1202IEEE TRANSACTIONS ON CONTROL SYSTEMS TECHNOLOGY, VOL. 20, NO. 5, SEPTEMBER 2012Fast and Global Optimal Energy-Efficient Control Allocation With Applications to Over-Actuated Electric Ground VehiclesYan Chen and Junmin Wang, Member, IEEEAbstract—This paper presents a fast and global optimization algorithm for an energy-efficient control allocation (CA) scheme, which was proposed for improving the operational energy efficiency of over-actuated systems. For a class of realistic actuator power and efficiency functions, a Karush-Kuhn-Tucker (KKT)-based algorithm was devised to find all the local optimal solutions, and consequently the global minimum through a further simple comparison among all the realistic local minima and boundary values for such a non-convex optimization problem. This KKT-based algorithm is also independent on the selections of initial conditions by transferring the standard nonlinear optimization problem into classical eigenvalue problems. Numerical examples for electric vehicles with in-wheel motors were utilized to validate the effectiveness of the proposed global optimization algorithm. Simulation results, based on the parameters of an electric ground vehicle actuated by in-wheel motors (whose energy efficiencies were experimentally calibrated), showed that the proposed global optimization algorithm was at least 20 times faster than the classical active-set optimization method, while achieving better control allocation results for system energy saving. Index Terms—Electrical ground vehicles (EGVs), energy-efficient control allocation (EECA), global optimality, in-wheel motors, Karush-Kuhn-Tucker (KKT) conditions, over-actuated systems.I. INTRODUCTIONOVER-ACTUATED systems, in which the number of actuators is greater than the degrees of freedom, have attracted increasing attention in recent years [1]–[5]. Many existing physical systems, such as marine vessels [6], [7], airplanes [8]–[10], and ground vehicles [11]–[16], [34], can be classified as over-actuated systems because redundant actuators are utilized to improve system performance, reliability, and reconfigurablity. In order to coordinate the redundant actuators and control the over-actuated systems in an elegant configuration, the main challenge is how to handle the actuator redundancy and physical constraints simultaneously. Control allocation (CA),Manuscript received October 06, 2010; revised February 20, 2011; accepted July 05, 2011. Date of publication August 12, 2011; date of current version June 28, 2012. This work was supported by Office of Naval Research (ONR) Young Investigator Award under the Grant N00014-09-1-1018, Honda-OSU Partnership Program, and OSU Transportation Research Endowment Program. Recommended by Associate Editor F. Basile. The authors are with the Department of Mechanical and Aerospace Engineering, The Ohio State University, Columbus, OH 43210 USA (e-mail: wang. 1381@). Color versions of one or more of the figures in this paper are available online at . Digital Object Identifier 10.1109/TCST.2011.2161989as a feasible and promising method, is commonly employed in over-actuated systems to optimally allocate the desired virtual (generalized) controls among all the available actuators within their respective constraints, see surveys [6], [9], and the references therein. Several different CA algorithms, such as direct allocation [17], daisy-chain allocation [18], (redistributed) pseudo-inverse allocation [14], [19], optimization-based allocation [8], and adaptive control allocation [20] (which was later applied to vehicle control simulations in [21]) have been proposed based on different methods of distributing the virtual controls to the redundant actuators. Although the aforementioned CA methods have different strengths and limitations, numerical optimization-based algorithms are becoming more and more widely used in CA [7], [8], [13], [22]–[26]. Sørdalen [7] adopted the magnitude of actuation as a secondary optimization term in CA formulation in order to achieve minimum control effort. Bodson [8] summarized and evaluated error minimization, control minimization, and mixed error and control optimization problems for optimization-based CA methods. Härkegård [22] suggested a dynamic CA method by penalizing virtual control at a previous sampling time in the secondary optimization term. Zaccarian [24] described a dynamic CA by inserting a dynamic system as an allocator between the high-level controller and low-level actuators. Moreover, Johansen et al. [25] resolved a CA problem on ship control by decomposing a non-convex thrust region into multiple convex sets, solved by mixed-integer-like convex quadratic programming. Within the aforementioned optimization-based allocation methods, the optimal algorithms for solving various CA problems were similar although the problem formulations were different. Standard numerical algorithms, such as quadratic programming, active-set, and fixed-point, were applied to solve various nonlinear programming problems of CA. However, local minima are often obtained by adopting these standard numerical algorithms for general nonlinear programming CA problems. On the other hand, the selections of the initial conditions of these standard algorithms also strongly influence the acquisition of the global minimum. Although there are no uniform optimization algorithms for general nonlinear programming problems, this paper proposes a global and initial-condition-independent algorithm for energy-efficient CA [27] based on the Karush-Kuhn-Tucker (KKT) conditions. In the expression of energy-efficient CA problem, a special nonlinear programming is considered since the secondary optimization term, standing for the system power consumption,1063-6536/$26.00 © 2011 IEEECHEN AND WANG: FAST AND GLOBAL OPTIMAL ENERGY-EFFICIENT CONTROL ALLOCATION1203consists of polynomial and/or fractional functions. By utilizing this kind of characteristics, the nonlinear programming procedure for energy-efficient CA can be transferred into classical eigenvalue problems, which are initial-condition-independent, based on the KKT conditions. Thus, all the physically meaningful eigenvalues are obtained as local minima of the energy-efficient CA problem. Through simple comparisons and exclusions, a global minimum can be obtained. The main contributions of this paper are: 1) a novel energy-efficient control allocation scheme which can explicitly incorporate actuator efficiencies and actuator operating modes into the control allocation for over-actuated systems and 2) a fast and global algorithm for solving the non-convex optimization problems associated with the proposed energy-efficient CA scheme. The remainder of this paper is organized as follows. In Section II, both single-mode and dual-mode energy-efficient CA schemes are described. The KKT condition-based algorithm is proposed to find the global minimum for the energy-efficient CA in Section III. In Section IV, numerical examples are given to verify the effectiveness of the proposed global optimization methods. In Section V, simulations based on longitudinal motion control of an electric ground vehicle actuated by in-wheel motors are presented to show the effectiveness of the proposed optimization algorithm in comparison to the standard active-set optimization algorithm. Conclusive remarks are presented in Section VI. II. ENERGY-EFFICIENT CONTROL ALLOCATION In this section, the main ideas and formulations of singlemode and dual-mode energy-efficient CA are briefly described. For more details and discussions on the energy-efficient CA, the reader can refer to the authors’ work [27]. A. Single-Mode Energy-Efficient CA A general over-actuated dynamic system is described as follows: (1) , where the system state vector is represented by is the system output vector, is the virtual control is the control effectiveness matrix, and vector, is the control input vector. Note that when the relaand is nonlinear, the control effectivetionship between ness matrix can be obtained by local approximation with an affine mapping through linearization at each sampling instant holds as the number [28]. For over-actuated systems, of actuators is greater than the number of virtual control signals and the number of the controlled system outputs. Thus, there is no unique solution for in general and CA is often utilized to optimal mapping problem. address the For over-actuated systems in which each actuator has only one actuation mode and one corresponding energy efficiency function, the energy-efficient control allocation is formulated as (2) s.t.where is the total instantaneous power consumption by all the actuators. The actuator amplitudes’ lower and upper bound and in a component-wise vectors are represented by fashion, respectively. Such magnitude bounds can also incorporate the actuator rate limitations based on the given system sampling period and difference formulation, see [22] and [27] for details. A small positive parameter is used to balance the efforts between reducing CA errors and the power consumptions. Note that smaller parameter will yield less CA errors [8]. Small CA errors may not noticeably affect the overall system control performance. Moreover, a high-level robust controller can help to overcome/mitigate the influence of the limited CA errors in the mixed optimization formulation. The system power consumption can be a function of actuator force or torque values based on the corresponding actuator efficiency functions, which can be expanded as follows: (3) Within (3), and are the output power function and the efficiency function of the th actuator, respectively. and represents Therefore, the division between . Thus, the formulathe th actuator’s power consumption tion of single-mode energy-efficient CA is developed by (2) and (3). B. Dual-Mode Energy-Efficient CA When actuators in physical systems have dual operating modes, such as consuming and gaining energy, and different associated efficiency functions, the energy-efficient CA becomes more challenging. For such dual-mode actuators, their effects on the virtual control, magnitude/rate constraints, efficiencies, power consumption, or gain characteristics may vary with operating modes. Consequently, to achieve energy-efficient CA for systems involving dual-mode actuators, the actuator mode selections need to be seamlessly integrated into the CA scheme as well since the dual-mode actuators with different modes/amplitudes may construct the same virtual control signal with different energy consumptions. Thus, the energy-efficient CA scheme needs to dictate not only the magnitude but also the operating mode for each of the redundant actuators in an optimal way. Such a requirement cannot be satisfied by the existing standard CA schemes and thus represents a key challenge for extending the foregoing energy-efficient CA to physical systems whose actuators have dual actuation modes. Here, a virtual actuator concept is introduced for tackling such a challenge. The expression of over-actuated dynamic systems described in (1) is correspondingly augmented as (4) A virtual actuator control input vector , is introduced to represent the of actuators who have dual operating modes in the system. This virtual actuator augment systematically incorporates the dual-mode actuators into the energy-efficient CA scheme. Such a method enables the CA to seamlessly dictate both actuators’ magnitudes and operating1204IEEE TRANSACTIONS ON CONTROL SYSTEMS TECHNOLOGY, VOL. 20, NO. 5, SEPTEMBER 2012modes, resolving the aforementioned challenge. The augis the new control mented matrix effectiveness matrix for the new system. Note that the matrix are determined by the dual-mode actuators. The energy-efficient CA scheme (2) is modified for the new augmented system (4) with dual-mode actuators as (5) s.t. Since a particular dual-mode actuator can only operate in one of the two operating modes at a given time instant, the added third term in the constraints ascertains that only one operating mode is assigned to a physical actuator by the energy-efficient CA. Electric motors such as the in-wheel motors equipped on electric ground vehicles can be classified as dual-mode actuators. In electric ground vehicles, the electric motors can either work in motor (driving) mode, in which motor consumes onboard electric energy or generator (regenerative braking) mode, in which motor produces electric energy from the vehicle kinetic energy. thus can include Within (5), the total power consumption the instantaneous power consumption and gain with respect to the dual operating modes for each of the physical actuators. For example, if denotes the actuator energy consuming mode and represents the actuator energy gaining mode, then the total power consumption of all the actuators in different modes is formulated as (6) Within (6), and denote the actuator output power and and efficiency at the energy consuming mode while represent the actuator input power and efficiency at the energy gaining mode, respectively. The energy gaining mode is inferred by the minus sign in front of virtual actuator power consumption. Therefore, the formulation of the dual-mode energy-efficient CA is built by (5) and (6). III. KKT-BASED GLOBAL OPTIMIZATION ALGORITHMS FOR ENERGY-EFFICIENT CONTROL ALLOCATION For both single-mode and dual-mode energy-efficient CA, the authors adopted a standard nonlinear optimization method, active-set algorithm, to obtain the solutions [27]. The adopted active-set algorithm, however, cannot guarantee the global optimization and is sensitive to the selections of initial conditions. Based on the KKT conditions, a global and initial-condition-independent optimization algorithm for the energy-efficient CA is proposed as follows. A. KKT Conditions and Algorithm for the Single-Mode Energy-Efficient CA In order to develop the optimization algorithm for the singlemode energy-efficient CA, the expressions (2) and (3) are combined and modified as (7)s.t. The square modification for CA errors is convenient for derivatives of the Lagrangian function defined later. are the vector forms of output power functions and efficiencies, respectively. Define the following Lagrangian function: (8) Within (8), nonnegative vectors and are the Lagrangian multipliers. Based on the KKT conditions [29], the optimal solution with certain Lagrangian multipliers and satisfy the following conditions:(9)Remark 1: Generally, the KKT conditions are necessary conditions for local minima. They can also be utilized to sufficiently characterize the global optimal solution when the objective/cost function and the constraint set are convex. Although the cost function in (7) is not in a convex form, further examinations can be fulfilled to exclude the maximum and the local minima. The calculations on optimal control are categorized by the values of Lagrangian multipliers and . When the Lagrangian mul, the control can be calculated within tipliers and/or boundaries by solving the algebraic equations (9). For the special form of energy-efficient CA, the secondary optimization term is a sum of fractions, whose denominators and numerators can be characterized by low-order polynomial functions of power and efficiency, making (9) be easily solvable. When the and/or , the control Lagrangian multipliers has to be equal to the boundary values and/or based on the KKT conditions. Thus, a few local minima including the boundary values can be obtained from the above two categorized steps. At last, further comparisons among these local minima based on the criteria of the least power consumption can give the global minimum. In sum, the algorithm is written in a pseudo-code form as follows. Inputs: and are from the system model and high-level • controller. , , , and are from actuator models • and experimental calibrations. and are adjustable parameters for optimization. • Steps: 1) Set , solve algebraic equations (9) and obtain all the nontrivial local minima .CHEN AND WANG: FAST AND GLOBAL OPTIMAL ENERGY-EFFICIENT CONTROL ALLOCATION12052) Set , consequently obtain and , can be obtained by solving the other the remaining algebraic equations in (9), like the process in step 1. , consequently obtain 3) Similar to step 2, set and , the other can be obtained by solving the left algebraic equations in (9), like the process in step 1. 4) Calculate the total power consumptions (the second term obtained from the above three steps, in (7)) for all the by comparison. and obtain the global optimal B. KKT Conditions and Algorithm for the Dual-Mode Energy-Efficient CA Similar to the single-mode energy-efficient CA case, the expressions (5) and (6) are combined and modified so as to develop the optimization algorithm for the dual-mode energy-efficient CA(10)s.t.Remark 2: Again, although the KKT conditions, which are necessary for local minima, generally cannot guarantee the global optimal solution of the non-convex cost function (10), further examinations can be fulfilled to exclude the maximum and and the local minima. The calculations on control are categorized by the values of Lagrangian multipliers , , , and . When the Lagrangian multipliers and/or ( and/or ), the control can be calculated within boundaries by solving the algebraic (12). The special secondary optimization term in (10) consists of a sum of polynomials and fractions, whose denominators and numerators can also be characterized by low-order polynomial functions of power and efficiency, making (12) be readily solvable. Moreover, the complementary condition determines that and cannot be nonzero simultaneously, which can be further applied to reduce the computational cost and . When the Lagrangian of (12) by setting multipliers and/or ( and/or ), has to be equal to the boundary values. the control Thus, a few local minima including the boundary values can be obtained from the above two categorized steps. At last, further comparisons among these local minima based on the criteria of the least power consumption can give the global minimum. Similar to the single-mode CA case, the algorithm for dual-mode CA is given in a pseudo-code form as follows: Inputs: • and are from the system model and high-level controller. , , , , , , and • are from actuator models and experimental calibrations. • and are adjustable parameters for optimization. Steps: , solve algebraic equations 1) Set and . (12) and obtain all the nontrivial local minima can be Note that the complementary condition applied to simplify the calculation of (12) by categorizing , and , the whole process into ( are trivial cases without solving (12)). 2) Set , consequently obtain and , through the complementary and furthermore . The other and condition can be obtained by solving the remaining algebraic equations in (12), like the process in step 1. , consequently obtain 3) Similar to step 2, set and , and furthermore through . The other the complementary condition and can be obtained by solving the left algebraic equations in (12), like the process in step 1. and , obtain 4) Repeat steps 2 and 3 for and . 5) Calculate the total power consumptions (the second term and obtained from the above in (10)) for all the and by steps, and find out the global optimal comparison.where are the vector forms of input power functions and efficiencies, respectively. Define the corresponding Lagrangian function as(11) Within (11), nonnegative vectors and , and , are the Lagrangian multipliers. Based on the KKT conditions and with certain Lagrangian [29], the optimal solutions , and satisfy the following conditions: multipliers , ,(12)1206IEEE TRANSACTIONS ON CONTROL SYSTEMS TECHNOLOGY, VOL. 20, NO. 5, SEPTEMBER 2012Fig. 1. Experimentally measured efficiency curve of an in-wheel BLDC motor and its controller. Fig. 2. Driving efficiency map of an in-wheel BLDC motor and its controller based on experimental data. TABLE I IN-WHEEL MOTOR EFFICIENCY FUNCTION PARAMETERSmotor speed also slightly affects the motor efficiency as can be seen from the measured experimental data shown in Fig. 2, the efficiency curves are similar within a large range of motor rotational speeds [31]. Moreover, motor rotational speed can be reasonably assumed to be a constant at each instantaneous time for CA due to the short sampling period. The power consumption of the in-wheel motor is given by It should be noted that as the number of the system redundant actuators increases, the computational effort and complexity of the preceding algorithms will grow as well. In addition, the actuator efficiency function characteristics also influence the complexity of the KKT-based energy-efficient CA algorithm. IV. NUMERICAL EXAMPLES AND DISCUSSIONS In this section, the aforementioned KKT-based algorithms for both single-mode and dual-mode energy-efficient CA are applied to numerical examples. These examples, generated from in-wheel motor models and experimental data, verify the assumption and validation of the aforementioned formulation. of an in-wheel brushless The output efficiency function DC (BLDC) motor and its controller is expressed by fitting the experimental data [31] shown in Fig. 1. Two linear functions are adopted to approximate the rising and falling potions of the entire experimentally measured data (13) , , , and are coefficients, listed in Table I. where Multiple reasons make the piece-wise linear function (13) as the efficiency fitting function. The first one is the consideration of computational effort. From either (9) or (12), the KKT conditions offer algebra equations or eigenvalue problems to obtain the optimal values. The simpler efficiency fitting function makes the computational cost less. It can be seen later that this piece-wise division actually makes the global optima obtainable. The second reason is due to the DC characteristics of the brushless DC motors. The piece-wise linear function (13) can sufficiently describe the rising and falling trends along with the increase of motor torque. Last but not the least, although the (14) is a given rotational speed and stands for the motor where torque. Without loss of generality, two in-wheel BLDC motors are considered for the longitudinal speed control of a bicycle vehicle model under straight line driving condition. Different scaling ratios were applied to the efficiency curve in Fig. 1 to generate different efficiency functions for two motors and two operating modes. In the case of single-mode actuation (both in-wheel motors drive), the control effectiveness matrix is . In the case of dual-mode actuation (both in-wheel motors can perform driving and regenerative braking), the control . The boundaries of the effectiveness matrix is 0 Nm and 100 Nm for driving, and actuator are 100 Nm and 0 Nm for regenerative braking. A. Single Mode Energy-Efficient CA In the numerical example for single-mode energy-efficient 400 rpm, CA, the motor rotational speed was selected as which is about 50 km/h for a passage car with a common tire effective radius around 0.3 m. The penalty coefficient was set was set to be an idento be 0.001 and the weighting matrix tity matrix for the optimization problem. The scaling ratio for efficiency of the second in-wheel motor was 0.9. for nonSubstituting (14) into (9) and letting trivial solutions, the following equations are obtained:(15)CHEN AND WANG: FAST AND GLOBAL OPTIMAL ENERGY-EFFICIENT CONTROL ALLOCATION1207Fig. 3. Space of the total power consumption for single-mode energy-efficient CA with two in-wheel motors.Fig. 4. Power consumption of single-mode energy-efficient CA with two in-wheel motors with different given virtual control values.Since the efficiency function (13) is piece-wise linear, the equations in (15) have to be solved by combining different efficiency functions. Basically, four pairs of two algebraic equations are obtained. In order to solve two algebraic equations with two variables, like (15), variable cancellation can obtain the equivalent problem of finding the roots of a polynomial due to the simple efficiency and power expressions (13) and (14). The obtained polynomial can be taken as a characteristic polynomial of a classical eigenvalue problem. Thus, the optimization problem are transferred into four (two pairs of) eigenvalue problems to and . Compared with the trivial solutions find optimal (boundary values of actuators), the global minimum can be finally obtained. In order to show the real global minimum is achieved, the space/surface of the total power consumption, the secondary optimization term in (7), is plotted in Fig. 3. As shown in Fig. 3, the total power consumption generally becomes larger with the increase of motor torque. However, due to the non-convex characteristics of the surface, it is hard to find the global minimum by using standard optimization algorithms. Through the piece-wise fitting for the efficiency functions of the actuators, the non-convex space/surface is actually , each of which is convex divided into four portions within its own definitional area. Thus, in each area, the corresponding global minimum can be obtained through KKT conditions, which is equivalent to an eigenvalue problem. Then, a simple comparison among these global minima and boundary values of different areas can suggest the true global minimum in the entire non-convex space of the power consumption. The five lines on the surface stand for different virtual control values, which are the intersection curves between the non-convex power consumption surface and vertical planes, corresponding to dif. These curves define the solution sets of ferent problem (7) under different virtual control signals without considering the power minimization. Given different virtual control , we can clearly observe the global optimization of the distribution torques and from the plot in Fig. 4. Fig. 4 shows the power consumption of optimization problem (7) by inserting efficiency and power expressions in (13) and (14). Each curve in Fig. 4 represents different virtual controlsignals from 10, 20, 40, 60, to 80 Nm. On each curve, the corresponding virtual control is equal to the sum of distributions onto the two motors. All the labels in Fig. 4 represent the global optimization points. Although these global minimum points vary on the non-convex curves, the proposed KKT-based algorithm accurately found all of them from the equivalent eigenvalue problems and simple comparisons with boundary values. If a standard active-set algorithm is applied to solve the nonlinear optimization problem, the global minimum points may not be found by inappropriate choices of initial conditions and only local minima can be obtained. It is interesting to observe the 40 Nm. The global optimum dictates that the equal case of torque distribution gives the lowest power consumption, which happens to be the distribution result of a standard CA method [27]. This is because the most efficient operating point lies on 20 Nm, which can be seen from Fig. 1 and the efficiency scaling factor. B. Dual-Mode Energy-Efficient CA The same values for motor rotational speed , weighting maand the penalty coefficient as those in the single-mode trix case were adopted for the numerical example in dual-mode energy-efficient CA. Assuming that two in-wheel motors (front and rear) have similar efficiency profiles, the scaling ratio for the driving efficiency of the rear motor was 0.9. Moreover, the motor regenerative braking efficiency is usually less than the driving efficiency. Thus, the scaling ratio for the regenerative braking efficiency of each motor was set as 0.9 of the corresponding driving efficiency. Substituting (14) into (12) and letting for nontrivial solutions, the following expressions are obtained:1208IEEE TRANSACTIONS ON CONTROL SYSTEMS TECHNOLOGY, VOL. 20, NO. 5, SEPTEMBER 2012Fig. 5. Space of the total power consumption for dual-mode energy-efficient CA with two in-wheel motors.Fig. 6. Power consumption of dual-mode energy-efficient CA with two in-wheel motors with different given virtual control values.(16)Fig. 7. EGV longitudinal speed tracking control structure.Note that although four algebra equations and two complemenand tary equations are listed in (16), the four equations for or and cannot be held simultaneously due to two complementary equations. Thus, the two complementary equations can be logically applied to reduce the computational cost since only two algebra equations are solved together. Like the process in the single-mode case, similar steps are implemented to solve (16) for local minima by plugging in the piece-wise linear efficiency function (13). Then, comparisons among local minima and trivial solutions (boundary values) will give the globally optimal solutions. In order to show the real global minimum is achieved, the space/surface of the total power consumption, the secondary optimization term in (10), is plotted in Fig. 5. As shown in Fig. 5, the power consumption space becomes more non-convex than that in the single-mode case. Since each virtual actuator has two piece-wise linear efficiency potions, the total non-convex power space consists of sixteen convex . However, for a certain virtual control value, areas only parts of the convex areas are meaningfully involved in. Fig. 5 shows that seven convex surfaces are physically achiev20 Nm. Thus, similar to the single-mode case, a able when simple comparison among global minima and boundary values of different areas can suggest the true global minimum in the entire non-convex space. The four lines on the surface stand for different virtual control values, which are plotted in Fig. 6 to clearly show the global minimum for the torque distribution and . among Fig. 6 shows the power consumption of the optimization problem (10) by inserting efficiency and power expressions in (13) and (14). Each curve in Fig. 6 represents different virtual control values from 4, 8, 10, to 20 Nm. On each curve, the corresponding virtual control is equal to the sum of distributions on the two motors. All the labels in Fig. 6 represent the global optimal points. Although these global minimum points vary on the non-convex curves, the proposed KKT-based algorithmaccurately finds all of them from the equivalent eigenvalue problems and simple comparisons with the boundary values. V. SIMULATIONS OF ELECTRIC GROUND VEHICLES In this section, the proposed KKT-based algorithm for both single- and dual-mode energy-efficient CA are applied to an electrical ground vehicle (EGV) model equipped with front/rear in-wheel motor sets. Electric concept cars with independently actuated in-wheel motors such as the GM Geo Storm and Volvo ReCharge have already proceeded into the prototyping and/or pre-market phases. Thus, in-wheel motors have potential applications in automotive industry, see [13], [27], and reference therein. Compared with the conventional vehicle drivetrain architectures where driving and braking of different wheels are coupled, EGVs with independently actuated in-wheel motors can offer higher control flexibility and many potential advantages. Consequently, the efficiency and control allocation problem are meaningful to be investigated. Note that the proposed energy-efficient CA schemes are applicable for many over-actuated systems, well beyond the electrical/hybrid vehicles. Moreover, such general energy-efficient CA methods are different from the power management methods for hybrid vehicles [32], [33] because of the distinctions in number and characteristics of actuators as well as the control purposes for over-actuated systems. The structure for vehicle longitudinal speed tracking control is shown in Fig. 7 for comparison between the common active-set method and the proposed KKT-based algorithm. The and , desired and measured vehicle speed are denoted as respectively. The high-level speed tracking controller is a proportional plus integral (PI) controller, which generates the virtual control for the energy-efficient control allocator. Note that other high-level speed tracking controllers can be employed as。
