error injection. If one has a preview of the reference signal, then some basis choices will result in less overall error than others. A clear example is when the reference signal lies within the tracking manifold MT , then our choice in the previous section of B = [Gr G⊥ r ] would yield zero tracking error, while any other choice would yield more. Additionally if a path following method is employed then depending on the path the error injection direction can be adjusted to coincide with the direction in which the path speed can be modulated to compensate, see [1],[8]. Another option is to use a system model in strict feedback form as opposed to normal form. The Trackability Filter can also be modeled to similarly to the plant so that better intuition can be maintained with respect to design parameters and physical states e.g. x˙ 1 x˙ 2
x˙ r
= f1 (x1 ) + g1 (x1 )x2 = f2 (¯ x2 ) + g2 (¯ x2 )x3 .. . = fr (x) + gr (x)up (34)
where x¯i = Filter as d˙1 d˙2 d˙r
[xT1
xT2
···
xTi ]T .
Then design the Trackability
= Ad,1 + g1 (x1 )d2 −1 T = −Pd,2 g1 Pd,1 d1 + Ad,2 d2 + g2 (¯ x2 )d3 .. . −1 T = −Pd,r gr−1 Pd,r−1 dr−1 + Ad,r dr + ud
mentioned throughout the paper. We employ the Bicycle Model for steering dynamics [10]. Now let us look at the car steering problem as it fits the vectorial strict feedback form. (Rolling resistance is ignored). cos ψ − sin ψ 0 vu cos ψ 0 vv ξ˙1 = sin ψ (36) 0 0 1 vψ −Cd vu2 + 2vv vψ c1 0 ˙ξ2 = −2vu vψ + c1 Fvr (ξ2 ) + 0 c1 Fu (37) Fvf c3 Fvr (ξ2 ) 0 c2 c1 = 2/m, c2 = 2a/I, c3 = − 2b/I (38) Where m = 1533 kg is the vehicle mass, a = 1.04 m is the distance from the center of gravity (CG) to the front axle, b = 1.65 m the distance from CG to rear axle, and I = 2712 kg m2 is the moment of inertia. A lumped parameter Cd = 0.000321 contains the coefficient of aerodynamic drag and vehicle frontal cross-sectional area. Note that G2 is a constant matrix. An important point is our choice of the forward tire force Fu and lateral tire force Fvf as control variables. Of course wheel torque and wheel angle are the real control variables. We utilize the Pacejka tire model [13] which is a static nonlinear mapping between vehicle states and wheel angle to tire forces Fu and Fvf . For the region of normal operation, where the tire is not sliding, this function is invertible. Thus, provided our controller only commands forces achievable by the tire through a choice of wheel angle and torque, then we are free to work with whichever variable is most convenient. ξ˙1 ξ˙2
(35) This results in a block diagonal Pd matrix, and the Hurwitz Ad,i matrices can be chosen to weight error injection in different states specifically. This system and the normal form system are algebraically equivalent, however while this new form affords more intuitive selection of design parameters, the calculations are more cumbersome. While this paper focuses on a continuous control law from the practical perspective that many mechanical actuators cannot produce switching inputs, such switching controls, like switching between different basis choices with a hybrid supervisory control can be accommodated through dynamic extension. The systems can be extended to relative degree r + 1 or r + 2 and a switching design applied at that stage guaranteeing that the control law is C 0 or C 1 respectively. It is enticing to use switching approaches on the d system as it is not bound by physical limitations, but care must be taken so that discontinuities are not imposed on up if its actuators cannot supply them. IV. E XAMPLE : F RONT WHEEL STEERING CAR An example of the Trackability Filtering design method presented above was applied to the car steering problem
= G1 (ξ1 )ξ2 = f2 (ξ2 ) + G2 up
(39)
The model is in strict feedback form which is easy to place in normal form through the diffeomorphism x = T (ξ) based on the following Lie derivatives. h(ξ1 ) = ξ1 = x1 Lf h(ξ) = G1 ξ1 = x2 ∂G1 ξ2 G1 ξ2 + G1 f2 L2f h(ξ) = ∂ξ1 Lg Lf h(ξ) = G1 G2
(40)
Which gives us the normal form representation x˙ 1 x˙ 2
= x2 = Fr (T −1 (x)) + Gr (T −1 (x))up
(41)
First we are going to choose our basis B = [Bp Bd ] where 1 0 Bp = 0 0 0 1 0 Bd = 1 0
Preprint submitted to 2008 American Control Conference. Received September 21, 2007.
CONFIDENTIAL. Limited circulation. For review only. Which means we are partitioning our control goal vector v as v vp
= Bp vp + Bd vd = BpT v
vd
= BdT v (42)
Our control law is then up
= BpT v
ud v
= Bd (BdT Gr BpT − BdT )v = −Fr + Yu + Kd d + Ke e
(43)
The simulation results of this design are shown in the following two figures. Fig. 1 shows the trajectory (solid line) Fig. 2. Tracking error trajectory with respect to time. Top: Position error. Bottom: Heading angle error.
car. The method allowed the designer the freedom to choose where the tracking error got injected demonstrating the practical utility of the method. R EFERENCES
Fig. 1. Vehicle trajectory in the (x,y) plane. Heading is shown by a triangle within the car image.
of the vehicle compared to the reference (dashed line) for a double lane change maneuver. In our design we chose our basis such that all error got injected into the lateral (Y) distance state. Fig. 2 shows clearly that the angular error and X direction error exponentially converge to zero, while the only nonzero error is in the lateral deviation. V. F UTURE W ORK In the future we plan to develop an optimal trackability filter which yields the trackable reference which is closest to the nominal signal in some sense. After that we will use this method to address tracking controller designs for systems with unstable zero dynamics. We will also explore other forms of trackability filters for systems with state and control limitations.
[1] E. E. Sandoz and P. V. Kokotovi´c, Continuous Path Following Control for Underactuated Systems With Bounded Actuation, Proc. of Intelligent Vehicles Symposium 2007, June 13-15, Istanbul. [2] R. Skjetne and T. I. Fossen and P. V. Kokotovi´c, Robust Output Maneuvering for a Class of Nonlinear Systems, Automatica J., vol. 40, 2004, pp 272-283. [3] P. Falcone and F. Borrelli and J. Asgari and H.E. Tseng and D. Hrovat, Predictive Active Steering Control for Autonomous Vehicle Systems, IEEE Transactions on Control Systems Technology, vol. 15, no. 3, May 2007, pp. 566-580. [4] M. M. Seron and J. H. Braslavsky and P. V. Kokotovi´c and and D. Q. Mayne, Feedback Limitations in Nonlinear Systems: From Bode Integrals to Cheap Control, IEEE Transactions on Automatic Control, vol. 44, no. 4, April 1999. [5] M. M. Seron and J. H. Braslavsky and G. C. Goodwin, Fundamental Limitations in Filtering and Control, Springer, London; 1997. [6] H. K. Khalil, Nonlinear Systems, Prentice Hall, New Jersey; 2002. [7] A. Isidori, Nonlinear Control Systems, Springer, London; 1995. [8] D. B. Daˇci´c and M. V. Subbotin and P. V. Kokotovi´c, Path-Following Approach to Control Effort Reduction of Tracking Feedback Laws, Automatica J., vol. 40, 2004, pp 373-383. [9] D. B. Daˇci´c and M. V. Subbotin and P. V. Kokotovi´c, ”Path-Following for a Class of Nonlinear Systems with Unstable Zero Dynamics”,43rd IEEE Conf. on Decision and Control, 2004, pp 4915-4920. [10] U. Kiencke and L. Nielsen, Automotive Control Systems, Springer, Berlin; 2000. [11] F. Rehman, Discontinuous Steering Control for Nonholonomic Systems With Drift, Nonlinear Analysis, vol. 63, 2005, pp 311-325. [12] M. Kristi´c and I. Kanellakopoulos and P. Kokotovi´c, Nonlinear and Adaptive Control Design, Wiley Interscience, New York, 1995. [13] E. Bakker and L. Nyborg and H. B. Pacejka, ”Tyre Modelling for Use in Vehicle Dynamics Studies”, SAE Technical Paper Series, 1987, pp 1-15.
VI. C ONCLUSIONS A Trackability Filter based tracking control design was developed and necessary conditions for a stable design were presented. The method was tested in simulation on a practical example of an underactuated system, a front wheel steering
Preprint submitted to 2008 American Control Conference. Received September 21, 2007.
Trackability Filtering for Underactuated Systems Eric E. Sandoz and Petar V. Kokotovi´c and Jo˜ao P. Hespanha
Abstract— Underactuated systems are commonplace and present a challenge in designing tracking controllers. Foremost among these are vehicles, like passenger cars and aircraft. Such systems with n outputs and p < n control inputs are not right invertible and continuous control laws cannot be designed to track arbitrary reference signals YN (t) ∈ Rn . While switching controllers can be designed to make the tracking error converge in the average, many physical actuators cannot produce the discontinuous motion required. We present an Trackability Filter design which produces an augmented reference signal as close as possible to the original reference which the underactuated system can track.
I. I NTRODUCTION As more effort is placed on control of autonomous vehicles there is a growing need for control design methods which are computationally efficient, robust and command feasible control effort by the actuators. Often for cost reasons or simple historical inertia, many systems, vehicles in particular, are underactuated, their physical actuators can only produce bounded continuous motion, and the entire position state of the vehicle is of interest. The last issue precludes the relegation of some states to zero dynamics, as their trajectories cannot be ignored. Model Predictive Control can address all of these issues [3], but at the cost of significant computational complexity. Switching methods often require control actions which are not feasible by many actuators. One could solve inverse dynamics problems to find all possible signals that the system can track, but this is an extensive offline calculation. A simpler approach would be to develop a filter which takes as input a nominal tracking signal and produces an augmented version which the underactuated system can track with zero error. This paper proposes such a Trackability Filter design. We will begin by looking at how we would design a tracking controller if our system was fully actuated. This will then guide us in dealing with the limitations imposed by an underactuated system. For the general nonlinear system ξ˙ = f (ξ, up ) x1 = h(ξ)
(1)
This work supported by Ford Motor Co. and NSF J.P. Hespanha is with Faculty of Electrical and Computer Engineering, University of California, Santa Barbara, USA
[email protected] P.V. Kokotovi´c is with Faculty of Electrical and Computer Engineering, University of California, Santa Barbara, USA [email protected] E.E. Sandoz is a graduate student at University of California, Santa Barbara, USA, [email protected]
with state ξ, input up and output x1 , who’s normal form is z˙ x˙ 1 x˙ 2 x˙ r
= = = .. . =
φ(z, x) x2 x3 Lrf h(T −1 (x)) + Lg Lf h(T −1 (x))up n
(2)
p
Where each xi ∈ R , i = 1, . . . , r and up ∈ R , p < n. The system has uniform vector relative degree r and and it is assumed that Lg Lf h(·) has uniform full column rank p and is Lipschitz continuous on a domain D with Lipschitz rank L. Because Lg Lf h(·) is not square, it has no inverse and the system is not right invertible thus tracking of arbitrary signals YN (t) ∈ C r is not possible. The normal form of system (1) shown in (2) is produced through the diffeomorphism x = T (ξ) where h(ξ) x1 Lf h(ξ) x2 T (ξ) = (3) = .. .. . . Lr−1 h(ξ) f
xr
See [7][6] for details. Only systems with input to state stable (ISS) zero dynamics will be addressed here, as they may be neglected henceforth without changing the control design. When confronted with a system which is not right invertible as is the case when the dimension of up is p < n ¯ 1 ) ∈ Rp . it is common to choose a new output function h(x Then compute a new normal form for a system of dimension rp which is right invertible, leaving an (n − p)r dimensional system of zero dynamics. How does one choose which states ¯ 1 ) and or combination of states to ignore when picking h(x will the zero dynamics be stable? What if one cares about all states in x1 and has performance goals for them? A good example of this problem exists in both ground and aerial vehicles. In ground vehicles the position states consist of X and Y position in a plane and a heading or yaw angle ψ. For straight line maneuvers (i.e. reference signals) it would ¯ = [Y ψ]T , but what about a 90◦ be sufficient to choose h turn? All three variables change during a turn maneuver, and ignoring any one of them may produce unacceptable deviation from a nominal maneuver path. It is commonplace in the active steering/steering control literature [3] to consider only the lateral deviation state Y and heading angle ψ and to ignore X (i.e. pushing X into zero dynamics). This is valid for maneuvers like lane changes where the heading angle is only perturbed slightly, but this approach is not sufficient for large heading angle maneuvers. Similarly for aerial vehicles
Preprint submitted to 2008 American Control Conference. Received September 21, 2007.
CONFIDENTIAL. Limited circulation. For review only. it is commonplace to push the roll and pitch angle dynamics into the zero dynamics, as for most maneuvers these states should be bounded in the neighborhood of zero. But when a loop or a roll maneuver is commanded these states become the ones which must track a reference and others become less important. Our approach is to provide a tool which can smoothly handle all of these cases without moving to a hybrid switching design between a set of controllers for different operational cases. We wish to solve a tracking problem and assume that we have a reference signal YN (t) and its first r time derivatives. Our goal is to choose a control law for up such that x1 (t) → YN (t) as t → ∞. Let us for convenience rename the reference signal and its derivatives. Y1 Y˙ i Y˙ r
:= YN := Yi+1 , i = 1, . . . , r − 1 := Yu
(4)
Which is in the same block chain of integrators form as (2). To solve the tracking problem we form the tracking error coordinates as ei
:= xi − Yi , i = 1, . . . , r
(5)
Then our tracking error system becomes e˙ 1 e˙ 2 e˙ r
= e2 = e3 .. . = Fr (x) + Gr (x)u
(6)
linearization control approach [7] choosing our control law to be u ˆp = Gr (x)−1 (Ke − Fr (x)) e := (eT1 eT2 · · · eTr )T
(9)
Which would produce the following exponentially stable error system e˙ = Ae e
(10)
Which, for the positive definite radially unbounded Lyapunov function Ve = eT Pe e would produce the following Lyapunov function derivative V˙ e
=
2eT Pe Ae e = −eT Qe e < 0, ∀x 6= 0
(11)
Which implies that V (t) converges to zero and since Pe is positive definite, this implies that our tracking error e1 (t) → 0 as t → ∞. Unfortunately our actual control signal is not of dimension n, the matrix Gr (x) is not square and thus its inverse does not exist. We cannot impliment the control law of (9) as a result. However we will use (9) with our modified Trackable Filter system presented in the next section. This paper is orgainized into the following five sections: Section 2 discusses the concept of Trackable Filtering followed by Section 3 which constructively develops the control law for the Trackable Filtering concept. Section 4 presents a simulation example of the design method applied to a standard front wheel steering car model which is a common example of an underactuated system. Future work is presented in Section 5 and concluding remarks are made in Section 6.
where x := (xT1 xT2 · · · xTr )T (7) Fr (x) := Lrf h(T −1 (x)), Gr (x) := Lg Lf h(T −1 (x)) Henceforth, a vector without a numbered subscript like x refers to the stack of all of the n-dimensional sub-state vectors as in (7). a feedback gain matrix Ke = Choose Ke,1 Ke,2 · · · Ke,r , Ke,i ∈ Rn×n such that the matrix 0 In 0 ··· 0 .. 0 0 In . (8) Ae = . .. In Ke,1 Ke,2 Ke,3 · · · Ke,r is Hurwitz and Pe = PeT > 0 is the solution to the Lyapunov equation Pe Ae + ATe Pe = −Qe for some positive definite symmetric matrix Qe . At this point if our system was fully actuated then our control signal would be u ˆp ∈ Rn (as opposed to our actual p control up ∈ R , p < n) and Gr (x) was uniformly invertible over all x in a domain of interest, then following the feedback
II. T RACKABILITY F ILTERING C ONCEPT Systems with n outputs and p < n inputs are not right invertible and continuous control laws cannot be designed to track any arbitrary reference signal Y1 (t) ∈ C r . The best that we can hope for is follow Y1 (t) as closely as possible and maintain a bounded tracking error. Furthermore, we would like to design a controller which allows the control designer the freedom to choose in which states or combination of states the tracking error is injected in and to be able to compute for known reference signals a tracking error tollerance a priori. In [1] a method utilizing nonlinear damping for dealing with this problem is presented but it does not allow as much freedom to control to which states the tracking error goes. Motion our system output will lie on a p-dimensional manifold in n-dimensional space. This is clear from the zero dynamics evolving on their own. Ignoring the ISS zero dynamics of (2) φ(z, x) we construct a new normal form with ¯ the output function h(x) ∈ Rp . We use the same form of the coordinate transformation T (x) as in (3) and the control law (9), with the only difference being dimension. The result is an exponentially stable tracking error system and ¯ z , e, Y ) which possess unknown stability zero dynamics φ(¯
Preprint submitted to 2008 American Control Conference. Received September 21, 2007.
CONFIDENTIAL. Limited circulation. For review only. properties and are driven by the reference signal Y which acts as a disturbance to the zero dynamics. ¯ z , e, Y ) z¯˙ = φ(¯ e¯˙ = A¯e e¯
(12)
Only the r-dimensional error vector output e1 can be con¯ z , 0, Y ). From this trolled. When tracking, e ≡ 0 and z¯˙ = φ(¯ it is clear that the zero dynamics are completely determined by signals outside our control. Thus we can only move our system output in an p-dimensional manifold. What we would like to do is construct a reference signal lying within this manifold which is as close as possible to our nominal reference signal. Such a signal can be tracked with zero error and represents the best that our underactuated system can do. Let YN (t) be the nominal reference signal which we would like to track, but which may not lie within our trackable pdimensional manifold. Then let the trackable reference signal be described as YT
= YN + d1
(13)
Where d1 represents the vector between YN and the trackable manifold, and is the output of an ISS relative degree r filter. d˙1 d˙2 d˙r
overdetermined problem and can easily satisfy the condition in (17). Since our total control vector now is u = [uTp uTd ]T ∈ n+p R and we have an excess of degrees of freedom, there are multiple ways to solve this problem. We want to choose ud carefully as it will act as a disturbance to the ISS trackablity filter. Furthermore, as ud will inevitably be a function of d, a poor choice could render the trackablity filter unstable. Intuitively we would like the plant control signal up to do as much work as it possibly can, as anything not done by up must be done by ud which will only inject error between our nominal reference signal and the one which we can track. Thus minimizing ud is desirable. For clarity let Gr (x)up − ud Gr (x)up − ud
The plant control up can only effect vectors lying in the range space of Gr (x). To make sure up does as much as it can, let us decompose v into the sum of two vectors with one lying in the range of Gr (x). We will now drop functional dependencies where they may clutter equations. Let v T G⊥ r Gr T ⊥ G⊥ r Gr
= d2 = d3 .. . = Kd d + u d
(14)
We begin our design by constructing new error variables. Let = Yi + di = xi − YT,i = xi − Yi − di
(15)
Then our error system becomes
e˙ r
= e2 = e3 .. . = Fr (x) + Gr (x)up − Yu − Kd d − ud
(16)
From our ideal control law in (9), we would like to render Fr (x) + Gr (x)up − Yu − Kd d − ud
=
(19)
0n−p×p
= In−p
Where is orthogonal to Gr and each column is of unit length such that B = [Gr G⊥ r ] forms a basis set of vectors, enabling us to perform the decomposition. We now have Gr u p − u d
= Ke e (17)
as that would make (16) equivalent to the exponentially stable system (10) which guarantees that x1 (t) → YT (t) as t → ∞. While up lacks sufficient degrees of freedom on its own to accomplish this task, ud ∈ Rn thus we have an
= Gr vp + G⊥ r vd
(20)
Moving the Gr vp to the left hand side of the equation we get Gr (up − vp ) − ud
= G⊥ r vd
(21)
Choosing up = vp our problem reduces to
III. C ONSTRUCTIVE T RACKABLE F ILTER D ESIGN
e˙ 1 e˙ 2
= Gr vp + G⊥ r vd
G⊥ r
Where ud ∈ Rn is a control term to be chosen later. The Trackable Filter concept simply produces a filter which generates, from the nominal reference YN (t) a signal YT (t) which lies within the trackable p-dimensional manifold of motion for our plant, and the plant controller up is designed to track YT (t).
YT,i ei
= −Fr (x) + Yu (t) + Kd d + Ke e = v(e, d, t) (18)
ud = −G⊥ r vd
(22) T
We now compute vd and vp . Left multiplying (19) by G⊥ r removes the vp term due to the orthogonality of Gr and G⊥ r T and yields vd = G⊥ v. Following a similar approach to find r vp we left multiply (19) by GTr producing GTr Gr vp = GTr v. By assumption Gr has uniform full column rank, or rank p, the p × p matrix GTr Gr is thus uniformly invertible, and vp = (GTr Gr )−1 GTr v. Our two control laws are summarized below (GTr Gr )−1 GTr v
up
=
ud v
⊥ = −G⊥ r Gr v = −Fr + Yu + Kd d + Ke e
T
(23)
Applying these two control laws to (16) and (14). We get e˙ 1 e˙ 2 e˙ r
= e2 = e3 .. . = Ke e
Preprint submitted to 2008 American Control Conference. Received September 21, 2007.
(24)
CONFIDENTIAL. Limited circulation. For review only. which is exponentially stable, and equivalent to e˙ = Ae e. The trackablity filter system is now d˙1 d˙2 d˙r H
= d2 = d3 .. . = Kd d − H(−Fr + Yu + Kd d + Ke e) =
V˙ ≤ −µe |e|2 − (µd − c1 − c2 L)|d|2 + c2 L|d||e| + |d|(c2 LY¯ + c3 Y¯u ) reassigning constants c4 = µd − c1 − c2 L, c5 = c2 L, and c6 = c2 LY¯ + c3 Y¯u simplifies to
(25)
⊥T G⊥ r Gr
V˙ ≤ −µe |e|2 − c4 |d|2 + c5 |e||d| + c6 |d| κc2
The cross term can be bounded by c5 |e||d| ≤ 25 |e|2 + κ1 |d|2 where κ > 0 is a free parameter to assist the analysis
Which we can rewrite as d˙ = Ad d + Bd ∆ 0 .. Bd = . ∆
Collecting terms, particularly with respect to |d|2 we get
In = −H(−Fr + Yu + Kd d + Ke e)
(26) (27) (28)
Where Pd = PdT > 0 is the positive definite solution to the Lyapunov equation Pd Ad + ATd Pd = −Qd for some positive definite symmetric matrix Qd .
V˙
= −eT Qe e − dT Qd d −2dT Pd Bd H(−Fr + Yu + Kd d + Ke e) (29)
The product Prd = Pd Bd selects the last n columns of Pd , or the rth block column of Pd . We now search for upper bounds on our remaining terms, all constants are positive. Let c1 = 2kPdr HKd k if this tighter bound can be computed. Generally is a function H(x, YT , d) and not a constant, however it is uniformly rank p and norm 1. If the first bound cannot be computed, then use the more conservative bound c1 = 2kPdr kkKd k ≥ 2kPdr HKd k. Next, by assumption Fr (x) is Lipschitz thus kFr (x)k ≤ L|x| on some domain D ⊂ Rrn and we can bound the second term with 2kPdr HFr k ≤ c2 L(|e| + |Y | + |d|) where c2 = 2kPdr k ≥ 2kPdr Hk. Again tighter bounds are possible if structure of H is known for all possible x. Since x = e+Y +d its norm can be upper bounded by |x| ≤ |e|+Y¯ +|d|, where |Y (t)| ≤ Y¯ , ∀t. Let c3 = 2kPrd Ke k. Next we have to consider our disturbance term 2kPdr HYu k ≤ c2 Y¯u where |Yu (t)| ≤ Y¯u . Let µe , µd > 0 be the smallest eigenvalues of Qe , Qd respectively. Now we can upper bound our Lyapunov function derivative. V˙ ≤ −µe |e|2 − µd |d|2 + c1 |d|2 + c2 L|d|(|e| + Y¯ + |d|) + c3 |d|Y¯u
(30)
≤
V˙ V˙
≤
−c7 |e| − c8 |d|2 + c6 |d| κc2 c7 = µe − 5 2 1 c8 = c4 − (31) κ Finally arranging terms to examine the bounded disturbance c6 we have
A. Stability Analysis Our last step is to ensure that the trackability filter is ISS with respect to bounded input Yu (t) ∈ L∞ ∩ C 0 and exponentially decaying input e(t) ∈ L2 ∩ L∞ . Of clear concern is the term −HKd d which is potentially working against the stabilizing term Kd d. The matrix H has rank n − p, thus has p eigenvalues at 0, and its norm is unity kHk = 1. With the Lyapunov function V = eT Pe e + dT Pd d its derivative is
V˙
−c7 |e| − |d|(c8 |d| − c6 ) c6 ≤ 0, when |d| ≥ c8
(32)
Our Lyapunov function will decrease along the trajectories of e(t) and d(t) when |d| > cc68 . The necessary conditions for a stable design on a domain D are summarized below µd
> c1 + c2 L +
1 κ
κc22 L2 2 Rewriting the necessary conditions in terms of the system and design parameters a stable design requires µe
>
µd
> 2kPdr HKd k + 2kPdr HkL +
µe
>
κkPdr Hk2 L2
=
⊥T G⊥ r Gr
H
1 κ (33)
for any κ > 0. In summary when the control law of (23) is employed, the necessary conditions of (33) must be met for a stable design. B. Design Extensions Recall that our choice of ud was non-unique and many different choices can be made. For the example of car steering they system has three outputs and two control inputs. This means that ud needs to produce only one dimension of information. We can choose a different basis set of vectors than Gr and G⊥ r . One could choose to inject tracking error only in the lateral speed direction for example, thereby having up focus exclusively on on the forward and angular velocity directions. Such an approach guarantees that the tracking error is predictably injected only into one of the three states that make up d1 . This gives the designer a good deal of flexibility to control how and where the tracking error goes. The stability analysis doesn’t change, only the contents of the H matrix do.
Preprint submitted to 2008 American Control Conference. Received September 21, 2007.
CONFIDENTIAL. Limited circulation. For review only. Another appealing option is to make the basis B a uniformly orthonormal function of d. Then designing a supervisory control which changes which states or combination of states gets the error injection. If one has a preview of the reference signal, then some basis choices will result in less overall error than others. A clear example is when the reference signal lies within the tracking manifold MT , then our choice in the previous section of B = [Gr G⊥ r ] would yield zero tracking error, while any other choice would yield more. Additionally if a path following method is employed then depending on the path the error injection direction can be adjusted to coincide with the direction in which the path speed can be modulated to compensate, see [1],[8]. Another option is to use a system model in strict feedback form as opposed to normal form. The Trackability Filter can also be modeled to similarly to the plant so that better intuition can be maintained with respect to design parameters and physical states e.g. x˙ 1 x˙ 2
x˙ r
= f1 (x1 ) + g1 (x1 )x2 = f2 (¯ x2 ) + g2 (¯ x2 )x3 .. . = fr (x) + gr (x)up (34)
where x¯i = Filter as d˙1 d˙2 d˙r
[xT1
xT2
···
xTi ]T .
Then design the Trackability
= Ad,1 + g1 (x1 )d2 −1 T = −Pd,2 g1 Pd,1 d1 + Ad,2 d2 + g2 (¯ x2 )d3 .. . −1 T = −Pd,r gr−1 Pd,r−1 dr−1 + Ad,r dr + ud
mentioned throughout the paper. We employ the Bicycle Model for steering dynamics [10]. Now let us look at the car steering problem as it fits the vectorial strict feedback form. (Rolling resistance is ignored). cos ψ − sin ψ 0 vu cos ψ 0 vv ξ˙1 = sin ψ (36) 0 0 1 vψ −Cd vu2 + 2vv vψ c1 0 ˙ξ2 = −2vu vψ + c1 Fvr (ξ2 ) + 0 c1 Fu (37) Fvf c3 Fvr (ξ2 ) 0 c2 c1 = 2/m, c2 = 2a/I, c3 = − 2b/I (38) Where m = 1533 kg is the vehicle mass, a = 1.04 m is the distance from the center of gravity (CG) to the front axle, b = 1.65 m the distance from CG to rear axle, and I = 2712 kg m2 is the moment of inertia. A lumped parameter Cd = 0.000321 contains the coefficient of aerodynamic drag and vehicle frontal cross-sectional area. Note that G2 is a constant matrix. An important point is our choice of the forward tire force Fu and lateral tire force Fvf as control variables. Of course wheel torque and wheel angle are the real control variables. We utilize the Pacejka tire model [13] which is a static nonlinear mapping between vehicle states and wheel angle to tire forces Fu and Fvf . For the region of normal operation, where the tire is not sliding, this function is invertible. Thus, provided our controller only commands forces achievable by the tire through a choice of wheel angle and torque, then we are free to work with whichever variable is most convenient. ξ˙1 ξ˙2
(35) This results in a block diagonal Pd matrix, and the Hurwitz Ad,i matrices can be chosen to weight error injection in different states specifically. This system and the normal form system are algebraically equivalent, however while this new form affords more intuitive selection of design parameters, the calculations are more cumbersome. While this paper focuses on a continuous control law from the practical perspective that many mechanical actuators cannot produce switching inputs, such switching controls, like switching between different basis choices with a hybrid supervisory control can be accommodated through dynamic extension. The systems can be extended to relative degree r + 1 or r + 2 and a switching design applied at that stage guaranteeing that the control law is C 0 or C 1 respectively. It is enticing to use switching approaches on the d system as it is not bound by physical limitations, but care must be taken so that discontinuities are not imposed on up if its actuators cannot supply them. IV. E XAMPLE : F RONT WHEEL STEERING CAR An example of the Trackability Filtering design method presented above was applied to the car steering problem
= G1 (ξ1 )ξ2 = f2 (ξ2 ) + G2 up
(39)
The model is in strict feedback form which is easy to place in normal form through the diffeomorphism x = T (ξ) based on the following Lie derivatives. h(ξ1 ) = ξ1 = x1 Lf h(ξ) = G1 ξ1 = x2 ∂G1 ξ2 G1 ξ2 + G1 f2 L2f h(ξ) = ∂ξ1 Lg Lf h(ξ) = G1 G2
(40)
Which gives us the normal form representation x˙ 1 x˙ 2
= x2 = Fr (T −1 (x)) + Gr (T −1 (x))up
(41)
First we are going to choose our basis B = [Bp Bd ] where 1 0 Bp = 0 0 0 1 0 Bd = 1 0
Preprint submitted to 2008 American Control Conference. Received September 21, 2007.
CONFIDENTIAL. Limited circulation. For review only. Which means we are partitioning our control goal vector v as v vp
= Bp vp + Bd vd = BpT v
vd
= BdT v (42)
Our control law is then up
= BpT v
ud v
= Bd (BdT Gr BpT − BdT )v = −Fr + Yu + Kd d + Ke e
(43)
The simulation results of this design are shown in the following two figures. Fig. 1 shows the trajectory (solid line) Fig. 2. Tracking error trajectory with respect to time. Top: Position error. Bottom: Heading angle error.
car. The method allowed the designer the freedom to choose where the tracking error got injected demonstrating the practical utility of the method. R EFERENCES
Fig. 1. Vehicle trajectory in the (x,y) plane. Heading is shown by a triangle within the car image.
of the vehicle compared to the reference (dashed line) for a double lane change maneuver. In our design we chose our basis such that all error got injected into the lateral (Y) distance state. Fig. 2 shows clearly that the angular error and X direction error exponentially converge to zero, while the only nonzero error is in the lateral deviation. V. F UTURE W ORK In the future we plan to develop an optimal trackability filter which yields the trackable reference which is closest to the nominal signal in some sense. After that we will use this method to address tracking controller designs for systems with unstable zero dynamics. We will also explore other forms of trackability filters for systems with state and control limitations.
[1] E. E. Sandoz and P. V. Kokotovi´c, Continuous Path Following Control for Underactuated Systems With Bounded Actuation, Proc. of Intelligent Vehicles Symposium 2007, June 13-15, Istanbul. [2] R. Skjetne and T. I. Fossen and P. V. Kokotovi´c, Robust Output Maneuvering for a Class of Nonlinear Systems, Automatica J., vol. 40, 2004, pp 272-283. [3] P. Falcone and F. Borrelli and J. Asgari and H.E. Tseng and D. Hrovat, Predictive Active Steering Control for Autonomous Vehicle Systems, IEEE Transactions on Control Systems Technology, vol. 15, no. 3, May 2007, pp. 566-580. [4] M. M. Seron and J. H. Braslavsky and P. V. Kokotovi´c and and D. Q. Mayne, Feedback Limitations in Nonlinear Systems: From Bode Integrals to Cheap Control, IEEE Transactions on Automatic Control, vol. 44, no. 4, April 1999. [5] M. M. Seron and J. H. Braslavsky and G. C. Goodwin, Fundamental Limitations in Filtering and Control, Springer, London; 1997. [6] H. K. Khalil, Nonlinear Systems, Prentice Hall, New Jersey; 2002. [7] A. Isidori, Nonlinear Control Systems, Springer, London; 1995. [8] D. B. Daˇci´c and M. V. Subbotin and P. V. Kokotovi´c, Path-Following Approach to Control Effort Reduction of Tracking Feedback Laws, Automatica J., vol. 40, 2004, pp 373-383. [9] D. B. Daˇci´c and M. V. Subbotin and P. V. Kokotovi´c, ”Path-Following for a Class of Nonlinear Systems with Unstable Zero Dynamics”,43rd IEEE Conf. on Decision and Control, 2004, pp 4915-4920. [10] U. Kiencke and L. Nielsen, Automotive Control Systems, Springer, Berlin; 2000. [11] F. Rehman, Discontinuous Steering Control for Nonholonomic Systems With Drift, Nonlinear Analysis, vol. 63, 2005, pp 311-325. [12] M. Kristi´c and I. Kanellakopoulos and P. Kokotovi´c, Nonlinear and Adaptive Control Design, Wiley Interscience, New York, 1995. [13] E. Bakker and L. Nyborg and H. B. Pacejka, ”Tyre Modelling for Use in Vehicle Dynamics Studies”, SAE Technical Paper Series, 1987, pp 1-15.
VI. C ONCLUSIONS A Trackability Filter based tracking control design was developed and necessary conditions for a stable design were presented. The method was tested in simulation on a practical example of an underactuated system, a front wheel steering
Preprint submitted to 2008 American Control Conference. Received September 21, 2007.
