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Mathematical Problems in Engineering
Volume 2012, Article ID 287195, 31 pages
http://dx.doi.org/10.1155/2012/287195
Research Article

Vehicle Sliding Mode Control with Adaptive Upper Bounds: Static versus Dynamic Allocation to Saturated Tire Forces

Department of Mechanical Engineering, Amirkabir University of Technology, 424 Hafez Avenue, Tehran 15914, Iran

Received 20 December 2011; Accepted 13 February 2012

Academic Editor: Alexander P. Seyranian

Copyright © 2012 Ali Tavasoli and Mahyar Naraghi. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

Nonlinear vehicle control allocation is achieved through distributing the task of vehicle control among individual tire forces, which are constrained to nonlinear saturation conditions. A high-level sliding mode control with adaptive upper bounds is considered to assess the body yaw moment and lateral force for the vehicle motion. The proposed controller only requires the online adaptation of control gains without acquiring the knowledge of upper bounds on system uncertainties. Static and dynamic control allocation approaches have been formulated to distribute high-level control objectives among the system inputs. For static control allocation, the interior-point method is applied to solve the formulated nonlinear optimization problem. Based on the dynamic control allocation method, a dynamic update law is derived to allocate vehicle control to tire forces. The allocated tire forces are fed into a low-level control module, where the applied torque and active steering angle at each wheel are determined through a slip-ratio controller and an inverse tire model. Computer simulations are used to prove the significant effects of the proposed control allocation methods on improving the stability and handling performance. The advantages and limitations of each method have been discussed, and conclusions have been derived.

1. Introduction

In recent years by rapid emergence of electronic control devices, employing all available actuators, or individual tire forces, for ground vehicle control has become possible [1]. The vehicle motion is governed by tire forces which are constrained based on friction circle notion. Thus, tire saturation constraints must be taken into account for a proper control design. The problem of optimal actuators selection to execute a control task, while minimizing effort and satisfying constraints, is known as Control Allocation (CA). To tackle actuator constraints in control design of a general over-actuated nonlinear system, two approaches of Static Control Allocation (SCA) and Dynamic Control Allocation (DCA) are proposed. In SCA, the total body forces/moments of a high-level controller are allocated to available actuators by optimizing a suitable cost function at each sampling time, whereas DCA generates a dynamic update law for actuators. No optimization problem needs to be solved by DCA.

In the field of Integrated Vehicle Dynamics Control (IVDC), Optimal Distribution of tire Forces (ODF) was introduced to meet various objectives, such as maximizing longitudinal acceleration [2], minimizing total tire workload usage [3], or adaptive-optimal coordination of braking and steering [4]. Attempts to account for actuator constraints were rare in these works.

To allocate control objectives to the actuators with limited amplitude/rate, SCA methods have been subject of different studies. Algorithms using various optimization methods, such as quadratic programming [5], multiparametric nonlinear programming [6], and fixed point [1], were developed. Kou [7] proposed an SCA scheme based on the model predictive control and fixed point method. The main problem in static control allocation is its computational burden for practical applications, due to numerical solution of a constrained optimization problem at each sampling instant.

To deal with this difficulty, Johansen [8] developed a dynamic control allocation method for a particular class of nonlinear systems. In this regard, a dynamic update law leads the desired actuator efforts to converge to the solution of a definite optimization problem, without solving the optimization problem. DCA was extended for systems with unknown parameters [9] and applied to vehicle yaw control [10].

This paper addresses nonlinear vehicle CA constrained to tire saturation conditions. Referring to schematic view of IVDC structure in Figure 1, the required body lateral force and yaw moment for vehicle motion are determined by a high-level sliding mode enhanced adaptive controller. The adaptive control methodology is utilized to update the perturbation and sliding mode control gains, so that the upper bounds of uncertainties are not required to be known in advance. Then, the body force and moment, along with the driver’s desired braking force, are allocated to individual lateral and longitudinal forces of each tire. Considering tires saturation induces nonlinear constraints in CA problem. To tackle this problem, we look into SCA and DCA methods for vehicle control. For static allocation purpose, the interior point is formulated and employed to solve the nonlinear inequality constrained optimization problem. To formulate dynamic allocation control for the proposed problem, a dynamic update law is derived and utilized in the IVDC scheme. The desired lateral force of each tire by CA modules is mapped into the corresponding active steering angle through an inverse tire model. In addition, the desired longitudinal forces are tracked by a low-level slip-ratio control scheme. Simulation results are conducted to evaluate the effectiveness of each method.

287195.fig.001
Figure 1: Overall structure of the integrated vehicle dynamics control scheme.

The rest of the paper is organized as follows. The high-level control is described next. Section 3 presents the formulation of optimal distribution of tire forces in IVDC considering nonlinear tire saturation constraints. In Section 4, the SCA is formulated based on the interior-point method. In Section 5, the DCA approach is utilized to derive an optimized dynamic update law for individual tire forces. The low-level slip-ratio controller has been addressed in Section 6. Sections 7 and 8 are devoted to simulation results and concluding remarks.

2. High-Level Sliding Mode Control with Adaptive Upper Bounds

In this section we first design a high-level controller for vehicle handling and stability based on the conventional Sliding Mode Control (SMC), then an SMC with updated upper bounds of uncertainties is considered.

2.1. Conventional Sliding Mode Control

In general, vehicle handling and stability are achieved through the control of yaw rate and side-slip angle, respectively. The design procedure is based on the 2DoF vehicle model, where the basic equations are [11]̇𝐼𝑚𝑉𝛽+𝑟=𝑌,𝑧̇𝑟=𝑀,(2.1) where 𝑚 and 𝐼𝑧 denote the total mass and yaw moment of inertia, from which only the estimates of 𝑚 and 𝐼𝑧 are available, and 𝑉 is the vehicle velocity. 𝛽 and 𝑟 stand for the actual vehicle side-slip angle and the yaw rate, respectively. 𝑀 and 𝑌 are sum of external moments in the yaw direction and lateral forces acting on the vehicle, respectively. To account for the unmodelled dynamics and uncertainties in modelling the actual nonlinear vehicle dynamics, the unknown, but bounded, disturbance terms, 𝜔𝛽 and 𝜔𝑟, are embedded into each channel to geṫ𝑚𝑉𝛽+𝑟=𝑌+𝜔𝛽,𝐼(2.2)𝑧̇𝑟=𝑀+𝜔𝑟.(2.3) To design the total lateral force (𝑌), for a zero desired side-slip angle, the sliding surface,𝑠𝛽, is selected as𝑠𝛽=𝛽.(2.4) Differentiating this equation and considering (2.2),̇𝑠𝛽=𝑌Δ𝑚𝑉𝑟+𝛽𝑚,(2.5) where the term Δ𝛽=𝜔𝛽/𝑉 is assumed to be bounded by a known value Δ𝛽:||Δ𝛽||<Δ𝛽.(2.6) To guarantee the sliding condition [12]𝑠𝛽̇𝑠𝛽<0,(2.7) the desired body lateral force is considered as𝑌=𝑉𝑚𝑟+𝑣𝛽(2.8) in which 𝑚 is our estimate of 𝑚 and 𝑣𝛽 is to be designed. Insert (2.8) into (2.5) to get the left side of (2.7) as𝑠𝛽̇𝑠𝛽=𝑠𝛽𝑚1Δ𝛽+𝑣𝛽+𝑚𝑟,(2.9) where the mass estimation error 𝑚=𝑚𝑚 is assumed to satisfy||||=||||<𝑚𝑚𝑚𝑚,𝑚>0,(2.10) whose combination with (2.9) and (2.6) results in𝑠𝛽̇𝑠𝛽𝑚1𝑠𝛽𝑣𝛽+||Δ𝛽||||𝑠𝛽||+||||||𝑠𝑚|𝑟|𝛽||<𝑚1𝑠𝛽𝑣𝛽+Δ𝛽||𝑠𝛽||+||𝑠𝑚|𝑟|𝛽||.(2.11) To achieve (2.7), 𝑣𝛽 is considered to be𝑣𝛽=𝑘𝛽𝑠sgn𝛽,(2.12) where sgn() is the signum function, and𝑘𝛽>Δ𝛽+𝑚|𝑟|+𝜂𝛽,𝜂𝛽>0.(2.13) By substituting (2.12) into (2.8), the desired body lateral force is attained. To mitigate the problem of chattering, the sign function is replaced by saturation function with a boundary layer thickness of Φ𝛽>0. Thus, the final control law becomes𝑌=𝑉𝑚𝑟𝑘𝛽𝑠sat𝛽Φ𝛽.(2.14) In order to design the body yaw moment (𝑀), for tracking the desired yaw rate (𝑟𝑑), the sliding surface,𝑠𝑟, is adopted as𝑠𝑟=𝑟𝑟𝑑+𝜆𝑟𝑡0𝑟𝑟𝑑𝑑𝜏,𝜆𝑟>0,(2.15) where the integral term is used to mitigate the undesirable yaw angle offset and to ensure the desired vehicle heading. Differentiating (2.15) along with (2.3) leads tȯ𝑠𝑟=𝑀𝐼𝑧+𝜔𝑟𝐼𝑧𝜏,𝜏=̇𝑟𝑑𝜆𝑟𝑟𝑟𝑑.(2.16) The design process of the desired body yaw moment is similar to that of (2.14):𝐼𝑀=𝑧̇𝑟𝑑𝜆𝑟𝑟𝑟𝑑𝑘𝑟𝑠sat𝑟Φ𝑟,Φ𝑟>0,(2.17) where 𝐼𝑧 is an estimate of 𝐼𝑧 and𝑘𝑟>Δ𝑟+𝐼𝑧||̇𝑟𝑑𝜆𝑟𝑟𝑟𝑑||+𝜂𝑟,𝜂𝑟>0,(2.18) with Δ𝑟>0 and 𝐼𝑧>0 being the upper bounds for |𝜔𝑟| and |𝐼𝑧𝐼|=|𝑧𝐼𝑧|, respectively.

2.2. Sliding Mode Control with Adaptive Upper Bounds

From (2.13) and (2.18) it can be observed that the selection of the SMC gains 𝑘𝛽 and 𝑘𝑟 depends on upper bounds of uncertainties in vehicle dynamics and body mass and inertia, that is, Δ𝑟,Δ𝛽, 𝑚, and 𝐼𝑧. In practice, uncertainties and disturbances depend primarily on the highly nonlinear dynamics of vehicle and tire which are not completely known, and one cannot determine their exact bounds too. Therefore, no universal method is available yet to tune the controller gains and these gains should be tuned by trial and error approach in practical implementations. In this regard, the controller tends to be overconservative, which may induce poor tracking performance as well as undesirable oscillations in control signal. To overcome this drawback, adaptive control methodology with control parameters updated online is a promising approach. In this section, we use the adaptive control technique to attain a sliding mode controller with adaptive upper bounds. To design sliding mode controls with variable gains, the following modified control laws are established:̂𝑘𝑌=𝑉𝑚𝑟𝛽1+̂𝑘𝛽2𝑠|𝑟|sgn𝛽𝐼,(2.19)𝑀=𝑧̂𝑘𝜏𝑟1+̂𝑘𝑟2𝑠|𝜏|sgn𝑟,(2.20) where𝜏=̇𝑟𝑑𝜆𝑟𝑟𝑟𝑑(2.21) and the varying controller gains are updated as follows:̇̂𝑘𝛽1=𝛾1𝛽1||𝑠𝛽||,𝛾𝛽1̇̂𝑘>0,(2.22)𝛽2=𝛾1𝛽2|||𝑠𝑟|𝛽||,𝛾𝛽2̇̂𝑘>0,(2.23)𝑟1=𝛾1𝑟1||𝑠𝑟||,𝛾𝑟1̇̂𝑘>0,(2.24)𝑟2=𝛾1𝑟2||𝑠|𝜏|𝑟||,𝛾𝑟2>0.(2.25) Assume that there are positive constants𝑘𝑑𝛽1,𝑘𝑑𝛽2, 𝑘𝑑𝑟1, and 𝑘𝑑𝑟2 that satisfy𝑘𝑑𝛽1>||Δ𝛽||,𝑘𝑑𝛽2>||||𝑘𝑚,(2.26)𝑑𝑟1>||Δ𝑟||,𝑘𝑑𝑟2>||𝐼𝑧||.(2.27) It should be noted that we need only to assure that such constants exist without acquiring the knowledge of these upper bounds to use in control laws. Also, consider̃𝑘𝛽1=̂𝑘𝛽1𝑘𝑑𝛽1,̃𝑘𝛽2=̂𝑘𝛽2𝑘𝑑𝛽2̃𝑘,(2.28)𝑟1=̂𝑘𝑟1𝑘𝑑𝑟1,̃𝑘𝑟2=̂𝑘𝑟2𝑘𝑑𝑟2.(2.29) Then, the stability of the considered adaptive-sliding mode control laws can be shown through Lyapunov candidates:𝑉𝛽=𝑚2𝑠2𝛽+12𝛾𝛽1̃𝑘2𝛽1+12𝛾𝛽2̃𝑘2𝛽2,𝑉(2.30)𝑟=𝐼𝑧2𝑠2𝑟+12𝛾𝑟1̃𝑘2𝑟1+12𝛾𝑟2̃𝑘2𝑟2.(2.31) To prove the stability of side-slip angle under (2.19) with adaptation laws (2.22) and (2.23), first insert (2.19) into (2.5) so that𝑚̇𝑠𝛽=𝑚𝑟+Δ𝛽̂𝑘𝛽1+̂𝑘𝛽2𝑠|𝑟|sgn𝛽.(2.32) Then, differentiate (2.30) to geṫ𝑉𝛽=𝑚𝑠𝛽̇𝑠𝛽+𝛾𝛽1̃𝑘𝛽1̇̃𝑘𝛽1+𝛾𝛽2̃𝑘𝛽2̇̃𝑘𝛽2.(2.33) Replacing (2.32) in (2.33) and considering (2.28), we have thaṫ𝑉𝛽=𝑚𝑟+Δ𝛽̂𝑘𝛽1+̂𝑘𝛽2𝑠|𝑟|sgn𝛽𝑠𝛽+𝛾𝛽1̇̂𝑘𝛽1̂𝑘𝛽1+𝛾𝛽2̇̂𝑘𝛽2̂𝑘𝛽2𝛾𝛽1̇̂𝑘𝛽1𝑘𝑑𝛽1+𝛾𝛽2̇̂𝑘𝛽2𝑘𝑑𝛽2.(2.34) Using adaptation laws (2.22) and (2.23) in (2.34) and considering (2.26) results iṅ𝑉𝛽=𝑚𝑟+Δ𝛽𝑠𝛽𝑘𝑑𝛽1||𝑠𝛽||𝑘𝑑𝛽2||𝑠|𝑟|𝛽||||||||Δ𝑚|𝑟|+𝛽||||𝑠𝛽||𝑘𝑑𝛽1||𝑠𝛽||𝑘𝑑𝛽2||𝑠|𝑟|𝛽||=||||𝑚𝑘𝑑𝛽2||𝑠|𝑟|𝛽||+||Δ𝛽||𝑘𝑑𝛽1||𝑠𝛽||<0,(2.35) where we use 𝑠𝛽sgn(𝑠𝛽)=|𝑠𝛽|. Accordingly, the convergence of 𝑠𝛽 to zero and also boundedness of ̃𝑘𝛽1 and ̃𝑘𝛽2 are resulted by Barbalat’s lemma [12].

In an identical way, the stability of the yaw motion can be demonstrated, first, by combining (2.16) and (2.20) so that𝐼𝑧̇𝑠𝑟=𝐼𝑧𝜏+Δ𝑟̂𝑘𝑟1+̂𝑘𝑟2𝑠|𝜏|sgn𝑟.(2.36) Differentiate (2.31) and replace (2.36) for 𝐼𝑧̇𝑠𝑟 to geṫ𝑉𝑟=𝐼𝑧𝜏+Δ𝑟̂𝑘𝑟1+̂𝑘𝑟2𝑠|𝜏|sgn𝑟𝑠𝑟+𝛾𝑟1̇̂𝑘𝑟1̂𝑘𝑟1+𝛾𝑟2̇̂𝑘𝑟2̂𝑘𝑟2𝛾𝑟1̇̂𝑘𝑟1𝑘𝑑𝑟1+𝛾𝑟2̇̂𝑘𝑟2𝑘𝑑𝑟2.(2.37) Using adaptation laws (2.24) and (2.25) as well as the inequalities in (2.27), we have thaṫ𝑉𝑟=𝐼𝑧𝜏+Δ𝑟𝑠𝑟𝑘𝑑𝑟1||𝑠𝑟||𝑘𝑑𝛽2||𝑠|𝜏|𝑟||||𝐼𝑧||||Δ|𝜏|+𝑟||||𝑠𝑟||𝑘𝑑𝑟1||𝑠𝑟||𝑘𝑑𝑟2||𝑠|𝜏|𝑟||=||𝐼𝑧||𝑘𝑑𝑟2||𝑠|𝜏|𝑟||+||Δ𝑟||𝑘𝑑𝑟1||𝑠𝑟||<0.(2.38) Thus, according to the Barbalat’s Lemma, the system state can be driven to the sliding surface 𝑠𝑟 and the controller gains ̃𝑘𝑟1 and ̃𝑘𝑟2 will be bounded. Furthermore, to tackle the chattering problem saturation function is used to derive the final adaptive control lawŝ𝑘𝑌=𝑉𝑚𝑟𝛽1+̂𝑘𝛽2𝑠|𝑟|sat𝛽Φ𝛽,𝐼𝑀=𝑧̂𝑘𝜏𝑟1+̂𝑘𝑟2𝑠|𝜏|sat𝑟Φ𝑟.(2.39)

3. Control Allocation in Vehicle System

The total body lateral force and yaw moment, as well as the braking acceleration command by driver, are generated by longitudinal and lateral forces of each tire. In this paper, a 4-wheel vehicle system with each wheel being braked/derived and steered independently is considered. Such a full tire-actuated vehicle can be available through X-by-wire systems [1, 13]. Thus, the overall control system contains eight actuators and only three control objectives, raising an overactuated control system. A general approach to resolve redundancy is to optimize a cost function for specific performance. The well-accepted cost function in IVDC is the sum of work load of four wheels:𝑓=4𝑖=1𝐴𝑖𝑋2𝑖+𝑌2𝑖𝜇𝑖𝑍𝑖2,(3.1) where 𝑖 denotes wheel number, 𝑋𝑖 and 𝑌𝑖 stand for desired values of longitudinal force, 𝐹𝑥𝑖, and lateral force, 𝐹𝑦𝑖, 𝑍𝑖 is the vertical load, all defined in the vehicle body fixed coordinate system, as shown in Figure 2, 𝐴𝑖 is the weighting coefficient and 𝜇𝑖 is the friction coefficient, of the 𝑖th tire. Defining the 8 × 1 actuator vector of 𝐮 as𝑋𝐮=1𝑋2𝑋3𝑋4𝑌1𝑌2𝑌3𝑌4𝑇,(3.2) the cost function is written in matrix form𝑓(𝐮)=𝐮𝐓𝐖𝐮(3.3) in which 𝐖𝟖×𝟖 is a diagonal weighting matrix. To get 𝑌 and 𝑀, by assuming small steering angles in Figure 2, individual tire forces must satisfy the equality constraints:𝑌=4𝑖=1𝑌𝑖,(3.4)𝑀=2𝑖=1𝐿𝑓𝑌𝑖𝐿𝑟𝑌(𝑖+2)+𝑑22𝑖=1𝑋(2𝑖)𝑋(2𝑖1).(3.5) Also, the longitudinal acceleration, 𝑎𝑥, by driver is generated by longitudinal forces𝑋=𝑚𝑎𝑥=4𝑖=1𝑋𝑖.(3.6) Equations (3.4)–(3.6) can be expressed in linear matrix form as𝐀𝐮=𝐯,(3.7) where 𝐀3×8 is a constant matrix and the vector of generalized forces/moment, 𝐯, is given by[]𝐯=𝑋𝑌𝑀𝑇.(3.8) On the other hand, the resultant force of each tire is constrained to friction circle𝜇𝑖𝑍𝑖2𝑋2𝑖+𝑌2𝑖0(𝑖=1,,4).(3.9) Using (3.2), (3.9) can be written as𝜇𝑖𝑍𝑖2𝑢2𝑖+𝑢2𝑖+40(𝑖=1,,4).(3.10) or𝐂𝐈(𝐮)𝟎,(3.11) where 𝐂𝐈 is a 4 × 1 vector with the 𝑖th component𝐶𝐼𝑖𝑢𝑖=𝜇𝑖𝑍𝑖2𝑢2𝑖+𝑢2𝑖+4(𝑖=1,,4).(3.12) The optimization problem is summarized as follows:minimize𝑓(𝐮)=𝐮𝐓𝐖𝐮,subjecttolinearequalityconstraints,𝐀𝐮=𝐯,nonlinearinequalityconstraints,𝐂𝐈(𝐮)𝟎.(3.13)

287195.fig.002
Figure 2: Schematic view of forces acting on the vehicle.

4. Static Control Allocation: Application of Interior-Point Method

SCA solves the optimization problem (3.13) at each sampling instant. A powerful approach for nonlinear programming is the set of interior-point (IP) methods. A benefit of such methods is that the distance from the optimum is always known, so that one can terminate the optimization algorithm when the solution reaches within desired tolerance. Convergence is also uniform toward the optimal solution. The description of IP algorithm in this paper is an excerpt from [14, 15].

4.1. Karush-Kuhn-Tucker Conditions

At the outset to avoid infeasible solutions, the equality constraints are embedded in cost function to get𝐽=𝜌𝐮𝐓𝐖𝐮+(𝐀𝐮𝐯)𝐓(𝐀𝐮𝐯),𝜌>0,(4.1) which is written in the quadratic form1𝐽=2𝐮𝐓𝐆𝐮+𝐜𝐓𝐮+𝐡,(4.2) where𝐆=2𝜌𝐖+𝐀𝐓𝐀,𝐜=2𝐀𝐓𝐯,𝐡=𝐯𝐓𝐯.(4.3) IP methods use barrier logarithmic functions to satisfy the inequality constraints. In this regard, the optimization problem turns into1minimize𝐿=2𝐮𝐓𝐆𝐮+𝐜𝐓𝐮+𝐡𝜂4𝑖=1𝐶log𝐼𝑖𝐂,𝜂>0,subjectto𝐈(𝐮)𝐩=𝟎,𝐩𝟎.(4.4) Applying KKT formulation [16] to (4.4), the optimality conditions can be expressed as follows,𝐆𝐮+𝐜+𝐀𝐓𝐈𝝀=𝟎,𝐀𝐈(𝐮)=𝐂𝐈(𝐮),𝐂𝐈[](𝐮)𝐩=𝟎,𝐏𝝀𝜂𝐞=𝟎,𝐞=1111𝑇,𝐩>0,𝝀>0,(4.5) where 𝝀 is the vector of Lagrange multipliers and 𝐏 is a 4 × 4 diagonal matrix whose diagonal elements are the components of the vector 𝐩.

4.2. Primal-Dual Path-Following Method Steps

The Primal-dual path-following IP method steps for IVDC are as follows:

(1) Newton’s Step Direction
In each iteration, the step direction {Δ𝐩,Δ𝐮,Δ𝝀} is obtained by applying Newton’s method to KKT conditions (4.5): 𝐫𝐜+𝐆+𝚲𝟐𝚫𝐮+𝐀𝐓𝐈𝐫𝚫𝝀=𝟎,𝐛𝐀𝐈𝐫𝚫𝐮𝚫𝐩=𝟎,𝐬+𝐏𝚫𝝀+𝚲𝚫𝐩=𝟎,(4.6) where the residuals 𝐫𝐜, 𝑟𝐛, and 𝑟𝐬 are obtained as 𝐫𝐜=𝐆𝐮+𝐜+𝐀𝑇𝐈𝝀,𝐫𝐛=𝐂𝐈(𝐮)𝐩,𝐫𝐬𝑝=𝐏𝝀𝜂𝐞,(4.7)𝐏=diag1,𝑝2,𝑝3,𝑝4𝜆,𝚲=diag1,𝜆2,𝜆3,𝜆4,𝚲𝟐=2diag(𝚲,𝚲).(4.8) Equation (4.6) is solved to achieve the step directions Δ𝐮=𝐆+𝚲𝟐+𝐀𝐓𝐈𝐏𝟏𝚲𝐀𝐈1𝐀𝐓𝐈𝐏1𝐫𝐬+𝚲𝐫𝐛𝐫𝐜,Δ𝝀=𝐏1𝐫𝐬+𝚲𝐫𝐛+𝐏1𝚲𝐀𝐈Δ𝐮,Δ𝐩=𝐫𝐛𝐀𝐈Δ𝐮.(4.9)

(2) Step Length Calculation
To satisfy nonnegativity condition (𝐩,𝝀)𝟎, the new iteration (𝐮+,𝐩+,𝝀+) is calculated as 𝐮+=𝐮+𝛼𝑠maxΔ𝐮,𝐩+=𝐩+𝛼𝑠max𝚫𝐩,𝝀+=𝝀+𝛼𝜆max𝚫𝝀,(4.10) where 𝛼𝑠max]𝛼=max{𝛼𝜖(0,1𝐩+𝛼Δ𝐩(1𝜏)𝐩},𝜏𝜖(0,1),𝜆max]=max{𝛼𝜖(0,1𝝀+𝛼Δ𝝀(1𝜏)𝝀}.(4.11)

(3) Updating the Barrier Parameter 𝜂
The sequence of barrier parameters {𝜼𝐤} converges to zero by the update law 𝜂𝑘+1=𝜎𝑘𝐩𝐓𝐤𝝀𝐤𝟒,𝜎𝑘𝜖[]0,1,(4.12) where to update the centring parameter𝜎𝑘, first, the predictor (affine scaling) direction {Δ𝐮aff,Δ𝐩aff,Δ𝝀aff} and the corresponding longest step lengths 𝛼𝑠aff and 𝛼𝜆affare calculated by setting 𝜂=0 in (4.9) and (4.11). Then the value of complementarity along the affine scaling 𝜂aff step is defined to be 𝜂aff=𝐩𝐤+𝛼𝑠affΔ𝐩aff𝑇𝝀𝐤+𝛼𝜆affΔ𝝀aff.(4.13) The centring parameter is updated as follows: 𝜎𝑘=𝜂aff𝐩𝐓𝐤𝝀𝐤/𝟒3.(4.14)

(4) Stopping Criteria
The algorithm is terminated, when the following error function with 𝜂=0 has converged sufficiently close to zero 𝐸(𝐮,𝐩,𝝀,𝜂)=max𝐆𝐮+𝐜+𝐀𝑇𝐈𝝀,𝐂𝐈(𝐮)𝐩,𝐏𝝀𝜂𝐞.(4.15) All of these steps have been summarized in Pseudocode 1. To reduce the time of control allocation, the last allocated 𝐮 is chosen as the initial value of the control input 𝐮𝟎. Then, 𝐮𝟎 is modified (reduced) to satisfy tire saturation constraints.

pseudo1
Pseudocode 1: Pseudocode for interior-point algorithm.

5. Dynamic Control Allocation

DCA uses an optimizing update law for system inputs. Since it is not required to solve the optimization problem at each sampling time, the main advantage of DCA is its computational efficiency. In what follows, the procedure explained in [8] is applied to the proposed integrated vehicle control scheme. However, the interested reader might refer to [8, 9] for convergence study and detailed analysis.

5.1. Dynamic Control Allocation Applied to IVDC

DCA is formulated by introducing the following Lagrangian based on (3.13):=𝐮𝐓𝐖𝐮+(𝐯𝐀𝐮)𝐓𝝀dyn𝜔dyn4𝑖=1𝐶log𝐼𝑖𝑢𝑖,𝑢𝑖+4(5.1) with 𝜔dyn>0and𝝀dyn being a 3-component vector of Lagrange multipliers. DCA updates 𝐮 and 𝝀dyn in the form of the Newton-like update laẇ𝐮̇𝝀dyn=𝛾dyn𝑇+𝜀dyn𝐈1𝜕𝜕𝐮𝜕𝜕𝝀dyn+𝜻𝝓,(5.2) where 𝛾dyn>0 and 𝜀dyn0, and for our problem is written as𝜕=2𝜕𝐮2𝜕(𝐀𝐮)𝐓𝜕𝐮𝜕(𝐀𝐮)𝟎𝜕𝐮.(5.3) In (5.2) the feedforward-like terms 𝜻 and 𝝓 are chosen so that the following scalar algebraic equation holds:𝜶𝐓dyn𝜻+𝜷𝐓dyn𝝓+𝛿dyn=0(5.4) with𝜶dyn𝜷dyn=𝜕𝜕𝐮𝜕𝜕𝝀dyn,𝛿dyn=(𝐀𝐮𝐯)𝑇̇𝐯.(5.5)

Assume that the stability of the high-level control could be shown through a Lyapunov function𝑉0(𝑡,𝑟,𝛽), then, by the Lyapunov function𝑉dyn𝑡,𝑟,𝛽,𝐮,𝝀dyn=𝜎dyn𝑉01(𝑡,𝑟,𝛽)+2𝜕𝑇𝜕𝐮𝜕+𝜕𝐮𝜕𝑇𝜕𝝀dyn𝜕𝜕𝝀dyn,𝜎dyn>0,(5.6) global exponential convergence to optimality conditions is achieved.

5.2. Discussion and Modification

Consider the equality-constrained optimization problem with the Lagrangian (5.1). Then the optimizing conditions are𝜕𝜕𝐮𝜕𝜕𝝀dyn=𝟎.(5.7) Applying Newton’s conditions to (5.7) results in optimizing Newton’s steps Δ𝐮 and Δ𝝀dyn𝜕2𝜕𝐮2𝜕𝜕𝝀dyn𝜕𝜕𝐮𝑇𝜕𝜕𝐮𝜕𝜕𝝀dyn𝜕2𝜕𝝀𝟐dyn𝚫𝐮𝚫𝝀dyn=𝜕𝜕𝐮𝜕𝜕𝝀dyn(5.8) or𝚫𝐮𝚫𝝀dyn=𝜕2𝜕𝐮𝟐𝜕(𝐀𝐮)𝑇𝜕𝐮𝜕(𝐀𝐮)𝟎𝜕𝐮1𝜕𝜕𝐮𝜕𝜕𝝀dyn=1𝜕𝜕𝐮𝜕𝜕𝝀dyn.(5.9) By setting𝜀dyn=0, ignoring the terms 𝜻 and 𝝓, and considering the symmetry of , it can be demonstrated that the update law (5.2) represents the solution of Δ𝐮 and Δ𝝀dyn in (5.9), that is, one Newton’s step towards the optimum point. Therefore, (5.9) can be interpreted as one Newton’s step from the solution of the control allocation problem at the current sampling time for the solution at the next instant, for which the terms 𝜻 and 𝝓 are to compensate time-varying optimum solution.

Despite the log-barrier term in (5.1), depending on the values of𝛾dyn and𝜔dyn, there is no guarantee that the Newton-like step (5.2) will satisfy the inequality constraints (3.9). As the 𝑖th inequality constraint is infringed, 𝐶𝐼𝑖 becomes negative andlog(𝐶𝐼𝑖) and correspondingly the Lagrangian (5.1) turn meaningless. To tackle this problem, in this paper, a line search is adopted for the coefficient 𝛾dyn, so that the resulting 𝐮 and 𝝀dyn at the next sampling time satisfy the inequality constraints 𝐂𝐈>𝟎 and𝝀dyn>0. The idea of the line search approach is taken from the second step of the interior-point stages, described in the previous section.

Another problem with DCA is that (5.2) could induce a Newton’s step from an infeasible point, violating the inequality constraints (3.9), because of time-varying nature of these constraints. This arises because the term(𝜇𝑖𝑍𝑖)2 in (3.9) is time varying. Consequently, it is possible for the feasible solution at the current time to violate the inequality constraints of the next sample time. In such conditions, (5.2) needs to be modified for Newton’s steps from infeasible points [15].

After desired longitudinal and lateral forces of each tire are computed by either of the proposed control allocation algorithms, the active steering angle, 𝛿𝑖, at wheel 𝑖 can be determined as follows:𝛿𝑖𝐿𝛽+𝑓𝑟𝑣𝑥𝑖𝛼𝑖𝛿,𝑖=1,2,𝑖𝐿𝛽𝑟𝑟𝑣𝑥𝑖𝛼𝑖,𝑖=3,4,(5.10) where 𝑣𝑥𝑖 is the longitudinal velocity of the 𝑖th tire and 𝛼𝑖 is the side-slip angle of the 𝑖th tire and is obtained using the inverse of a simple tire model as𝛼𝑖=𝑌𝑖𝐶𝑖,𝑋2𝑖+𝑌2𝑖𝜇𝑛𝑖𝑍𝑖2,0<𝑛<1,𝜇𝑖𝑍𝑖𝐾arctan𝑖𝑌𝑖𝜇𝑖𝑍𝑖𝐶𝑖,𝑋2𝑖+𝑌2𝑖𝜇>𝑛𝑖𝑍𝑖2,𝑖=1,,4,(5.11) in which 𝐶𝑖 denotes the cornering stiffness of the 𝑖th tire and 𝐾𝑖s are chosen to approximate the saturation conditions. Longitudinal forces are fed into the low-level control unit.

6. Low-Level Slip-Ratio Control Design

The longitudinal force of each tire is related to the corresponding longitudinal slip ratio and is adjusted through slip-ratio control (SRC). The slip ratio of the 𝑖th tire,𝜎𝑖, is defined as𝜎𝑖=𝑅𝜔𝑖𝑣𝑥𝑖𝑣𝑥𝑖𝜎duringbraking,(6.1)𝑖=𝑅𝜔𝑖𝑣𝑥𝑖𝑅𝜔𝑖duringacceleration,(6.2) where 𝑅 denotes the radius and 𝜔𝑖 is the angular velocity of the 𝑖th wheel. In the case where longitudinal slip ratio is small, the longitudinal tire force is found to be proportional to the slip ratio. Then, it gains its maximum value at a typical value of 𝜎, after which it starts to lessen. Experimental studies have established that the tire lateral force decreases with increasing slip ratios greater than |𝜎| as well [17].

6.1. Description of the SRC Scheme

In this paper, when the tire slip ratio is smaller than 𝜎, by neglecting the wheel rotational inertia [3], the applied braking/traction torque, 𝑇𝑖, at wheel 𝑖 is obtained as𝑇𝑖=𝑅𝑋𝑖.(6.3) In this case, the SRC works for Desired Longitudinal Force Generation (DLFG). However, when the demanded 𝑋𝑖 is too high, applying (6.3) would increase the slip ratio beyond 𝜎, inevitably leading to wheel lock and lateral tire force drop. In such conditions, the idea of Antilock Braking System (ABS) is employed to keep the slip ratio of tires at 𝜎. This idea is utilized during both braking and traction. When traction torque applies, the proposed slip-ratio control is in the Traction Control System (TCS) mode. The SRC scheme is shown in Figure 3.

287195.fig.003
Figure 3: The SRC scheme applied to the vehicle system.
6.2. The ABS/TCS Design

The 𝑖th wheel rotational dynamics is stated as𝐼𝜔̇𝜔𝑖=𝑇𝑖𝑅𝐹𝑥𝑖,(6.4) where 𝐼𝜔 is wheel rotational inertia. To achieve slip-ratio differential equation when braking, by differentiating (6.1) and replacing for ̇𝜔𝑖 from (6.4) ̇𝜎𝑖=𝑅𝐼𝜔𝑣𝑥𝑖𝑇𝑖𝑟𝐹𝑥𝑖1+𝜎𝑖̇𝑣𝑥𝑖𝑣𝑥𝑖.(6.5) During decelerating, the longitudinal force can be stated as [17]𝐹𝑥𝑖=𝐶𝜎𝜎𝑖+Δ𝐹𝑥𝑖(6.6) in which the term 𝐶𝜎𝜎𝑖 represents the linear part, with 𝐶𝜎 being the tire longitudinal stiffness, and Δ𝐹𝑥𝑖 is the deviation from the linear part and is bounded by 𝐹𝑥𝑖:0<Δ𝐹𝑥𝑖<𝐹𝑥𝑖.(6.7) The goal of the ABS is to regulate 𝜎𝑖 around the constant value𝜎. Applying the sliding mode control design procedure to (6.5), an ABS is designed so that to get the sliding condition𝑑1𝑑𝑡2𝑠2𝜎𝜂𝜎||𝑠𝜎||,𝑠𝜎=𝜎𝑖𝜎,𝜎<0,𝜂𝜎>0,(6.8) the applied braking torque at the 𝑖th wheel is𝑇𝑖=𝑅𝐶𝜎𝜎+𝐼𝜔̇𝑣𝑥𝑖𝑅1+𝜎𝑖𝐼𝜔𝑣𝑥𝑖𝑅𝑘𝜎𝑠sat𝜎𝜙𝜎,𝑘𝜎>𝑅2𝐼𝜔𝑣𝑥𝑖𝐹𝑥𝑖+𝜂𝜎,𝜙𝜎>0.(6.9) Furthermore, the applied torque during acceleration, by TCS, is computed as𝑇𝑖=𝑅𝐶𝜎||𝜎||+𝐼𝜔̇𝑣𝑥𝑖𝑅1𝜎𝑖𝐼𝜔𝑣𝑥𝑖𝑅1𝜎𝑖2𝑘𝜎𝑠sat𝜎𝜙𝜎,𝑠𝜎=𝜎𝑖||𝜎||.(6.10)

7. Simulation Results

The considered methods are tested in several critical maneuvers. A 9DoF nonlinear vehicle model is used for simulation purpose [4]. The vehicle behavior is probed during an open-loop maneuver, with no driver model, and two closed-loop maneuvers, including the driver model validated experimentally in [18]. The overall control scheme for simulation is shown in Figure 4. Comparison is made with a well-recognized case [3], where tire saturation conditions are ignored in ODF.

287195.fig.004
Figure 4: Overall closed loop scheme of the proposed IVDC.

The simulations are performed by MATLAB/Simulink. For the sake of practical implementations, the time for each control allocation was computed. Using a PC based on a 2.6 GHz Intel Core 2 Duo T9500 processor, the maximum time for the total IP algorithm is about 0.002 seconds. This time can be still improved using efficient C-Programming and effective numerical techniques.

7.1. Open-Loop Cornering Maneuver on a Split-𝜇 Road

The vehicle behaviour is examined in an open-loop split-𝜇 manoeuvre, where the tire-road friction coefficients on the vehicle left and right are equal to 0.3 and 1, respectively. The vehicle is assumed to move with a velocity of 110 km/hr and to turn according to the steering angle in Figure 5. The deceleration demand is −0.1 g. In Figures 6 and 7, both Saturation Constrained ODF (SCODF) approaches, that is, SCA and DCA, have better yaw rate tracking and side-slip angle reduction. Using ODF, the allocated work load to the front-right wheel is substantially beyond its actual capacity, other tires have still margin to saturation, as Figure 8 shows. On the other hand, from Figures 9 and 10 it is seen that SCODF methods have balanced the workload distribution and managed to use the total capacity of tire forces by saturating all tires in critical conditions. Accordingly, as shown in Figures 11 and 12, these methods have generated larger lateral and longitudinal forces, compared to ODF. To compare SCA and DCA, it is noted (Figure 7) that SCA has less side slip at some points relative to DCA. This arises because DCA has failed to completely saturate the tire on rear left in this case. Furthermore, Figures 11 and 12 reveal that lateral and longitudinal forces by DCA have properly converged to the optimal solution of SCA, with minor errors in some cases.

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Figure 5: Driver steering command in split-𝜇 scenario.
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Figure 6: Yaw angle velocity in split-𝜇 scenario.
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Figure 7: Side-slip angle in split-𝜇 scenario.
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Figure 8: Tire work-load distribution by ODF in split-𝜇 scenario.
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Figure 9: Tire work-load distribution by SCA in split-𝜇 scenario.
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Figure 10: Tire work-load distribution by DCA in split-𝜇 scenario.
fig11
Figure 11: Actually generated tire lateral forces in split-𝜇 scenario.
fig12
Figure 12: Actually generated tire longitudinal forces in split-𝜇 scenario.
7.2. Close-Loop Single-Lane Change Maneuver

A closed-loop high-speed Single-Lane Changing (SLC) on a slippery road with the braking acceleration of −0.5 g is considered. The vehicle initial speed is 120 km/h and the tire-road friction coefficient is set at 0.5. By Figures 13, 14, 15, 16, 17, and 18, enhanced vehicle control in terms of vehicle trajectory, yaw rate, and side-slip angle, as well as balanced work load distribution by SCODF approaches are evident. Figure 19 illustrates that the generated tire lateral forces by ODF are smaller than SCODF approaches. For the tires at the vehicle rear, this is primarily due to the smaller work load assigned to these tires by ODF. However, reduction of the lateral forces at the front tires can be described referring to longitudinal forces and slip-ratio plots in Figures 20 and 21. The ODF method has designated larger braking forces to front tires at some points, compared with SCODF approaches. According to friction circle concept, this reduces the corresponding generated tire lateral forces by ODF. Accordingly, as Figure 21 demonstrates, higher longitudinal forces by ODF have increased the slip ratio of these wheels by 𝜎=0.2. Then, ABS is activated by SRC to regulate the front slip ratios at 𝜎=0.2. On the other hand, SCA and DCA moderate the role of longitudinal forces in vehicle control and leave out the use of ABS.

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Figure 13: Vehicle path in SLC.
fig14
Figure 14: Yaw rate in SLC.
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Figure 15: Side-slip angle in SLC.
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Figure 16: Work-load distribution by ODF in SLC.
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Figure 17: Tire work-load distribution by SCA in SLC.
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Figure 18: Tire work-load distribution by DCA in SLC.
fig19
Figure 19: Actually generated tire lateral forces in SLC.
fig20
Figure 20: Actually generated tire longitudinal forces in SLC.
fig21
Figure 21: Tire longitudinal slip-ratios in SLC.
7.3. Close-Loop Double-Lane Change Maneuver

Another case to evaluate the system is high-speed Double-Lane Changing (DLC) on a slippery road with driver’s braking. The maneuver conditions remain unchanged with respect to SLC. In Figures 22, 23, and 24, SCODF approaches are still superior in controlling the vehicle path, yaw rate, and side-slip angle.

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Figure 22: Vehicle path in DLC with 𝜇=0.5.
fig23
Figure 23: Yaw rate in DLC with 𝜇=0.5.
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Figure 24: Side-slip angle in DLC with 𝜇=0.5.

To increase the challenge for the control systems, DLC is repeated on a more slippery road, where the tire-road friction coefficient is 0.4. Figures 25, 26, and 27 show the results in this case. Although both SCA and DCA methods have almost the same performance of vehicle control, oscillatory yaw rate response as well as more side-slip angle by DCA are clear. These are mainly due to transient response of DCA, specifically to panic reactions of driver in both steering and braking in critical conditions. The transient response is an inherent characteristic of the dynamic update law in DCA. In this scenario, the transient response has been triggered when the DCA tries to compensate for sudden braking release demand by driver. Another reason for oscillatory response by DCA is its relatively low robustness against highly nonlinear characteristics of vehicle dynamics and tire forces which are more substantial in more critical conditions. The oscillations by DCA grow more considerable once sudden changes occur in tire/road friction coefficient, for example, when the vehicle suddenly enters an icy road. Oscillations can intensify also after an actuator fails in a specific manoeuvre, where DCA is used for actuator failure compensation, which is one of the main advantages of control allocation methods [1]. In general, oscillatory response by DCA increases as the manoeuvre conditions worsen.

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Figure 25: Vehicle path in DLC with 𝜇=0.4.
fig26
Figure 26: Yaw rate in DLC with 𝜇=0.4.
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Figure 27: Side-slip angle in DLC with 𝜇=0.4.

8. Conclusion

Control allocation techniques can be applied effectively to enhance vehicle stability. Nonlinear vehicle motion control is split into three tasks. First, the total body lateral force and yaw moment were computed through a high-level sliding mode controller with adaptive upper bounds, where the knowledge of the upper bounds of uncertainties is not required. To allocate the vehicle high-level control objectives to saturation-constrained tire forces, static and dynamic control allocation techniques were formulated. Interior-point method was used to handle the nonlinear constrained optimization problem in SCA; a dynamic update law was derived using DCA. Simulation results were conducted, and vehicle operation under each method was evaluated and compared with a well-recognized work in the literature. It was seen that both SCA and DCA approaches manage to utilize whole capacity of tire forces more properly by offering balanced work-load distribution and rational adjustment of longitudinal and lateral tire forces. Since no optimization problem is required in DCA, the main advantage of this method is its computational efficiency for real-time implementations. Furthermore, although by DCA the set of distributed tire forces properly approaches the optimal one, that is, that of SCA, there is yet an amount of error, depending on the severity of maneuver. This error of DCA is mainly due to its transient response to panic behavior of driver and its relatively low robustness to highly nonlinear characteristics of vehicle dynamics and tire forces. Therefore, in maneuvers with more critical conditions, the performance of the vehicle control task can be degraded using DCA compared to SCA.

Nomenclature

𝑎𝑥:Driver’s braking acceleration
𝐴𝑖:Weighting coefficients of 𝑖th tire
𝐶𝑖:Cornering stiffness of 𝑖th tire
𝐂𝐈:Vector of saturation constraints
𝐶𝜎:Tire longitudinal stiffness
𝑑:Vehicle tread
𝑓:Objective function
𝐹𝑥𝑖:Actual longitudinal force of 𝑖th tire
𝐅𝑥𝑖:Maximum deviation from linear part for 𝐹𝑥𝑖
𝐹𝑦𝑖:Actual lateral force of 𝑖th tire
𝐼𝑤:Moment of inertia of each wheel
𝐼𝑧:Yaw moment of inertia of vehicle
𝐼𝑧:Estimation of yaw moment of inertia
𝐼𝑧:Estimation of yaw moment of inertia error
𝐼𝑧:Upper bound for 𝐼𝑧
𝐽:Objective function in quadratic form
𝑘𝑟, 𝑘𝛽, 𝑘𝜎:Sliding mode control gain
̂𝑘𝛽1, ̂𝑘𝛽2, ̂𝑘𝑟1, ̂𝑘𝑟2:Adaptive gains
𝑘𝑑𝛽1,𝑘𝑑𝛽2, 𝑘𝑑𝑟1, 𝑘𝑑𝑟2:Uncertainty upper bound
𝐿:Lagrangian
𝐿𝑓,𝑟:Distance between mass center and axle
𝑚:Vehicle mass
𝑚:Vehicle mass estimation
𝑚:Mass estimation error
𝑚:Upper bound for 𝑚
𝑀:Body yaw moment
𝑅:Wheel radius
𝑟:Yaw angle velocity
𝑟𝑏,𝑟𝑐,𝑟𝑠:IP residuals
𝑠𝑟:Sliding surface for yaw rate
𝑠𝛽:Sliding surface for side-slip angle
𝑠𝜎:Sliding surface for SRC
𝑇𝑖:Applied torque at the 𝑖th wheel
𝐮:Vector of allocated tire forces
𝑉:Vehicle velocity
𝑣𝑥𝑖:Longitudinal velocity of the 𝑖th tire
𝑉𝛽,𝑉𝑟,𝑉dyn,𝑉0:Lyapunov candidate
𝐯:Vector of virtual control input
𝐖:Matrix of weighting coefficients
𝑋:Body longitudinal force
𝑌:Body lateral force
𝑋𝑖:Allocated longitudinal force to 𝑖th tire
𝑌𝑖:Allocated lateral force to 𝑖th tire
𝑍𝑖:Vertical load of 𝑖th tire.
Greek Letters
𝛼𝑖:Side-slip angle of 𝑖th tire
𝛼𝑠max,𝛼𝜆max:Newton’s step size
𝛽:Side-slip angle
𝛿𝑖:Steering angle of 𝑖th tire
Δ𝛽, Δ𝑟:Uncertainty upper bound
𝜂𝑘:Barrier parameter
Φ𝛽, Φ𝑟, 𝜙𝜎:Boundary layer thickness
𝝀:Vector of Lagrange multipliers
Λ:Matrix of Lagrange multipliers
𝜎:Centering parameter
𝜎𝑖:Slip ratio of the 𝑖th tire
𝜎:Reference value of slip ratio
𝜇𝑖:Friction coefficient of 𝑖th tire
𝜔𝑟, 𝜔𝛽:Disturbance terms
𝜔𝑖:Angular velocity of 𝑖th tire.
Subscripts/Superscripts
aff:Affine scaling
𝑓:Front
𝑟:Rear
𝑑:Desired value
+:N iteration
dyn:Dynamic.
Coordinate System
(𝑥,𝑦,𝑧):Moving coordinate attached to vehicle centre.

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