Chapter 1 Toward a Systematic Design for Turbocharged Engine Control

Transcription

1 Chapter 1 Toward a Systematic Design for Turbocharged Engine Control Greg Stewart, Francesco Borrelli, Jaroslav Pekar, David Germann, Daniel Pachner, and Dejan Kihas Abstract The efficient development of high performance control is becoming more important and more challenging with ever tightening emissions legislation and increasingly complex engines. Many traditional industrial control design techniques have difficulty in addressing multivariable interactions among subsystems and are becoming a bottleneck in terms of development time. In this article we explore the requirements imposed on control design from a variety of sources: the physics of the engine, the embedded software limitations, the existing software hierarchy, and standard industrial control development processes. Decisions regarding the introduction of any new control paradigm must consider balancing this diverse set of requirements. In this context we then provide an overview of our work in developing a systematic approach to the design of optimal multivariable control for air handling in turbocharged engines. 1.1 Introduction The goal of this chapter is to present the emerging problems facing development of control for increasingly complex turbocharged engines and to discuss potential solutions. In turbocharged diesel engines ever-tightening emissions legislation drives the incorporation of new sensors [27, 33], actuators, and subsystems such as multistage turbochargers, complex exhaust gas recircu- Greg Stewart, David Germann, Dejan Kihas Honeywell Automation and Control Solutions, 500 Brooksbank Avenue, North Vancouver, BC, Canada, V7J 3S4 Francesco Borrelli Department of Mechanical Engineering, University of California, Berkeley, CA Jaroslav Pekar, Daniel Pachner Honeywell Prague Laboratory, V Parku 2326/18, Prague 4, Czech Republic 1

2 2 Stewart et al. lation (EGR) topologies [31], and aftertreatment devices (such as selective catalytic reduction [26, 32], lean NOx traps, and diesel particulate filters [21]). These changes are all introduced in the context of difficult analysis and decisions regarding the tradeoff of development and product cost, reliability, fuel economy, drivability, and emissions. The rising complexity of engines and the demand of tighter performance is increasing the complexity of the control functionality that is required to manage the engine. It is widely recognized that the development of the control is becoming a bottleneck in the development of engines and systematic approaches to developing performant controls would be welcome provided they mitigate these burdens. Typically most engine makers require control design techniques that provide some combination of improved closed-loop performance and reduced development effort. The relative importance of each consideration depends to a large extent on the business model of the particular control system developer. In this chapter we will discuss a wide view of the problem of developing such a systematic approach with the goal of integration into industrial practice. In so doing we need to consider the interaction of many practical issues; including engine physics and its resulting nonlinearities and multivariable interactions, the desired closed-loop performance, engine variability due to production dispersion and ageing, the restrictions in CPU time and memory due to the embedded electronic control unit (ECU) platform, the hierarchical software structure into which the final control function must integrate, the existing development process for engine control, and the range of personnel with whom any new process must interact. The scope of a solution to such a problem is large, but is aided by the significant body of previous research - both academic and industrial - in the domain. The requirements associated with engine modeling and closedloop control performance are considered in [28, 20, 15, 34, 16] and references therein. As will be introduced below, our approach includes a model predictive control (MPC) component and previous MPC engine control examples may be found in [23, 17]. The interplay of control design and computational restrictions 1 (CPU and memory) were noted in [17] where a fully nonlinear MPC (NMPC) is proposed which demonstrates improved performance with respect to linear state feedback and input-output linearization approaches. As pointed out in [17], the main drawback of most NMPC techniques comes from the fact that they typically require far more computational power than is available on modern automotive ECUs. To address these issues we have implemented several practical simplifications and have separately explored the question of online implementation of MPC in depth. This approach has allowed us to successfully implement a multivariable MPC controller in a production ECU [29]. 1 More detailed discussions of the very important subject of requirements and management of engine control software design and its innovations may be found in references such as [25, 30].

3 1 Toward a Systematic Design for Turbocharged Engine Control 3 Selected aspects of the underlying techniques are outlined in Section 1.4 below. Section 1.2 presents the requirements to be met by control design, Section 1.3 describes how the proposed modeling and model predictive control techniques are suited to address these requirements, Section 1.4 presents some more detailed results on one of the many technical aspects that are needed to be overcome in order to achieve an industrial quality systematic approach to control design. 1.2 Engine Control Requirements In this section we present the specific requirements that must be addressed by any engine control design process. These are discussed with reference to a high-level description of a typical engine control design process which has evolved over many years to address the industrial need of creating controllers for highly nonlinear engines which are to be hosted on ECUs and deployed across a fleet of thousands to millions of vehicles that may stay in active service as long as 20 years. As will be seen below, the support of all phases of this process requires a wide range of skills that include an understanding of engine physics, performance specifications (including emissions legislation), embedded software issues, and calibration and post-release support. Typically these activities involve separate skills and groups within a company and it is important that any proposed technology change anticipates and addresses all of these areas. While specific engine control design processes will vary site to site, many similarities exist and a high level outline of a typical process is illustrated in Figure 1.1 which is similar to those described in references such as [25] Steady-state engine calibration Also known as base mapping in this pre-control phase an engine is calibrated in order to produce the actuator positions at a coarse grid of engine speed and load points such that certain desired steady-state criteria are traded off. This step is often performed by executing a design of experiment that sweeps the relevant engine actuators at a selection of speed and load points while recording the engine s performance in terms of fuel consumption and emissions. Upon completion of the experiment the desired engine steady-state operating points (outputs and actuator positions as a function of exogenous variables such as speed and injected fuel quantity) are determined by optimizing with respect to requirements imposed by legislated drive cycle

4 4 Stewart et al. Steady-state engine calibration long-path functional iteration Control functional development Functional testing (simulation, testbench, vehicle) Software development (specification, coding, testing) Integration (testing and debugging) short-path functional iteration software iteration Calibration (simulation, testbench, vehicle) Certification and Release Fig. 1.1 Simplified illustration of the engine control development process. limits. This is a heavily experimental stage and requires much insight into the engines behavior Control functional development In this step control engineers decide on the partitioning of the engine functionality and, where needed, propose and configure new control functions with the goal of approximately delivering the steady-state engine calibration while simultaneously providing enough flexibility to meet the certification and drivability requirements that are evaluated over transient driving conditions. This stage is often performed with the aid of rapid prototyping tools that support the porting of high-level code such as Matlab or Simulink into an ECU bypass system. Commercially available examples include [12, 2, 22].

5 1 Toward a Systematic Design for Turbocharged Engine Control 5 Figure 1.2 is helpful for understanding the environment into which a controller must be designed. An engine is a highly nonlinear plant (see for example [28, 20, 15, 16, 19]) and in most cases requires the development of nonlinear control strategies. For example, when considering the response of airside parameters such as compressor flow and boost pressure to the variable geometry turbine (VGT) actuator, the control designer must address a steady-state gain that changes sign and the fact that the nonlinearities are a function of the engine speed and fuel injection quantity and thus may change more quickly than the dynamics of the air loop which are often dominated by the turbocharger inertia [20, 29]. In addition one must often consider the fact that multiple engine configurations may be required to be addressed by relatively minor configuration changes to a single control strategy. Figure 1.2 illustrates the hierarchical software environment into which the developed controller function must be integrated. An individual subcontroller is typically configured to be responsible for an engine subsystem and a given subcontroller may be responsible for fuel injection control, aftertreatment control, exhaust gas recirculation (EGR) control, or turbocharger (wastegate or VGT) control. The illustrated higher level function is responsible for engine monitoring and distribution of signals to the subcontrollers which typically includes information about special modes (cold start, regeneration of aftertreatment, etc.), setpoints, feedforward actuator values, and time varying constraints for actuators and engine states. Actuator constraints typically include the enforcement of minimum and maximum bounds on actuator positions and a common example of a state constraint is the imposition of an upper bound on the turbocharger speed in order to prevent damage to the turbocharger wheels. In our work we are concentrating on developing techniques that enable the user design multivariable controllers that may be substituted for one or more subcontrollers, initially focusing on unifying the control of the air induction subsystems and coordinating their interaction with fuel injection and aftertreatment subcontrollers. Multivariable interactions among subsystems have typically been neglected in traditional engine control design, but are rapidly becoming crucial to consider as performance requirements become more stringent due to legislation and the emergence of aftertreatment devices that are most effective in certain operating windows of temperature, flow, and composition of the exhaust gas. The techniques and tools required in the control functional development stage are those that enable the developer to systematically create models and design controllers for integration into the existing control hierarchy that will provide acceptable performance for a wide variety of engine configurations and performance specifications.

6 6 Stewart et al. ECU y h (t) sensor processing y 1 (t) y N (t) high level (monitoring and decision making) f 1 (t) f N (t) subcontroller 1 subcontroller N u 1 (t) u N (t) actuator processing Engine from sensors to actuators Fig. 1.2 Illustration of relative position of subcontroller algorithms within the ECU software hierarchy. The symbols y i (t) and u j (t) represent the i th and j th subsets of sensor and actuator information respectively. The symbols f k (t) denote the information transferred to the k th subcontroller and may include setpoints, feedforward actuator values, and time varying constraints for actuators and engine states. The techniques discussed in this chapter consider controller design for the replacement of one or more subcontrollers with an optimal multivariable controller Functional testing In this phase the developing control functionality may be progressively tested in simulation, in the engine test cell, in vehicle, and over a range of ambient conditions of temperature and altitude. This step is typically performed iteratively with the control functional development and the developer must evaluate how well the designed control function interacts with the engine and any other existing control loops to deliver the desired performance Software development This phase may be performed in-house by the engine maker or may involve a third party software provider. The software developers receive a functional specification from the above step and then specify, code, and test the desired function into the embedded software environment which may include the reformulation of the code to use fixed-point arithmetic. In this phase a key requirement is to respect the memory and processor limitations of the

7 1 Toward a Systematic Design for Turbocharged Engine Control 7 ECU. Furthermore, as this phase is potentially long and expensive, wherever possible one prefers to use control algorithm structures with the flexibility to enable control changes without requiring re-entry into the software development phase. The duration and expense of this phase has motivated the development of autocoding tools and techniques which enable the conversion of high-level languages such as Matlab and Simulink into production-quality embedded code. Such tools have begun to make inroads into various production applications [30] Integration The engine developer then integrates the developed code which may be tested in a simulated engine environment before proceeding to testing and debugging in the test cell and vehicle Calibration This is a detailed phase in which the numerous free parameters of the engine controller are tuned or calibrated to provide the desired steady-state and transient performance. This phase requires that controller tuning tools be efficient, intuitive, and well-integrated into the existing calibration environment. This phase also includes all of the calibration required for the diagnostics portions of the software which are indirectly related to the control functionality Certification Refers to the set of tasks related to preparing, documenting and performing the certification testing that is relevant to the vehicle class and geographical region in which it will be used Release and post-release support Once the software and control have been accepted, then the engine or vehicle is released to market. Subsequent control calibration or other fixes may be required in the post-release support of the product. Since post-release revision

8 8 Stewart et al. to the control is limited and expensive, the controllers designed and calibrated throughout the above steps must perform well over a fleet of engines each with differences in the manufactured components, and must maintain system performance over the lifetime as the engine components age and change in different ways Iteration loops Figure 1.1 illustrates the above steps along with some standard iteration paths. The short-path functional iteration between functional testing and control functional development has been made relatively rapid by modern rapid prototyping systems. It is these iterations in which the major decisions regarding overall controller structure are made and thus it is critical that the developer be provided tools that direct or otherwise assist control structure decisions to be made systematically. In this stage it is noted that the designer must design a controller to work with the engine physics of its designated task, and also to interact appropriately with neighboring control functions. In [29], the hierarchical nature of engine control is described and it is pointed out that any developed control function must be integrated into this hierarchy where the supervisory levels deliver signals including setpoints, actuator feedforward, time-varying constraints for actuators or engine states to the lower-level components of the control hierarchy. The illustrated software iterations on the lower-right portion of the diagram indicates any rework that is required for fixing bugs or other errors in the software coding. The long-path functional iteration illustrated on the left side of the diagram indicates the required iteration path when controller code is found to be unacceptable at the functional level at a later stage of development. Since this path contains the entire software development cycle, it may incur a cost of several months of development time 2 using existing development tools and processes, and is thus considered undesirable in the majority of cases. The above discussion has presented a high-level description of the engine control development process from which it is clear that several key requirements to be met by any control technology include: (i) to maintain performance of the family of highly nonlinear and multivariable engines that lie within the uncertainty bounds of a production fleet, (ii) integration into the hierarchical control structure (including time varying constraints), (iii) to fit within the tight processor and memory requirements for implementation in an ECU, and, 2 The time and effort incurred in the long-path functional iteration is sometimes reduced by the use of autocoding tools such as [30] in the Software Development phase.

9 1 Toward a Systematic Design for Turbocharged Engine Control 9 (iv) to provide a systematic, efficient development process to facilitate software development and enable a reduced calibration burden. 1.3 Modeling and Control for Turbocharged Engines Section 1.1 discussed the trends and pressures facing the development of modern engine control while Section 1.2 overviewed the scope of the requirements for any strategy - new or old. In this section we discuss the merits of using optimal model predictive control technology to address these issues. In light of the above requirements, MPC shows several strong advantages: MPC is ideally suited for handling the signals from the higher level of the hierarchy illustrated in Figure 1.2. Particularly the setpoints and time varying input and output constraints [29]. MPC has a general algorithmic structure which can cover many different control problems (e.g. unconstrained or constrained, single or multivariable) without requiring software structure changes. The tuning of MPC (while not quite as straightforward as is often claimed in MPC literature) can be made intuitive with the appropriate software tools to allow practical usage by people with a wide variety of technical backgrounds and interests [14]. Figures 1.3 and 1.4 contain representative examples of the on-engine results achieved using model predictive control on two very different engine control problems. Application details may be found in [29]. Figure 1.3 is an illustration of the simultaneous control of measured EGR flow and engine-out NOx concentration in a heavy-duty diesel engine using the EGR valve and variable cylinder valve as actuators. Such a control configuration could have application in coordinated engine-aftertreatment control. Figure 1.4 represents a more standard control problem where the intake manifold pressure and the compressor flow are simultaneously controlled to respective setpoints using the VGT vanes and EGR valve while the engine traverses the indicated transient in engine speed and injected fuel quantity. On the other hand, we will have to consider and address the challenges faced by MPC to achieve the desired reduction in development time and effort. These include: Modeling: one is obliged to develop an efficient and reliable process to generate the control oriented models that are required by all advanced control techniques Computation: MPC - and especially nonlinear MPC - typically require too much computing power and memory for implementation on a modern ECU (see for example [17]). This topic is discussed in some detail in Section 1.4 below.

10 10 Stewart et al. 100 EGR flow [%] NOx [%] time [sec] VVA [%] EGR valve [%] time [sec] time [sec] time [sec] Fig. 1.3 Simultaneous control of EGR flow and engine-out NOx concentration using the EGR valve and variable cylinder valve actuators in a heavy duty diesel engine. The setpoints are indicated by the dashed blue lines and the sensor measurements in red in the upper two subplots. The actuator constraints are illustrated as the broken magenta lines and the actuator positions as the solid green lines in the lower two subplots. The axes have been scaled for deidentification considerations. Usability: for deployment, advanced control techniques must be developed to the point of industrial quality, such that they do not require such detailed knowledge of MPC or modeling that disqualifies all but a small group of specialists for their use Modeling It is almost accepted as a truism that model based control design techniques are used in order to cut down on the development time. A better phrasing of

11 1 Toward a Systematic Design for Turbocharged Engine Control 11 boost pressure [%] air flow [%] VGT [%] 50 EGR valve [%] Fuel flow [%] Engine Speed [%] Time [seconds] Time [seconds] Fig. 1.4 Simultaneous control of boost pressure and compressor flow using the VGT and EGR valve actuators in a small diesel engine. The MPC was implemented on a production ECU (Motorola MPC555). In the upper two subplots the setpoints are represented by the solid blue lines and the measured sensor signals by the dashed green lines. The actuator constraints are constant at 5% and 95% respectively. The axes have been scaled for deidentification considerations. that statement may be that one should take care to ensure that the modeling results in a reduction of the overall the control development effort. Traditional, non-model-based approaches to engine control design have an ironic advantage of avoiding a potentially onerous and difficult modeling process. Of course traditional approaches are acknowledged to have serious limitations in the complexity of control problem they can address and furthermore engine models have many benefits beyond their use in control synthesis (see for example [11]).

12 12 Stewart et al. The models we develop in the course of our design are highly nonlinear and the reliable identification of their free parameters is a serious challenge, the detailed description of which is outside the scope of this chapter. We present a high level overview of the key issues here and will publish the technical details elsewhere. A practical modeling approach needs to be configurable to a wide variety of engines including single and multistage turbochargers, low and high pressure EGR, various actuators such as valves, throttles, wastegates, VGT vanes, variable valve actuation, and various sensor selections and locations on the engine. With an eye on their intended use, one must implicitly trade off model complexity and precision since low order models are preferable for model based control design. Figure 1.5 illustrates two different engine layouts which have been built from our library of components. The dynamic response is governed by the states associated with the intake and exhaust manifolds and also the turbocharger speed(s). In more complicated engine layouts, such as the illustrated multiple turbocharger example, correspondingly more states are required to include the associated dynamics. air filter compressor charge air cooler intake manifold turbocharger shaft EGR cooler EGR valve engine VGT turbine exhaust manifold air filter LP compressor HP compressor throttle intake manifold LP turbocharger shaft HP turbocharger shaft EGR cooler EGR valve engine aftertreatment LP turbine HP turbine (VGT) exhaust manifold Fig. 1.5 Example engine layouts for a standard single-stage turbocharged and a series turbocharged engine both with high-pressure exhaust gas recirculation (EGR).

13 1 Toward a Systematic Design for Turbocharged Engine Control 13 Next the configured model must be calibrated to the engine such that it matches the true nonlinear and dynamic input-output response across all operating conditions. The model identification is typically required to work with some mix of steady-state and transient data (see e.g. [16]). To address this nonlinear model identification problem we take a two-step approach. First the individual components (listed above and illustrated in Figure 1.5) are fit one by one to component maps (if available) and recorded engine data. In practice, the overall model quality achieved from an assembled collection of components is typically insufficient for capturing the required input-output behavior of the engine and the second step of the model identification is performed by executing a global nonlinear optimal fitting of the available parameters. In practice, this step typically results in a dramatic improvement in model accuracy and a representative example is illustrated in Figure 1.6. On a technical level the model identification is further complicated by the fact that turbocharged engine models have inherent feedback paths via the EGR and the turbocharger shaft. Still more feedback paths are often added during standard modeling procedures [20, 16]. Thus the sensitivity and even stability of the model with respect to its tuning parameters must be treated with due care to preserve the desired match to the true engine response. This modeling process results in a continuous-time nonlinear model in which f and h denote the state update and output functions, respectively: ẋ(t) = f(x, u, v), y(t) = h(x, u, v) (1.1) where y represent the controlled variables, the array u represents the actuator setpoints to be computed by the controller, and v represents the exogenous inputs to the system. The content of these variables is problem specific and configurable. The controlled variables y may include some subset of boost pressure, compressor air flow, EGR flow, turbocharger speed, engineout NOx, exhaust temperature, etc. The actuators u may include variable geometry turbine (VGT) vanes, variable (cylinder) valve actuators (VVA), EGR valve, intake or exhaust throttles. The exogenous inputs v will typically include engine speed and fuel injection quantity as a baseline and may be further refined by coolant temperature, and ambient pressure and temperature, etc. The following section will use the nonlinear model (1.1) as input to an MPC design and will discuss the simplifications required for practical controller synthesis.

14 14 Stewart et al Oxygen Concentration Intake Manifold 400 Temperature Post Charge Air Cooler model model data data model 4.5 x Intake Manifold Pressure model 1.6 x High Pressure Turbocharger Speed data x data Fig. 1.6 Representative accuracy plots obtained from modeling a two-stage turbocharged engine with the proposed medium-fidelity (7-11 dynamic states in this example) nonlinear control oriented model. Steady-state data from the full range of engine speeds and loads is represented. The blue circles indicate the model results when each component has been fit locally and the red squares illustrate the improved model results following the developed optimization-based simultaneous fitting of all model parameters. The straight lines indicate error bounds of 5%, 10%, and 15% with respect to the measured engine data. 1.4 Model Predictive Control and Computational Complexity Explicit Predictive Control We consider a piecewise affine (PWA) discrete-time approximation of the system dynamics (1.1)

15 1 Toward a Systematic Design for Turbocharged Engine Control 15 x(k + 1) = A σ x(k) + B σ u(k) + Bσv(k) v + Bσ w w(k) + f σ y(k) = C σ x(k) [ + Cσv(k) v + Cσ w w(k) + g σ xu ] (1.2) for (k) C σ v w where x(k), u(k), y(k), v(k), w(k) are the state, input, output, measured and unmeasured disturbances, respectively at time kt s where T s is the sampling time, f σ and g σ are constant vectors. The natural number σ(k) {1, 2,..., M} is the operating point at time kt s and it is a function of inputs u(k), states x(k) and disturbances v(k). The set {C σ } M i=1 is a polyhedral partition of the state, input and measured disturbance set. System (1.2) is subject to the following time varying constraint on inputs and outputs for all k 0. u(k) U(u min (k), u max (k)), y(k) Y(y min (k), y max (k)) (1.3) where U(u min (k), u max (k)) and Y(y min (k), y max (k)) are polyhedra for all k 0. Consider the problem of letting the output of system (1.2) track a given reference y ref,k while satisfying input and output constraints (1.3). Assume that estimates/measurements of the state x(k) and disturbances v(k) are available at the current time k and consider the following cost function J k (x, v, w, U, ɛ) := y Hp+k y ref,hp+k) P k+h p 1 t=k (y t y ref,t ) Q 2 + δu t R 2 + ρɛ 2 (1.4) where v M 2 at time k, = v Mv. Then, the finite time optimal control problem is solved

16 16 Stewart et al. min J k(x(k), v(k), w(k), U Hc, ɛ) (1.5a) U H c,ɛ s.t. x t+1 = A σ x t + B σ u t + B v σv t + B w σ w t + f σ w t+1 = A w σ w t + B u σu t y t = C σ x t + Cσv v t + Cσ w w t + g σ [ xu ] if C σ t = k,..., k + H p 1 v w k (1.5b) (1.5c) u t = δu t + u t 1 u t U(u min (k), u max (k)), y t Y(y min (k), y max (k)) ɛ t = k,..., k + H c 1, ɛ > 0 (1.5d) (1.5e) x k+hp X f (1.5f) δu t = 0, t = k + H c,..., k + H p 1 (1.5g) v t = v(t 1), t = k + 1,..., k + H p (1.5h) u k 1 = u(k 1) x k = x(k), w k = w(k), v k = v(k) (1.5i) (1.5j) where the column vector U Hc := [δu k,..., δu H c 1 ] and ɛ are the optimization vectors, H p and H c denote the output prediction horizon and the control horizon and X f is the terminal region. Note that the subscript notation is used to distinguish between the variables of the optimization problem (1.4)-(1.5) and the state, input, disturbances and outputs of the system model (1.2). Let UH c = {δu k,..., δu k+h } and c 1 ɛ be the optimal solution of (1.4)- (1.5) at time k. Then, the first sample of UH c (obtained from UH c and u(k 1)) is applied to the system: u(k) = u k. (1.6) The optimization (1.4)-(1.5) is repeated at time k + 1, based on the new state x k+1 = x(k + 1), measured disturbances v k+1 = v(k + 1), additive unmeasured disturbances w k+1 = w(k + 1), input and output constraints, yielding a moving or receding horizon control strategy. In (1.4) we assume that Q = Q 0, R = R 0, P 0. In problem (1.4)-(1.5) the following assumptions are used A1 H p > H c and the control signal is assumed constant for all H c k H p. This allows the reduction of the computational complexity of the MPC scheme. A2 The exogenous disturbance v is assumed constant over the horizon. If PWA prediction models for v(k) are available they could be included in the MPC formulation (1.4)-(1.5). A3 The region σ is constant over the horizon (1.5c).

17 1 Toward a Systematic Design for Turbocharged Engine Control 17 A4 Soft Constraints on outputs, i.e. Y(y min (k), y max (k)) ɛ := {y y + ɛ Y(y min (k), y max (k))}. Remark 1. Assumption (A3) basically implies that for any given time we simply implement a linear MPC for one member of the set of linear systems. Ideally the assumption should be removed in order to predict switches between affine dynamics over the horizon H p. This would improve both performance and attractivity region of the closed loop system. Nevertheless, we have been forced to use assumption (A3) by the current limitations of automotive ECUs. In fact, by removing (A3), problem (1.4)-(1.5) becomes a mixed integer quadratic program (MIQP) whose explicit solution [7] requires more floating point operations for its evaluations and more memory for its storage. The optimization problem (1.4)-(1.5) can be recast as a quadratic program (QP). 1 min U H c,ɛ 2 U H c H σ U Hc + H ɛ,σ ɛ 2 + φ(k) F σ U Hc (1.7) subj. to G σ U Hc W σ + E σ φ(k) where φ(k) := [x(k) u(k 1) v(k) w(k) y ref,k y ref,k+hp u min (k) u max (k) y min (k) y max (k)] and φ(k) R n p. Problem (1.7) is a multiparametric quadratic program that can be solved by using the algorithm presented in [5]. Once the multiparametric problem (1.7) has been solved, the solution UH c = UH c (φ(k)) of problem (1.4)-(1.5) and therefore u (k) = u (x(k)) is available explicitly as a function of the set of parameters φ(k) for all φ(k) X 0. X 0 R n p is the set of initial parameters φ(0) for which the optimal control problem (1.4)-(1.5) is feasible. The following result [5] establishes the analytical properties of the optimal control law and of the value function. Theorem 1. [5] The control law δu (k) = f σ (φ(k)), f σ : R n p R m, obtained as a solution of (1.7) is continuous and piecewise affine on polyhedra f σ (φ) = F i σφ + g i σ if φ CR i σ, i = 1,..., N r σ (1.8) where the polyhedral sets CR i σ = {φ R n p H i σφ K i σ}, i = 1,..., N r are a partition of the feasible polyhedron X 0. As discussed in [5] the implicit form (1.7) and the explicit form (1.8) are equal, and therefore the stability, feasibility, and performance properties are automatically inherited by the piecewise affine control law (1.8). Clearly, the explicit form (1.8) has the advantage of being easier to implement, M lookup tables, one for each operating point σ, are uploaded on the ECU and at each time step k the MPC resorts to selecting the current operating point

18 18 Stewart et al. σ, searching for the region CR j σ containing the current vectors of parameters φ(k) and implementing the corresponding controller F j σφ(k) + g j σ. The following lists some of the major practical issues which have been encountered while implementing the proposed MPC on an automotive ECU. In this chapter we will focus on only the final listed issue. State estimation. Estimation of the state x(k) and w(k) of system (1.2) is a nontrivial task. We have used a bank of M Kalman filters which run in parallel in order to smoothen the estimation during switch between different operating points. Time-varying actuator and state constraints. The proposed formulation has been designed to address constraints that vary arbitrarily as a function of time and thus independently of the state variables or gain scheduling parameters. Constraint satisfaction under steady state disturbances. For a range of steady-state exogenous disturbance v(k) = v the reference y ref might be infeasible for the given input and output constraints and thus not be trackable. At steady-state, the objective function will be composed of two terms with conflicting objectives: satisfy the constraints (ρɛ 2 ) and track y ref. In addition, model uncertainty and high ρ weights (usually used to strictly enforce soft constraints) can lead to oscillating behavior and poor performance. A possible solution to this problem based on computing regions of attraction and switching tuning has been given in [29] Limited ECU memory. We had to modify the explicit implementation in order to be able to run the MPC (1.4)-(1.6) in an industrial ECU (even for short horizons). We have used the Karush-Kuhn-Tucker conditions which lead to (1.8) in order to reduce the memory required for storing F σ and g σ in (1.8). More details can be found in [8]. In the next section we will provide more details on the last steps. The interested reader is refereed to [29] for a thorough treatment of the aforementioned topics On the Complexity of Explicit MPC Control Laws The efficient solution of the optimization problem (1.7) depends on the problem properties and on the hardware platform. In [9, 10] we have compared the computational time and storage demand associated to (i) active-set QPs for solving (1.7) and to (ii) the evaluation of the explicit solution (1.8). In particular we have shown that there might be alternative ways for solving the optimization problem (1.7) which are more efficient than evaluating the explicit linear MPC (1.8). However, the corresponding state-feedback controllers do not have the nice piecewise affine closed-form solution as the ex-

19 1 Toward a Systematic Design for Turbocharged Engine Control 19 plicit solution presented in [5, 4]. Next we provide a brief overview of the main results and refer the reader to [9, 10] for a more formal and detailed discussion. The evaluation of the explicit solution (1.8) in its simplest form would require: (i) the storage of the list of polyhedral regions and of the corresponding affine control laws, (ii) a sequential search through the list of polyhedra for the i-th polyhedron that contains the current state in order to implement the i-th control law. Since verifying if a point φ belongs to a critical region means to verify primal and dual conditions, then the on-line search for the polyhedron containing φ can be compared to the main steps of a QP solver. In fact in [9] we have shown that an active set QP solver requires more operations at each iteration than an explicit solver. This is obtained at the price of increased memory requirement. In fact, for the evaluation of (1.8) the polyhedral partition and the gains have to be stored which, in general, largely surpass the memory required for an active set QP (simply the matrices of the QP (1.9)). The main idea behind alternative approaches can be simply explained as follows. Rewrite the optimization problem (1.7) for a fixed σ compactly as [5] min z 1 2 z H z subj. to G z z b z (φ) (1.9) where z is the optimization variable, and b z (φ) is an affine function of φ. Let I be the set of constraint indices. Consider a subset A of the constraints index A I. Given a matrix M, M A denotes the submatrix of M consisting of the rows indexed by A. Consider the solution z (φ) and λ (φ) when the set A is active at the optimum [9]: z = Hz 1 G z,a (G z,ah 1 G z,a ) 1 b z,a (φ) = T A b A (φ) λ = (G z,a Hz 1 G z,a ) 1 b z,a (φ) = S A b A (φ) (1.10) The alternative class of algorithms presented in [9, 10], computes and stores the matrices S A and T A off-line for all optimal sets of active constraints A. Then, the online steps are: (1) compute b z (φ), verify duality conditions by computing λ from (1.10), compute the optimizer candidate z by using (1.10) and verify primal feasibility conditions G z z b z (φ). In conclusion, one can identify three classes of algorithms as illustrated in Figure 1.7: the explicit solvers (upper-left in the figure), the QP solvers (lower-right in the figure) and hybrid solvers which trade-off memory and computation in a different way and which, for certain classes of problems, can be more efficient than both QP solvers and explicit solvers [9, 10]. Remark 2. We remark that there exist other very efficient approaches appeared in the literature for solving predictive control problems for linear and PWA systems [18, 3, 6, 24, 13]. The comparison of our approach to other approaches would be problem dependent and requires the simultaneous anal-

20 20 Stewart et al. Explicit MPC Solvers Memory Hybrid MPC Solvers Active-Set MPC Solvers Number of Flops per Iteration Fig. 1.7 Comparison of various algorithms for solving QP (1.9). ysis of several issues such as speed of computation, storage demand and real time code verifiability. This is an involved study and as such is outside of the scope of this chapter. We refer the reader to [1] for a good review of explicit predictive control. 1.5 Summary and Conclusions The demand for systematic and efficient techniques for the development, calibration, and deployment of control algorithms is undisputed in modern industrial engine design. The key issues are the achieved closed-loop performance and also the time, effort, and expense that are required to achieve it. Within the context of a standard industrial engine control development process we surveyed the set of control requirements at a high level, including: to maintain performance of the family of highly nonlinear and multivariable engines that lie within the uncertainty bounds of a production fleet, to integrate into the hierarchical control structure (including time varying constraints), to fit within the tight processor and memory requirements for implementation in an ECU, and, to provide a systematic, efficient development process to facilitate software development and enable a reduced calibration burden.

21 1 Toward a Systematic Design for Turbocharged Engine Control 21 With respect to these requirements we discussed our recent work in attempting to address this set of requirements in the form of a systematic process and set of software tools that allow the user to configure a model using components from a library, to automatically and robustly fit this model to engine and component data, to use this model in the synthesis of both the feedforward and feedback components of a multivariable control strategy. We selected model predictive control (MPC) as the underlying technology due to its ability to address the multivariable interactions among subsystems, its ability to be integrated into the existing control software hierarchy found in industrial electronic control units (ECUs) - including its straightforward accommodation of time varying input and output constraints, and its general algorithmic structure which can cover many control problems without requiring software structure changes. We next overviewed some of the issues involved when considering implementing MPC for a nonlinear plant within the limited resources of a modern ECU, particularly paying attention to balancing the tradeoff between memory and processor usage. We presented examples of the overviewed control being used to control the air handling on two very different engine applications. Acknowledgements This work would not have been possible without the support of Honeywell. References 1. A. Alessio and A. Bemporad. A survey on explicit model predictive control. In F. Allgower D.M. Raimondo L. Magni, editor, Nonlinear Model Predictive Control: Towards New Challenging Applications, volume 384 of Lecture Notes in Control and Information Sciences, page Springer-Verlag, ETAS ASCET M. Baotić, F. Borrelli, A. Bemporad, and M. Morari. Efficient on-line computation of constrained optimal control. SIAM Journal on Control and Optimization, 47: , A. Bemporad, F. Borrelli, and M. Morari. Min-max Control of Constrained Uncertain Discrete-Time Linear Systems. IEEE Transactions on Automatic Control, 48(9): , September A. Bemporad, M. Morari, V. Dua, and E.N. Pistikopoulos. The explicit linear quadratic regulator for constrained systems. Automatica, 38(1):3 20, L.T. Biegler and V.M. Zavala. Large-scale nonlinear programming using ipopt: An integrating framework for enterprise-wide dynamic optimization. Computers & Chemical Engineering, 33(3): , Selected Papers from the 17th European Symposium on Computer Aided Process Engineering held in Bucharest, Romania, May F. Borrelli. Constrained Optimal Control of Linear and Hybrid Systems, volume 290 of Lecture Notes in Control and Information Sciences. Springer, 2003.

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