Engine Idle Speed Control Using Actuator Saturation

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IEEE TRANSACTIONS ON CONTROL SYSTEMS TECHNOLOGY, VOL. 8, NO. 1, JANUARY 2000

Engine Idle Speed Control Using Actuator Saturation Paul Herman and Matthew A. Franchek

Presente nted d in this this paper paper is the design design and experi experi- Abstract— Abstract—Prese mental validation of a saturating engine idle speed controller for a Ford V-8 fuel injected engine. The nonmeasurable external torque disturbance perturbing engine speed is delivered from the power steering steering pump. The performan performance ce specifi specificatio cation n is an allowable allowable engine speed tolerance of 140 r/min about a desired set speed of 600 r/min. The controlled input is a voltage to the bypass air valve (BPAV) which regulates the air ingested into the engine. The BPAV voltage available for engine speed regulation is bounded by 0.8 V. A frequency domain controller design methodology is used to design the controller. The performance objective is satisfied using saturation control during large engine speed transients. During steady-state operation, the controlled input is not saturated. Index Terms—Actuator saturation, engine speed control, regulating systems, robust control.

I. INTRODUCTION

I

IDLE SPEED regulati regulation on of internal internal combusti combustion on (IC) engines engines is one testbe testbed d used used by the contro controll commun communit ity y to demons demonstra trate te new controller design methodologies ([1 ([ 1], [10], 10], [16], 16], and references therein). This attraction to engine speed regulation is primarily due to the isolated structure of the engine idle speed control loop within the engine control module (ECM). The bypass air valve (BPAV) which regulates the air flow into the engine is primarily intended for engine speed control. Therefore, the idle speed control loop is separate from the rest of the engine control system. Another aspect which makes engine speed control attractive is the complex dynamics of IC engines. Engines are highly nonlinear, time-varying, time-varying, mechanical, electrical, thermal, fluid, and chemical systems whose dynamics are dominated by an induction-to-power delay. delay. Furthermore, the fundamental dynamics of the combustion event are either unknown or not well understood. Collectively these characteristics challenge any controller design methodology. During idle conditions, there are two basic controlled inputs used to regulate speed: air control and ignition timing. The ignition timing loop has a much greater bandwidth than the air induction loop. The bandwidth potential of the air loop is limited by the intake manifold filling dynamics. However, the control authority of the air control loop is much greater than that of the ignition timing loop. Typically Typically,, these two inputs are used in parallel allel to achie achieve ve enginespeed enginespeed contro controll [ 17]. 17]. This parallel parallel feedback feedback structure effectively combines the higher bandwidth characteristics of the ignition timing loop with the large control authority of the air loop. loop. Conseq Consequen uently tly,, both both tight tight engine engine speed speed regul regulati ation on

Manuscript received October 27, 1997. Recommended by Associate Editor, M. Jankovic. This work was supported by the National Science Foundation under Grant CMS-9634809. The authors are with the School of Mechanical Engineering, Purdue University, West Lafayette, IN. Publisher Item Identifier S 1063-6536(00)00996-9.

and total disturbance rejection can be achieved. Owing to the stringent regulations of tailpipe emissions, the role of ignition timing for engine speed control is diminishing. This investigation will focus only on intake air control of IC engines. The major source of air into the intake manifold during idle is from the BPAV [2 [2]. Essentially, the BPAV is an electronic valve which siphons air around the closed throttle plate from the intake runner into the intake manifold. The electronic part of the BPAV is a linear motor with a plunger on its end. The relative distance between the plunger and the valve seat determines the effective resistance to air flow. The air flow will increase until choked flow conditions occur. The larger the relative distance between the plunger and valve seat, the larger the air flow is into the manifold. Changes in the air flow result in fueling adjustments to meet stoichiometric combustion conditions. This increase of fueling and air charge leads to a larger torque torque produc productio tion n from from the engine engine thereb thereby y compen compensat sating ing for the additional torque load. For this investigation, the engine speed perturbations due to an external torque load are bounded by a hard time domain tolerance. In addition, the control effort available able for for spee speed d regu regula lati tion on is limi limite ted. d. It will will be show shown n that that the the perperformance tolerance on engine speed can only be satisfied under actuator saturation. A large amount of work exists exists concerning the the subject of of actuator ator satura saturatio tion n within within feedba feedback ck contro controll system systems. s. The majori majority ty of works focus on mitigating the effects of actuator saturation [ 3], [4], [6], [14], 14], [15], 15], [18], 18], through various antiwindup schemes (AWS’s). (AWS’s). These works provide useful results for the implementation of AWS for systems subject to actuator saturation. The unifying theme is to reduce the degradation of output performance due to actuator saturation. However, the performance benefits which can be achieved via actuator saturation are not yet explored. This work presents an experimental experimental validation of a controller design methodology which employs actuator saturation. In particula ticularr, a linear linear robust robust feedba feedback ck contro controlle llerr is design designed ed which which satsaturates for large external disturbances. However, for smaller external disturbances the controller will maintain linear operation. Such a controller appropriately links the external disturbance size to the degree of actuator input as specified by the time domain performance tolerance. II. DESIGN METHODOLOGY O VERVIEW Presented in this section is an overview of the controller design methodology to be used in this investigation [ 9]. The class of systems is limited to linear time invariant (LTI) single inputsingle single output (SISO) (SISO) uncertai uncertain n regulating regulating systems systems subject subject to delays described as

1063-6536/00$10.00 © 2000 IEEE

(1)


For this system, isa vectorrepresenting a compact set of uncertain system parameters, denotes the output variations about the desired set point, is the scalar controlled input operating about some nominal effort, is the plant dynamics, is the disturbance dynamics, and is an external step disturbance. Here denotes stable, strictly proper, real coefficient transfer functions. The output performance specification and actuator saturation constraint are and

(2)

In the absence of saturating the actuator with , the closed-loop transfer functions from the disturbance to the system output and actuator response are (3) and (4) where denotes the controller to be designed. The performance specifications (2) are satisfied if the following three sufficient conditions are met. Condition 1: The following amplitude inequalities are satisfied. Output Performance:

(5) Actuator Constraint:

(6) where magnitude of the step disturbance ; [5]. Condition 2: The Nyquist encirclement condition is satisfied . Condition 3: The closed-loop impulse response and is of one sign [13]. If a controller can be designed such that these sufficientconditions are satisfied and , thentheoutput tolerance and control effort constraint of (2) will be satisfied without actuator saturation. Remark: Condition 3 simplifies the enforcement of time domain tolerances using frequency domain constraints and is included in the sense of completeness. This condition will not necessarily be satisfied when designing linear controllers that achieve temporary actuator saturation. Franchek and Herman [5] give a complete development concerning time-to-frequency domain connections for a general class of linear feedback systems. To realize the controller, Jayasuriya and Franchek [ 12] propose the following loop shaping approach. First, the parameter space is gridded into parameter value sets denoted as

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where . Next the frequency space is replaced with discrete frequencies denoted as where . Typically the set contains low, middle, and high frequencies where resonant system frequencies are included into the middle frequency set. Evaluating (5) and (6) on the gain-phase plane constructs upper and lower amplitude bounds that must satisfy to meet (2). Classical loop shaping realizes the controller. Although discrete frequencies and parameter sets are used to design the controller, Conditions 1–3 must be satisfied and . A. Conditions Leading to Controller Design for Actuator Saturation

This proposed controller design technique applies when Conditions 1–3 cannot be simultaneously satisfied via a linear controller design. In such a case, (5) of Condition 1 (output performance), and Condition 2 (closed-loop stability) will be the only conditions satisfied by the proposed design methodology. The violation of (6) in Condition 1, and/or Condition 3 could potentially lead to the temporary actuator output saturation from a linear controller provided (7) Satisfying (7) guarantees that the actuator output is not saturated at steady-state conditions due to a step disturbance of magnitude . Hence for temporary actuator saturation, the satisfaction of (7) will be assumed in this manuscript. This proposed design technique will utilize the loop shaping approach of Jayasuriya and Franchek [12] to design feedback controllers. However, a pseudolinearization method based on a describing function approach will be employed to realize the linear saturating controller. To apply this linearization technique, specific frequencies requiring pseudolinearization are identified using (6). For those frequencies requiring pseudolinearization, the input amplitude size, , used to pseudolinearize the saturation element at a given frequency must be determined. The open-loop frequency response comprised of the pseudolinearized element, the linear dynamics, and the controller is shaped using classical loop shaping tools such that the resulting frequency response satisfies (5) of Condition 1 and satisfies Condition 2. The specific details of the pseudolinearization technique and the controller design are presented herein. B. Pseudolinearization Technique

The describing function technique is one approximation method used to represent a nonlinear element in the frequency domain [11]. Typically describing functions are associated with a limit cycle analysis of feedback systems. However, a rigorous development of the sinusoidal input describing functions (SIDF’s) extends their application beyond limit cycle analysis [11]. For the saturation element, the SIDF is

(8)

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where input amplitude to the saturation element; maximum output of a unity slope saturation element. As detailed in Herman [8], using the standard describing function approach within feedback systems does not accurately quantify the degree of saturation due to a step input. Of course this is not a surprise since the describing function is based on a sinusoidal input and is not intended for step inputs. To retain the frequency-based loop shaping approach for controller design, an alternative pseudolinearization approach based on the SIDF is proposed. The difference between the SIDF method and the proposed pseudolinearization is the amplitude ratio, namely in (8), used to evaluate the describing function. In performing the pseudolinearization of the saturation element, violation of inequality (6) is used to identify those frequencies and parameter sets which require pseudolinearization. In particular, the parameter sets of which violate amplitude inequality (6) for the given controller at the discrete frequencies are identified as (9) for and . The discrete frequencies for which at least one parameter set exists construct a frequency vector . The parameter sets identified from (9) at are denoted as . The variable denotes a dummy variable for the parameter sets identified in (9) at each discrete frequency . It is proposed in Herman and Franchek [ 9] to replace the amplitude ratio in (8) with defined as

(10) where

for a

combination. Hence

(11) Owing to the fact that describing function analysis is an approximate technique, a rigorous justification of replacing in (8) with to give (11) is notyet developed. Currently, applications of the pseudolinearization technique are used to test the hypothesis on a case-by-case basis. For the purpose of controller design, the nominal open-loop transfer function can be defined in the compact form as (12) where if if

Fig. 1.

Schematic of experimental facility.

where denotes the nominal plant. The resulting open-loop transfer function in (12) is realized through loop shaping to satisfy (5). III. EXPERIMENTAL F ACILITY Model development, controller design, and controller implementation were performed at the Advanced Engine Control Laboratory of the R. W. Herrick Laboratories at Purdue University. A schematic of the experimental setup is shown in Fig. 1. A complete description of the engine facility can be found in Hamilton and Franchek [7]. The Advanced Engine Control Research Facility includes a multichannel open architecture control system that is used for system identification, controller design, and controller implementation. This controller facility includes a Dell OptiPlex GXMT 5166 computer equipped with two Keithley Metrabyte data acquisition boards. These boards are dedicated A/D (DAS-1600, 16 channels) and D/A (DDA-06, 6 channels) which maximize the sample rate to 20 KHz. The support software is MATLAB, SIMULINK, and Real-Time Workshop from MathWorks, Inc. The combination of these software packages produces the open architecture control system that facilitates rapid prototyping. The engine used for thisinvestigation isa 1992Ford 4.6 L V-8 fuel injected engine. Engine speed is measured with a 0.6 /pulse optical encoder whose output is a square wave. This wave is processed through a frequency to voltage converter which produces a dc voltage representing engine speed. The maximum input frequency from the engine to the frequency voltage converter is 1.0 KHz, which is below the linear operating range of the converter which is 0–10 KHz. The linear relationship between engine speed and dc voltage is [V/r/min] where denotes absolute engine speed and denotes the dc voltage. The dc voltage is sent through a 3-pole Wavetek antialiasing filter with a cutoff frequency of Hz for digital sampling. The sampling rate for this investigation is


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Hz. Using this experimental facility, a controller is implemented. To implement this controller, the actuator signal amplifier (Fig. 1) is used to actuate engine speed via the BPAV. A torque load disturbance at idle can be generated through four primary sources: 1) the power steering pump; 2) transmission shifts; 3) alternator loads; and 4) air conditioning compressor loads. For this investigation, the power steering pump (PSP) will be used as the nonmeasurable, external torque disturbance denoted . The load generated via the PSP is modeled as a step disturbance with a magnitude of 20 Nm. A. Model Development

The development of the torque load to engine speed transfer function, , and the BPAV to engine speed transfer function, , is based on time domain (step response) information. Because is not in the feedback loop and does not affect closed-loop stability, it is modeled as a linear transfer function with fixed coefficients. However, is contained in thefeedback loop and therefore it is modeled as a linear transfer function with parametric uncertain coefficients. To develop the model for , the response of engine speed to a torque load of 20 Nm is measured. The engine speed is approximately 600 r/min when the PSP develops a 20 Nm step torque disturbance load to the engine. Based on this engine speed response, a transfer function consisting of a zero and a pair of complex poles is chosen for . The resulting model is

(13)

The development of is based on the transient response of engine speed subjected to a 1.0 V step input to the BPAV. This voltage input, which is larger than the 0.8 V saturation value, is used for the model identification phase to ensure that the dynamic behavior of the engine is captured beyond the operating range of interest. The engine is unloaded and idling at an approximate speed of 600 r/min when the step input is applied to the BPAV. The resulting engine speed response is used to determine the transfer function . Using this engine response, a parametric uncertain transfer function with a pure delay is used to bound the actual step response of the engine. The resulting transfer function model is

(14)

where ; ; ; ;

Fig. 2.

Block diagram of the engine.

; ; ; . These levels of uncertainty are determined such that the step response of this family of linear systems bounds the actual measured engine speed transient response. IV. ROBUST C ONTROLLER D ESIGN The feedback structure for the engine operating at 600 r/min is shown in Fig. 2. The performance specifications are

rpm

V

Nm

(15)

These specifications are based on the investigation of Hamilton and Franchek [7] which had the performance constraints of a 150-r/min tolerance with a 1.0–V saturation level. A. Amplitude Bound Formulation

Using the feedback structure (Fig. 2) and (13)–(15), amplitude bounds are formulated for the design. Employing the technique presented in Jayasuriya and Franchek [12], upper and lower amplitude bounds are formulated at the discrete fre0, 1, 2.5, 5, 8, and 12 rad/s. The design regions quencies for 0, 1, 5, 8, and 12 rad/s exist and can potentially be satisfied by . However, the upper and lower amplitude bounds at rad/s (Fig. 3) do not form a design region. Specifically, the solid line represents a lower amplitude bound on the open-loop transfer function. This lower bound is based on the degree of system uncertainty, the size of the external disturbance, and the output performance tolerance (5). The dashed line represents an upper amplitude bound on the open-loop transfer function as defined by (6). As can be seen in Fig. 3, the lower amplitude bound is greater than the upper amplitude bound for all open-loop phase angles. Therefore an acceptable design region does not exist. Consequently, the design methodology presented in Herman and Franchek [9] which allows the control effort upper amplitude bounds to be violated will be used. If either the actuator saturation

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Fig. 3.


Amplitude bounds at

rad/s.

constraint or the output performance tolerance is increased, a design region would emerge and actuator saturation could potentially be avoided. For this study, a controller will be designed which violates the control effort upper amplitude bounds such that the output performance amplitude bound is satisfied. The nominal plant , denoted as , is defined as the plant for which and does not contain the delay term. Instead, the delay will be embedded into the amplitude bounds.

a first-order lead compensator which achieves a 65 phase lead. Next the DC gain of the controller is chosen as 0.0025 such that the resulting meets its respective lower amplitude bound. Evaluating at the other discrete frequencies reveals that the output amplitude bounds are satisfied at these frequencies. Owing to sensor noise, the remaining task is to make the controller strictly proper. A high frequency pole is included at − 250. The final controller is

B. Controller Design

The controller design is executed to meet (5) of Condition 1, and Condition 2. The upper and lower amplitude bounds of (5) and (6) are shown in Fig. 4. In general, the controller design begins by adding lead compensation at the frequencies where a reduction in the amount of open-loop phase lag is required. This reduction in phase lag is identified by the amplitude bounds. For this design, a frequency of 2.5 rad/s will first be the first frequency investigated since an acceptable design region does not exist. At this frequency, the nominal open-loop plant frequency characteristics are dB and . The design goal at this frequency is to reduce this phase lag to zero. Consequently, the difference in open-loop amplitude between the upper and lower bound conflict is near its minimal value (Fig. 4). The compensator zeros and poles selected to achieve a 65 lead compensator at rad/s are − 1, − 3, and − 5, − 9, respectively. These pole and zero locations and the controller order are not unique. However, a second-order controller was selected to achieve a 65 phase lead since the corresponding phase and gain modifications made by the controller are isolated within a tighter frequency band than

(16)

The resulting open-loop transfer function (the solid line) and the pseudolinearized open-loop transfer function (denoted by ) are shown in Fig. 4. V. STABILITY AND L IMIT C YCLES As detailed in Herman and Franchek [9], must not encircle the stability region developed on the Nichols chart to ensure closed-loop stability for linear operation. Because the transfer functions and Nm the feedback system is bounded input–bounded output stable. Therefore, the remaining stability issue to be addressed is whether a limit cycle will exist. To determine if there is a potential for a closed-loop limit cycle to exist, a harmonic balance study [ 11] is conducted using describing function analysis. This analysis is predicated upon the satisfaction of (17)


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Fig. 4. Final controller design on the Nichols chart.

where is the resulting equivalent gain based on describing function analysis of the saturation element. For the controller in (16) and (17) is not satisfied. Thus, it is likely that the closed-loop system will avoid limit cycling. Further confidence is gained in the limit cycle analysis via the simulation of the closed-loop system shown in Fig. 5.

VI. RESULTS Shown in Fig. 5 are the 27 closed-loop responses of the system due to a step disturbance of magnitude 20 Nm for the permutations of the parameter values, , . The controller satisfies the performance specifications in simulation. In fact, the design is conservative since r/min. This is attributed to the fact that the energy content of the saturated signal is larger than the energy content of the single harmonic used to approximate the frequency response of the saturation element [ 11]. Following the numerical simulation, the controller is implemented on the engine. Shown in Fig. 6 is the transient response of the engine for a 20 Nm step torque disturbance from the PSP. The performance specifications are satisfied including r/min. The difference between the simulation and implementation performances are attributed to the consequences of approximating a nonlinear system with a limited order linear uncertain model. In addition, the engine speed deviations due to the individual firing events, which are neglected in the engine

model, can be seen in Fig. 6 as the higher frequency noise like signature. Also shown in Fig. 6 is a steady-state limit cycling behavior. This limit cycling is due to the BPAV controller interacting with the fueling controller. As the BPAV controller compensates for the engine speed fluctuations, the fueling controller reacts so that the tailpipe emissions are met. This produces the limit cycling. To eliminate this oscillation in engine speed, the −250 polein couldbe movedcloserto the -plane origin. However, the magnitude of the engine speed oscillations is not audible by the driver.

VII. CONCLUSIONS A feedback controller design methodology is demonstrated which utilizes actuator saturation to achieve enhanced regulating performance while satisfying stringent output performance specifications. Enhanced regulating performance is achieved via the design of a linear controller which temporarily saturates the actuator for large step disturbances yet operates linearly for smaller step disturbances. The design methodology is based on a pseudolinearization technique which employs describing function analysis. The pseudolinearization technique is formulated such that controller design is executed in the frequency domain. Designing in the frequency domain facilitates the design of systems with parametric uncertainty and delays. Therefore, the methodology is well suited to the design of controllers for physical systems.

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Fig. 5. Simulated closed-loop step engine speed responses.

Fig. 6. Experimental closed-loop engine speed response.



Using the methodology, a controller is designed to regulate the idle speed of a Ford V-8 engine subject to a torque disturbance. The resulting controller is simulated and implemented which results in a level of performance unattainable with linear control. In conclusion, the presented technique is a novel approach for the design of saturating controllers. ACKNOWLEDGMENT The authors would like to thank F. Yeung and S. Smith of Ford Motor Co. for the experimental facility. REFERENCES [1] M. Abate, B. R. Barmish, C. Murillo-Sanchez, and R. Tempo, “Application of some new tools to robust stability analysis of spark ignition engines: A case study,” IEEE Trans. Contr. Syst. Technol. , vol. 2, no. 1, pp. 22–30, 1994. [2] R. Bosch, Automotive Electric/Electronic Systems : Soc. Automotive Eng. Publication, 1988. [3] P. J. Campo and M. Morari, “Robust control of processes subjected to saturation nonlinearities,” Comput. Chem. Eng. , vol. 14, no. 4/5, pp. 343–358, 1990. [4] J. C. Doyle, R. S. Smith, and D. F. Enns, “Control of plants with input saturation nonlinearities,” in Amer. Contr. Conf. , 1987, pp. 1034–1039. [5] M. A. Franchek and P. A. Herman, “Direct connection between time domain performance and frequency domain characteristics,” Int. J. Robust Nonlinear Contr. , vol. 8, pp. 1021–1042, 1998. [6] A. H. Glattfelder and W. Schaufelberger, “Stability analysis of single loop control systems with saturations and antireset-windup circuits,” IEEE Trans. Automat. Contr. , vol. AC-28, no. 12, pp. 1074–1081, 1983. [7] G. K. Hamilton and M. A. Franchek, “Robust controller design and experimental verification of I.C. engine speed control,” Int. J. Robust Nonlinear Contr. , vol. 7, pp. 609–627, 1997.

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[8] P. A. Herman, “Performance enhancement of regulating systems via actuator saturation,” doctoral dissertation, Purdue Univ., West Lafayette, IN, 1997. [9] P. A. Herman and M. A. Franchek, “Performance enhancement of fixed regulating systems via actuator saturation,” J. Dynamic Syst., Measurement, Contr. , vol. 121, no. 1, pp. 34–40, 1999. [10] D. Hrovat and B. Bodenheimer, “Robust automotive idle speed control design based on µ -synthesis,” in Proc. Amer. Contr. Conf. , San Francisco, CA, 1993, pp. 1778–1783. [11] J. C. Hsu and A. U. Meyer, Modern Control Principles and Applications . New York: McGraw-Hill, 1968. [12] S. Jayasuriya and M. A. Franchek, “A QFT-type design methodology for a parallel plant structure and its application in idle speed control,” Int. J. Contr. , vol. 60, no. 5, pp. 653–670, 1994. [13] S. Jayasuriya and J. W. Song, “On the synthesis of compensators for nonovershooting step responses,” J. Dynamic Syst., Measurement, Contr. , vol. 118, no. 4, pp. 757–763, 1996. [14] N. J. Krikelis and S. K. Barkas, “Design of tracking systems subject to actuator saturation and integrator wind-up,” Int. J. Contr. , vol. 39, no. 4, pp. 667–682, 1984. [15] M. C. Leu, S. Yang, and A. U. Meyer, “Antiwindup control of secondorder plants with saturation nonlinearity,” J. Dynamic Syst., Measurement, Contr. , vol. 115, pp. 715–720, 1993. [16] T. Vesterholm andE. Hendricks, “Advancednonlinear engine speedcontrol systems,” in Proc. Amer. Contr. Conf. , Baltimore, MD, 1994, pp. 1579–1580. [17] S. J. Williams, D. Hrovat, C. Davey, D. Maclay, J. W. Crevel, and L. F. Chen, “Idle speed control designusingan H approach,” in Proc. Amer. Contr. Conf. , Pittsburgh, PA, 1989, pp. 1950–1956. [18] M. Zheng, M. Kothare, and M. Morari, “Antiwindup for internal model control,” Int. J. Contr. , vol. 60, no. 5, pp. 1015–1024, 1994.

Engine Idle Speed Control Using Actuator Saturation

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