Trajectory Optimization with Memetic Algorithms: Time-to-Torque Minimization of Turbocharged Engines

Transcription

1 Trajectory Optimization with Memetic Algorithms: Time-to-Torque Minimization of Turbocharged Engines Dan Simon *, Yan Wang **, Oliver Tiber *, Dawei Du *, Dimitar Filev **, John Michelini ** * Cleveland State University, Cleveland, USA ** Ford Motor Company, Dearborn, USA d.j.simon@csuohio.edu Abstract A general memetic trajectory optimization method is introduced. The method is comprised of an evolutionary algorithm (EA) for global optimization, followed by local optimization. The global optimization algorithm is biogeography-based optimization (BBO), which is an EA motivated by the migratory behavior of biological organisms. For local optimization, we start with identifying a local linearized model within the region of the BBO solution by approximating the linear model with Jacobian matrix, and then optimize trajectory using gradient method. The process iterates Jacobian learning and optimization until an optimal trajectory is identified. We apply this memetic algorithm to a time-to-torque minimization problem for a gasoline turbocharged direct injection automotive engine. The optimized trajectory demonstrates significant improvement over the intuitive bang-bang controls that were originally thought to deliver the fastest transient torque response. Simulation results show that BBO decreases time-to-torque by 48% relative to bang-bang controls, and adaptive optimization decreases time-totorque by an additional 26%. These results have significant implications for improved automotive engine performance. I. INTRODUCTION Trajectory optimization is a broad field of research that began in the 1960s with the advent of optimal control and its implementation in digital computers. Trajectory optimization can be viewed as the problem of determining a set of time-varying control inputs for a dynamic system that minimizes a given cost function. The cost function may also be referred to as a performance index or return function. The problem may include constraints on the states or controls. Necessary mathematical conditions can be readily derived for broad classes of trajectory optimization problems. However, analytical solutions are available only for special types of problems, such as linear quadratic (LQ) problems in which the system dynamics are linear and the cost function is quadratic. If analytic solutions are not available, as is typical in real-world problems, then solutions can be found using numerical methods. Numerical trajectory optimization methods can be categorized as direct or indirect. Direct methods parameterize the controls or the states. The time points at which the controls and states are sampled are called collocation points, and direct trajectory optimization methods are sometimes called collocation methods. Direct methods have the advantage that they can be initialized with guesses for not only the controls but also for the states, which can often be easier to guess than initial estimates for the controls. Because of the conversion of the dynamics into constraints at each time point, direct methods can result in huge optimization problems, but this drawback can be alleviated by the sparseness of the matrices that are involved. One direct method is direct shooting, which parameterizes only the control and then integrates the system dynamics to satisfy the differential constraints [1]. Another direct method is state and control parameterization, which parameterizes both states and controls and converts the system dynamics into a set of constraints, thus removing the need for dynamic simulation [2]. Indirect methods simulate the dynamic system and then check if the necessary optimality conditions are satisfied. If not, the initial conditions or parameters of the simulation are modified with some numerical method, and the process is repeated. Indirect methods can suffer from poor numerical properties, such as high sensitivity of the optimality conditions with respect to the initial conditions. However, because of the use of simulation, every iteration results in a trajectory that satisfies the dynamic constraints, whereas in direct methods the constraints need to be satisfied before optimality can be pursued. Some popular indirect methods are steepest descent [3], conjugate gradient [4], and evolutionary algorithms [5]. This paper presents a framework for trajectory optimization using memetic algorithms. Our method is a direct method because we parameterize the control trajectories, followed by optimization of those parameters to obtain optimal trajectories. First we parameterize the trajectories using Fourier series or Gaussian kernels, and then optimize the Fourier series parameters or Gaussian kernel gains using an evolutionary algorithm (EA) called biogeography-based optimization (BBO). Then we further improve the trajectories by optimizing the Fourier series parameters or Gaussian kernel gains using a local optimization algorithm called adaptive optimization using iterative Jacobian learning. Finally, we apply our methods to a practical and important automotive problem, which is the minimization of time-to-torque of GTDI engines. This paper is organized as follows. Section 2 discusses the approach we propose for trajectory optimization, including parameterization of the control profiles using two different methods (Fourier series and Gaussian kernels); global optimization using BBO as a first step to find the neighborhood of the global optimum; and local optimization to search for the best solution within the globally-optimal neighborhood using adaptive optimization with Jacobian learning. Section 3 discusses the case study for our approach, which is a time-to-torque minimization problem for a GTDI engine. Section 4 contains conclusions and suggestions for future work. II. MEMETIC TRAJECTORY OPTIMIZATION WITH Researchers have found EAs to be attractive options for solving difficult control problems. For trajectory optimization problems, we can parameterize the control trajectories and then use the EA for parameter optimization. Previous research along these lines includes

2 GAs for nonlinear, second-order, two-point boundary value problems [6], for trajectory optimization [7], for missile guidance [8], and for robot control [9]. However, the weakness of EAs is their inability to home in on the global optimum. EAs have been shown to be good global optimizers, but they only find candidate solutions that are in the neighborhood of the global optimum. Therefore, further optimization within the neighborhood is required to find a more exact optimum. Such an algorithm combines evolutionary search with local search and is called a memetic algorithm. In this section we introduce our memetic algorithm for trajectory optimization. The first part of this section discusses parameterization of control trajectories using static parameters, the second part discusses global optimization of the static parameters with an EA to find the neighborhood of the global optimum, and the last part discusses local optimization to home in on the exact optimum within the neighborhood of the global optimum. A. Parameterization of Transient Profile The key to the approach in this section is the conversion of the trajectory optimization problem to a parameter optimization problem. In this section, we introduce two approaches to parameterize control profiles: Fourier series and Gaussian kernels. 1) Fourier Series The Fourier series is a general approach to parameterization that was first used for the optimization of structural systems [10]. We assume that the optimal profile of each control signal is continuous on the interval [0, T], where T is the fixed final time of the transient response. Although T is fixed here, the optimization cost function that we describe later is the time at which the transient response reaches steady-state, which, in general, is less than T. So even though T must be fixed in this approach, the approach can still be used for free-final-time problems. In order to solve the trajectory optimization problem, we parameterize each candidate control solution as a Fourier cosine series. = + (1) = half of control duration 1,, where s the number of control signals s the number of Fourier coefficients per control is the number of independent variables M is a tradeoff between search resolution and problem size. T is half of the control duration. We use the Fourier cosine series rather than the full Fourier series to reduce the dimension of the optimization problem by almost 50% [11]. That is, u k(t) is defined as an even function on the time interval t [ T, +T ], but we only use the control signal between time 0 and T when simulating the system. The value of T that we use is somewhat arbitrary. If we re trying to minimize time, then we set it to some value that is comfortably larger than our expected minimum transient response. 2) Gaussian Kernel Another approach to parameterize trajectories is to use a Gaussian kernel, which is defined in one dimension as follows:, = (2) where σ defines the width of the Gaussian. As described in [12], any trajectory U i(t) can be approximated by the sum of multiple kernels: = where k is the control index. For the k-th control, M i is the number of kernels, A kj is the j-th kernel gain, and σ k is the smoothness parameter. With this representation, the total number of tuning parameters for the optimal control problem is, where N is the number of control profiles. Note that the Fourier series above could use the same approach, in which we have a different number of coefficients for each control. B. Global Optimization With the parameterization approaches discussed above, the transient optimization problem has been converted to a parameter optimization problem. An EA can optimize parameters to find the neighborhood of the optimum. In this paper, BBO is selected as the global optimization method. It has shown good performance compared with many other EAs for a variety of problems [13], including benchmarks and real-world problems such as ECG classification [14], power system optimization [15], and image classification [16]. BBO is based on the science of species migration between habitats. Habitats have different degrees of suitability for species habitation. This is called the habitat suitability index (HSI). Habitats with a high HSI tend to have a large number of species, and habitats with a low HSI tend to have a small number of species. Species will immigrate to, and emigrate from, a habitat with a probability that is determined by how many species reside in the habitat. A habitat with a large number of species (high HSI) will tend to have a low immigration rate and a high emigration rate. Conversely, a habitat with a small number of species (low HSI) will tend to have a high immigration rate and low emigration rate. Figure 1 shows the migration curves for BBO. S1 S 2 Figure 1. BBO migration curves. This figure shows two candidate solutions, S1 and S2, for the same problem. S1 is a relatively poor solution, and S2 is a relatively good solution. Poor solutions are likely to receive features from other solutions, and unlikely to share features with other solutions. Good solutions are likely to share features with other solutions, and unlikely to receive features from other solutions. BBO treats each candidate solution as a biological habitat, and treats each species as a specific feature of that candidate solution. Candidate solutions are also called (3)

3 individuals, or simply solutions. Solution features are also called independent variables or decision variables. The number of solution features in each habitat is the dimension of the problem. Each candidate solution shares its features with other candidate solutions, and this sharing process is analogous to migration in biogeography. As migration occurs, the habitats become more suitable for their species, which corresponds to the improvement of candidate solutions. One iteration is often called a generation. We also implement common EA concepts in BBO such as elitism and mutation (Simon 2013b). Figure 2 shows the outline of the BBO algorithm. where k is the time step index, U(k) is the input vector at time k, Y(k) is the output vector which includes optimization objectives as well as constraints, and F(U(k)) is a nonlinear and smooth function of the correlation between the inputs and outputs. The objective of the closed-loop controller is to find the input vector U that minimizes the error (5) where Y d(k) is the desired output vector at time step k. 2) Surrogate Model Learning The core of adaptive optimization is an algorithm for accurate and accelerated estimation of the Jacobian that correlates the change of trajectories (parameterized as multiple calibrations) with the time-to-torque output and the constraints. For MIMO systems, the linearized system can be represented as a Jacobian matrix between the inputs and outputs: Y = J U, where J is the Jacobian. (6) Figure 2. Outline of BBO. N is the population size, that is, the number of candidate solutions, y is the population of candidate solutions, and z is a temporary population of candidate solutions so that migration can complete before the population is replaced for the next generation. zk(s) is the s-th feature of zk. C. Local Optimization Here we use adaptive optimization, which involves Jacobian learning. The main idea is to use an iterative process that includes the learning of a local surrogate model of a linearized time-varying approximation of a nonlinear process, followed by constrained gradient-based optimization with the surrogate model. 1) Surrogate Model Based Adaptive Optimization Surrogate model based adaptive optimization is illustrated by Figure 3. Controlling the plant output Y to match the reference trajectory Y d is achieved by an iterative process of online learning of the plant model, based on the input U and output Y, followed by optimizing the input U, until given stopping criteria are met. Figure 3. Model Based Adaptive Optimization Consider the optimization problem as a static system with the following representation: (4) Y is the n-element output vector, and X is the m-element input vector. Learning and updating the Jacobian can be viewed as a real-time least squares minimization problem: (7) where k is the time step index. For the purpose of learning, the MIMO linearized model is decomposed into q MISO subsystems, where q is the output dimension: (8) The n-element vector J s(k) is the s-th column of the Jacobian matrix for s [1, q]. We apply the Kalman filter implementation of the recursive least squares method to estimate each individual row of the Jacobian: 1 1 (9) 1 where vector w s(k) represents the inaccuracy of the linearized model of Equation (34) with covariance,, and v s(k) is measurement noise with zero mean and variance. The Kalman filter based expression for recursively learning the rows of the Jacobian estimate is given as 1 1 (10) (11) (12) The learning algorithm of Equation (11) minimizes the cost, where is the deviation between the predicted output and the actual output Y s(k): 1 (13) Q s in Equation (12) represents the drift factor that controls the rate of forgetting old data. It is analogous to

4 the forgetting factor in RLS [17] and can be estimated from the expected changes in the Jacobian. The advantage of using the drift factor versus the exponentially forgetting factor lies in cases when the system is not excited [18]. It forces the covariance matrix P s (which controls the variable learning rate of the Kalman filter) to grow linearly rather than exponentially. 3) Constrained Optimization Given the Jacobian estimate above, the next step is to find optimal inputs to minimize the cost in Equation (7). An optimal input update can be calculated from the pseudoinverse of the Jacobian: + 1 = + (14) = + = + (15) where I is the identity matrix and ρ is a small positive constant which is analogous to the Tychonoff regulation matrix [19] and which improves the numerical conditioning of the inverse system. Directly updating the input with Equation (14) from the pseudo-inversion of the Jacobian is not practical since it can affect the convergence of the mapping algorithm. Limiting the discussion to the more common case of n q we modify Equation (15) by a gain factor as follows: + 1 = + + = + = + (16) where diagonal matrix G represents the gain of the input update, and () is the estimated Jacobian from the learning algorithm of Equation (10). The input update of Equation (16) resembles the one-step calculation of the optimal direction of input change in the Levenberg- Marquardt method. One disadvantage of this input update rule is that it does not explicitly consider constraints on the inputs. Our solution to this problem is to write the input update as a constrained optimization problem: = argmin + 1 (17) = 1 + () ( 1) The penalty γ in the input update in Equation (17) is analogous to the gain matrix G in Equation (16) and improves the robustness of the algorithm due to the fact that () is a local approximation, and any large change of the inputs may lead to output oscillation for highly nonlinear functions of U(k). Since U(k) U(k 1) is always a solution to Equation (17), a sufficiently small value of γ guarantees that the mean square error will not increase from one iteration to the next, and consequently the Jacobian estimate () will approach the true Jacobian J(k). The update in Equation (17) can be performed using readily available QP solvers, such as FMINCON and LSQLIN in the Matlab Optimization Toolbox. III. CASE STUDY: TIME-TO-TORQUE OPTIMIZATION In this section, we apply the proposed approach to a time-to-torque optimization problem for GTDI engines. A. Gasoline Turbocharged Direct Injection Engines GTDI engine technology delivers higher specific power and torque than more traditional engines. The direct injection hardware cools the cylinder air-charge as the injected fuel evaporates. Charge cooling reduces spark knock and allows spark timing closer to the optimal time. In turn, GTDI designs utilize high compression ratio (CR). GTDI engine technology increases specific output, which translates to I4 engines with peak torque that rivals V6 engines. However, turbo hardware can take longer to reach full output torque; hence the term turbo-lag. Therefore, engine performance metrics include transient response, and specifically time-to-torque. This metric is obtained from engine hardware on dynamometers early in the design process to ensure acceptable performance. Increasing the size of a turbocharger increases peak power, but also increases transient response lag. The combination of transient and steady state metrics forces engine designs to consider both effects. B. Gasoline Turbocharged Direct Injection Engines A Simulink engine model is used in this work. The model represents a typical I4 GTDI engine with low pressure (LP) exhaust gas recirculation (EGR) (Wheeler 2013). It models the effects of actuators on brake engine torque output in both steady state and transient conditions. The model does not attempt to replicate exact performance on any given engine, but provides a simulation with the correct actuator effect on output torque. The model includes the following controls. throttle angle degrees ( ) [2,90] spark timing (degrees) [ 10,55] variable cam timing intake (degrees) [ 50,10] variable cam timing exhaust (degrees) [ 10,30] Exhaust gas recirculation valve (mm) [0,9] Air induction system throttle (degrees) [2,90] Waste gate valve position (close/open) [0,1] The model is depicted in Figure 4 and consists of induction system manifolds, where each manifold pressure derives from the ideal gas law and does not include heat transfer. The model integrates the sum of the mass flows entering (+) and exiting ( ) the manifolds to determine instantaneous mass. The induction system includes the manifold between the air induction system (AIS) throttle and the turbo compressor, the manifold between the compressor and the main throttle, and the main throttle and the intake valves (referred to as the intake manifold). The induction system includes the compressor by-pass valve (CBV), which opens during a back out when the desired torque decreases. Opening the CBV allows the pressure downstream of the compressor to decrease as the main throttle would close to reduce cylinder charge. The model uses the Heywood throttle flow model for the AIS, CBV, and main throttle [20]. The induction system includes a charge air cooler (CAC) in the manifold between the turbo compressor and main throttle, and in the intake manifold. The exhaust system consists of the manifold between the exhaust valves and the turbo charger, and a back-pressure model of the remaining exhaust.

5 Gasoline Turbocharged Direct Injection (GTDI) Engine 6) Air Induction Throttle (AIS) 1) Main Throttle 5) EGR Valve C A C Compressor Turbine 7) Waste-gate CAC 3) Intake VCT 4) Exhaust VCT Figure 4. Gasoline Turbocharged Direct Injection Engines The turbo charger model calculates turbine speed using turbo and compressor maps. It calculates turbine speed using the power balance between the compressor and the turbo, and the turbine inertia. The turbo charger model also includes a waste-gate to provided boost pressure control. The LP-EGR system consists of an EGR valve between the AIS and compressor. Exhaust flows from downstream of the turbo through a CAC into the induction system via the EGR valve. The model includes the transport delay of the EGR mass from the valve through the induction system to the cylinders. The actuators in the model include AIS, waste-gate, EGR valve, main throttle, intake variable cam timing (VCT), exhaust VCT, spark timing, and CBV. All of the actuators include characteristic response delays and time constants. Actuator position therefore lags behind commanded values during transients and converges to commanded values at steady state conditions. C. Time-toTorque Minimization In order to solve the minimum time-to-torque problem, we parameterize each candidate control solution as a Fourier cosine series. cos (18) where T = half the control duration, k [1, 7] (seven control signals), and M = 4. There are thus a total of 35 independent variables in the control problem. M is a tradeoff between search resolution and problem size. BBO solves the minimum time-to-torque problem by randomly generating a population {y k} of individuals for k = 1,, N p, where N p is the population size (see Figure 2). Each individual y k is a 35-element vector of Fourier coefficients. Each y k is used to generate a timevarying control vector. Based on the torque output from the simulation, time-to-torque is measured for each y k. After performing the above process for each of the N p BBO individuals, we have a list of N p time-to-torque values. We sort the values in order of increasing time-totorque, and sort the y k individuals correspondingly, from y 1 (smallest time-to-torque) to y N (largest time-to-torque). We assign emigration rates to each individual as follows: Intake Manifold & CAC Exhaust Manifold CAC Charge Air Cooler 2) Spark Timing (19) We demonstrate these methods on the problem of minimum-time torque transition from 27 Nm to 374 Nm. Although not discussed in this paper, methods for determining steady-state control solutions for the GTDI engine are available as a baseline for the techniques used in this paper. We find the steady-state controls as follows (listed in the same order as above): T 1 = 27 Nm u 1 = [7, 52, 10, 30, 6, 7, 1] T q= 374 Nm u q = [90, 9.5, -50, 0, 0, 90, 0] Suppose that we want to transition from torque T 1 to torque T q in minimum time. Intuitively, the quickest way is to step, or ramp very quickly, from u 1 to u q. This gives a 95% time-to-torque of 2.90 seconds, as shown in Figure 5. Torque (Nm) Time (seconds) Figure 5. If we use ramp controls starting at 27 Nm, it takes 2.90 seconds to reach 95% of the goal torque of 374 Nm. We initialize one of the BBO individuals to the ramp controls, which means that at the first generation, the best BBO individual will have a time-to-torque of 2.90 seconds or better. The rest of the BBO individuals are initialized to random Fourier series. Figure 6 shows the improvement of the best time to torque achieved over 30 generations (out of 60 individuals), where 0.82 seconds time-to-torque was achieved. This is a 72% decrease in the time-to-torque achieved by intuitive ramp controls. Minimum Time to Torque (seconds) Generation Figure 6. BBO reduces time-to-torque to 0.82 seconds after 30 generations. Figure 7 shows significant overshoot, undershoot, and torque fluctuations. Such torque response is not acceptable during driving. In order to alleviate this undesired behavior, we modify the BBO cost function as follows: Cost max min: max : and where W i are user-defined weights. This approach results in a time-to-torque of 1.52 seconds, worse than the previous value of 0.82 seconds, but reduces fluctuations.

6 EAs besides BBO, might provide better performance. In view of the multiple objectives in the optimization problem (for example, time-to-torque, overshoot, and undershoot), multi-objective optimization might provide good results. Parameterizing the controls with functions other than sinusoids might provide better results. Figure 7. BBO reduces time-to-torque to 0.82 seconds, but at the expense of undesirable fluctuations. These undesirable behaviors can be reduced or eliminated by modifying the BBO cost function. Next we use adaptive optimization, which converges to a time-to-torque of 1.13 seconds. We see that BBO was effective at finding the neighborhood of the global optimum, and local optimization was effective at finding a more precise optimum. BBO decreased time-to-torque by 48% relative to bang-bang controls. Adaptive optimization decreased time-to-torque by an additional 26%. Engine Torque (Nm) Base line BBO BBO+Adaptive time (sec) Figure 8. Time-to-torque with the modified cost function. BBO achieves a rise time of 1.52 seconds, and adaptive optimization reduces the rise time to 1.13 seconds. IV. CONCLUSION We have presented a memetic algorithm for general trajectory optimization. We parameterized the controls with Fourier series or Gaussian kernels, transforming the problem to a parameter optimization problem. Our memetic algorithm includes an evolutionary global optimizer called biogeography-based optimization (BBO) to find controls that are in the neighborhood of the global optimum. The global optimizer is followed by local optimization which uses Gaussian kernels to approximate the BBO-generated controls. The method iteratively identifies the local characteristics between the calibration parameters and the objectives and uses gradient-based optimization to optimize all calibration parameters. We applied our methods to the minimization of time-totorque for a gasoline turbocharged direct injection (GTDI) engine. 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