Robust SISO H∞ controller design for nonlinear systems

ARTICLE IN PRESS

Control Engineering Practice 13 (2005) 1413–1423 www.elsevier.com/locate/conengprac

Robust SISO H 1 controller design for nonlinear systems Grant A. Ingrama,, Matthew A. Franchekb, Venkataramanan Balakrishnanc, Gopichandra Surnillad a

Department of Mechanical Engineering, Purdue University, 140 S. Intramural Drive, IN 47907, West Lafayette, USA b Department of Mechanical Engineering, University of Houston, Houston, TX, USA c Department of Electrical and Computer Engineering, Purdue University, West Lafayette, IN, USA d Ford Motor Company, Dearborn, MI, USA Received 28 October 2003; accepted 22 December 2004 Available online 26 February 2005

Abstract Presented in this paper is a nonlinear SISO controller design methodology for a class of Hammerstein models. The design process is composed of standard system identification techniques integrated with an H 1 linear controller synthesis formulation. The system identification portion of this work first identifies the static, single-valued nonlinearity capturing the nonlinear behavior of the system. This nonlinearity is then inverted and serves as a precompensator to the system input. The frequency response function is then identified with the precompensator in place to capture the linear dynamics of the system. Errors associated with the nonlinear inversion are addressed in an unstructured uncertainty formulation. A robust H 1 controller is synthesized using the identified uncertain Hammerstein model and a systematic performance weighting selection process for a class of L1 constraints. Closed-loop performance and stability are assessed via sector bounds quantifying the maximum allowable precompensator error. Frequency domain conditions guaranteeing an L2 output provided the system input belongs to L2 are also presented. To illustrate the procedure, the design methodology is applied to synthesize a robust feedback controller to regulate the mass air flow of a 4.6 L V8 spark ignition engine equipped with an electronic throttle. r 2005 Elsevier Ltd. All rights reserved. Keywords: Control system analysis; Control system design; Engine control; H-infinity control; Nonlinear models; Robust control; Weighting functions

1. Introduction Today there exist many techniques that separately address system identification and controller design. However, these individual knowledge bases have not been harvested to produce a systematic controller design solution. Furthermore, after the controller has been determined, it is not always clear how to quantify the robustness of the design. El-Farra developed an approach to design SISO controllers for nonlinear systems with uncertainty and input constraints (El-Farra & Christofides, 2001). This Corresponding author. Tel.: +1 765 494 0231; fax: +1 765 494 0787. E-mail address: [email protected] (G.A. Ingram).

0967-0661/$ - see front matter r 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.conengprac.2004.12.007

method focused on a general Lyapunov-based design that did not provide a direct method of enforcing output constraints. Hedrick presented a multiple sliding surface method developed for a class of uncertain nonlinear systems (Hedrick, 1998). In this method, input and output constraints are not considered. Genetic Algorithms (Al-Duwaish & Bettayeb, 1997) and sinusoidalinput describing functions (Zhuang & Atherton, 1996) have also been used to design controllers for nonlinear systems. However, none of these methods have incorporated system identification techniques, input and output constraints, and an evaluation of modeling error into the total controller design process. The controller design methodology advanced in this manuscript addresses the robust SISO controller design problem for nonlinear systems, specifically uncertain

ARTICLE IN PRESS 1414

G.A. Ingram et al. / Control Engineering Practice 13 (2005) 1413–1423

linear plants preceded by static nonlinearities (Hammerstein models). Existing system identification techniques have been integrated with a controller design process in this methodology. The contributions of this work are four fold: (1) a methodology for robust SISO controller synthesis for class of nonlinear systems described by uncertain Hammerstein models, (2) a systematic method for selecting H 1 weighting functions for a class of L1 constraints, (3) a measure of system robustness specifically addressing the maximum allowable modeling and nonlinear inversion errors, and (4) frequency domain conditions guaranteeing an L2 output provided the system input belongs to L2 : The systematic robust SISO controller design process for nonlinear systems presented in this manuscript combines a common technique for system identification and a controller synthesis process into a complete systematic procedure. The process is broken into three distinct parts: system identification, controller synthesis, and system analysis. Modeling efforts are focused on an experimental method of system identification that captures nonlinear plant characteristics. The system identification process involves the identification of the plant nonlinearity followed by the identification of the uncertain linear plant dynamics. The errors between the nonlinear model and the actual system data are incorporated as unstructured uncertainty. Since uncertainty is present, a robust controller design methodology that guarantees system performance is required. A result of choosing the H 1 design methodology is that the closed-loop system performance is contingent upon the choice of the performance weighting functions. The selection of these weighting functions is often difficult, although, general guidelines for their development do exist. Many papers document techniques and guidelines for weighting function selection. General frequency domain guidelines are given in Meghani and Latchman (1992) and Grimble and Biss (1988). Chun and Hori (1996) provide typical weighting functions and tuning methods. A method for determining weighting functions which represent position and rate limit constraints of an actuator as well as a procedure for controlling the closed-loop system overshoot are provided in Hu, Unbehauen, and Bohn (1996), Hu, Bohn, and Wu (1999), Hu, Bohn, and Wu (2000). An experimental solution for weighting function selection, implementing orthogonal arrays, is presented by Yang, Ju, and Liu (1994). Genetic algorithms have also been employed to search for suitable weighting function solutions (Donha, Desanj, & Katebi, 1997). Further studies regarding methods of weighting function selection may be found in Postlethwaite, Tsai, and Gu (1990) and Beaven, Wright, and Seaward (1996). The technique chosen for the procedure described in this manuscript was developed by Franchek where the weighting

functions are chosen in a manner which enforces time domain tolerances (Franchek, 1996). This method of weighting function selection will be extended in this work to address the controller design objectives of maximizing the allowable reference step size to the system and maximizing the system tracking response. Following the controller synthesis process, the maximum allowable modeling and nonlinear inversion errors are determined from a stability analysis. Finally, provided certain frequency domain conditions are satisfied and the system input belongs to L2 ; an L2 output may be guaranteed. To illustrate this procedure, the design methodology is applied to synthesize a robust feedback controller to regulate the mass air flow (MAF) of an engine. In this application, a Hammerstein model of a 4.6 L V8 spark ignition engine from an electronic throttle input to engine MAF output is identified. An H 1 tracking controller is then designed to control engine MAF with zero steady-state error while addressing the nonlinear throttle characteristics and time delay. Experimental data validates successful closed-loop performance which includes noise and disturbance rejection while maintaining good transient and steady-state performance.

2. Problem statement and method of solution Consider the standard Hammerstein model given in ~ 2 L1 is the system input, nðÞ is a static Fig. 1 where uðtÞ ~ single-valued nonlinearity operating on uðtÞ; pðtÞ 2 L2 is an impulse response function, and yðtÞ 2 L1 is the measured system output (Ljung, 1999). It is assumed that both nðÞ and pðtÞ exist but are unknown a priori. The closed-loop tracking performance specifications for this class of systems includes a control effort constraint and an allowable tracking error constraint. The time domain control effort constraint about a nominal effort is given as juðtÞjpk

(1)

8t40

and the tracking deviation constraint is jeðtÞjpd

(2)

8t40;

where eðtÞ is the tracking error of the closed-loop system. All time domain specifications are known a priori and it is assumed the system is initially at rest. The goal specifically addressed in this work is to design a nonlinear feedback controller that meets the closed-loop performance specifications of Eqs. (1) and (2). ~u(t)

n(.)

y(t) p(t)

Fig. 1. General Hammerstein model.

ARTICLE IN PRESS G.A. Ingram et al. / Control Engineering Practice 13 (2005) 1413–1423

The method of solution is a controller design executed in two phases, system identification and H 1 controller synthesis, followed by system analysis. The system identification begins with a representation of nðÞ; ^ denoted as nðÞ; identified through steady-state testing. ^ is invertible For the class of nonlinearity considered, nðÞ and its inversion used as a precompensator (Fig. 2). An equivalent block diagram for the precompensated system where the errors of the inversion process are represented by an uncertainty in the linear dynamics, pðt; aÞ; is also shown in Fig. 2. The uncertainty of the linear plant dynamics is represented via the parameter a: Standard frequency response function (FRF) estimates using spectral density calculations identify pðt; aÞ: From these nonparametric models, a linear uncertain model, ^ aÞ; is chosen. denoted by Pðs; ^ With Pðs; aÞ identified, the mixed sensitivity problem is solved. The corresponding performance weights for the H 1 controller synthesis are selected by enforcing the prespecified time domain tolerances, given in Eqs. (1) and (2), using frequency domain constraints. After the H 1 controller design process is complete, the closed-loop stability of the entire system is quantified. It is desirable to determine the allowable error size due to modeling and nonlinearity inversion contained in pðt; aÞ subjected to closed-loop system properties. To this end, this error is assumed to be a memoryless timevarying nonlinearity, fðs; tÞ; which belongs to the sector [a,b]. To quantify this error, the Lur’e problem of Fig. 3 is analyzed using the well-known circle criterion (Sastry, 1999).

Unknown Nonlinear System ~u(t)

∧ [n(.)]-1

u(t)

u(t)

n(.)

p(t,α)

y(t)

p(t)

y(t,α)

Fig. 2. Block diagram of the dynamic system and precompensator.

u1(t)

+

e1(t)

y1(t) g(t)

+ y2(t)

φ (σ,t)

e2(t)

u2(t) +

Fig. 3. Lur’e feedback interconnection problem.

1415

Once the maximum sector bounds of the modeling and nonlinear inversion errors are determined, finitegain L2 stability between the actual system input and output is guaranteed provided certain conditions are satisfied (to be discussed). The details of each step of this process are presented in the following sections.

3. Main results Presented is a systematic controller design process for a large class of nonlinear systems, specifically uncertain linear plants preceded by static nonlinearities. In the following sections, a robust controller design process will be described including system identification, weighting function selection, controller design, and stability analysis. 3.1. Identification of the dynamic model A two-part system identification process is proposed to determine the control-oriented system model. First, the static single-valued nonlinearity, nðÞ; of the system is ^ identified, represented by nðÞ; to capture the nonlinear characteristics of the system. Once identified, a transfer function estimation of the combined system, shown in Fig. 2, is implemented to develop a dynamic model having unstructured uncertainty. The two-part system identification is detailed in the following subsections. 3.1.1. Nonlinear static mapping summary Steady-state testing can identify nðÞ; which is represented by an invertible parameterization denoted ^ ^ must be one to as nðÞ: The mapping represented by nðÞ one to ensure invertibility. System and measurement noise are mitigated during steady-state testing by averaging data. Black-box terms (such as spline-function coefficients and look-up tables) or physical parameters (such as saturation levels) may be used to parameterize nðÞ (Ljung, 1999). A polynomial expansion to describe nðÞ may be efficiently determined through a forward-regression orthogonal least-squares formulation. This procedure selects parameters from a set of possible candidate parameters such that the maximum increment to explained variance is achieved for each additional model parameter. Billings provides a complete description of this procedure (Billings, Chen, & Korenberg, 1989). Often times possible regressors to describe nonlinearities will be revealed using engineering insight (Ljung, 1999). Such black-box-based models are known as semi-physical models. However there exist some nonlinear functions which are difficult to represent using polynomial relationships. To mitigate this class of static nonlinearities, a ‘‘look-up’’ table may be developed to describe nðÞ:

ARTICLE IN PRESS G.A. Ingram et al. / Control Engineering Practice 13 (2005) 1413–1423

1416

With the representation of the static nonlinearity of ^ can be inverted and the nonlinear plant identified, nðÞ used as a precompensator to remove the nonlinear behavior of the system (Fig. 2). The errors associated with the nonlinear inversion process are represented through the uncertainty of the linear dynamics. Now pðt; aÞ may be identified. 3.1.2. System identification of the precompensator and plant To complete the system identification, the linear uncertain dynamic model of the plant will be estimated using standard FRF techniques. The equivalent linear uncertain system represented by pðt; aÞ in Fig. 2 and in equation form by ^ 1 Þ pðt; aÞ ¼ pðtÞ nð½nðÞ

(3)

is identified using the FRF calculation HðjoÞ ¼

S xy ðjoÞ ; S xx ðjoÞ

(4)

where HðjoÞ is the system FRF, S xy ðjoÞ is the crossspectral density of the input and output, and S xx ðjoÞ is the auto-spectral density of the input. This calculation provides an unbiased estimate of a SISO system with uncorrelated output noise (Bendat & Piersol, 1993). White noise (other frequency-rich signals may be selected) is sent as a command input to the precompensator. FRF estimates of the pseudo-linear system are obtained for a representative set of system operating conditions. These estimates create a family of experimental FRFs representing different operating conditions and plant-to-plant variability. Additive or multiplicative uncertainty may be chosen to capture the uncertainty of the plant and any errors associated with the inversion of nðÞ: Assuming multiplicative uncertainty, the uncertain plant model is ^ aÞ ¼ ð1 þ DW ^ 2 ðsÞÞPo ðsÞ; Pðs;

(5)

where Po ðsÞ is the chosen nominal plant, D is an allowable variable stable transfer function satisfying (6) kDk1 p1; ^ 2 ðsÞ is a fixed stable transfer function (Doyle, and W Francis, & Tannenbaum, 1992). Recalling that Pðjo; aÞ is measured, the multiplicative ^ 2 ðsÞ; may be deteruncertainty weighting function, W mined by Pðjo; aÞ pjW ^ 2 ðjoÞj 8o: 1 (7) P ðjoÞ o A further discussion may be found in Doyle et al. (1992). The uncertainty weighting provides an upper norm bound on the system uncertainty. It is advantageous to

choose Po ðsÞ such that it minimizes the uncertainty ^ 2 ðsÞ; of the plant description, ultimately weighting, W leading to a less conservative design. 3.2. Robust H 1 controller design Presented in this section is the synthesis of an H 1 controller for the equivalent linear uncertain system represented by pðt; aÞ in Fig. 2. For the purposes of ^ aÞ controller synthesis, a standard H 1 design on Pðs; will be presented. 3.2.1. Tracking control H 1 weighting function selection Presented in this section is the development of performance weights for the tracking control problem. Consider the tracking problem with multiplicative unstructured uncertainty shown in Fig. 4 where G R ðsÞ represents the reference dynamics (assumed stable), uðsÞ is the scalar controller output variation about a nominal effort, RðsÞ represents a step reference command of magnitude j; eðsÞ is the scalar error variation of the system about zero, qðsÞ and pðsÞ are the uncertainty input and output, respectively, KðsÞ is the controller, and Y ðsÞ is the scalar output of the system. The plant description, as described by Eq. (5), is assumed proper with no hidden unstable modes. Theorems 1 and 2 describe the performance weight selection to enforce the SISO tracking control effort and tracking deviation constraints, respectively. Fig. 5 incorporates the performance weights into the feedback structure for tracking. Theorem 1. Assuming the feedback system in Fig. 4 has zero initial conditions, GR ðsÞ 2 RH 2 ; Po ðsÞð1 þ ^ 2 ðsÞDÞ 2 RH 2 ; proper, and has no hidden unstable W modes, then juðtÞjpk for t40 when rðtÞ is a step input of magnitude j provided kW 2 ðjoÞT o ðjoÞk1 o1;

(8)

where W 2 ðsÞ is a stable, minimum phase transfer function (not necessarily proper) satisfying jG ^ R ðjoÞ ^ 2 ðjoÞj; þ jW jW 2 ðjoÞjX (9) kPo ðjoÞ ^ ¼ jL; L is a scaling constant which justifies the where j enforcement of time domain specifications through frequency domain amplitude constraints, and T o ðsÞ is

W2(s) R(s)

GR(s)

+-

e(s)

K(s)

u(s)

q(s)

Po(s)

Fig. 4. SISO feedback tracking block diagram.

p(s)

Y(s) ++

ARTICLE IN PRESS G.A. Ingram et al. / Control Engineering Practice 13 (2005) 1413–1423

W2(s) R(s)

+-

W1(s)

e(s)

K(s)

u(s)

q(s)

Po(s)

p(s)

++

Y(s)

Fig. 5. Feedback structure for tracking with uncertainty.

the complimentary sensitivity function of the nominal system, T o ðjoÞ ¼

KðjoÞPo ðjoÞ : 1 þ KðjoÞPo ðjoÞ

(10)

Proof. Applying a change of variables, the proof is similar to the proof given by Franchek (1996) for regulation control H 1 weighting function selection. & Theorem 2. Assuming the feedback system in Fig. 4 has zero initial conditions, G R ðsÞ 2 RH 2 ; Po ðsÞð1 þ ^ 2 ðsÞDÞ 2 RH 2 ; proper, and has no hidden unstable W modes, then jeðtÞjpd for t40 when rðtÞ is a step input of magnitude j provided kjW 1 ðjoÞSo ðjoÞj þ jW 2 ðjoÞT o ðjoÞjk1 o1;

(11)

where W 1 ðsÞ ¼

^ R ðsÞ jG ; d

(12)

W 2 ðsÞ and T o ðsÞ are defined in Theorem 1, and So ðsÞ is the sensitivity function of the nominal system, S o ðjoÞ ¼

1 : 1 þ KðjoÞ  Po ðjoÞ

(13)

Proof. Applying a change of variables, the proof is similar to the proof given by Franchek (1996) for regulation control H 1 weighting function selection. & The weighting function (W 2 ðsÞ) is contingent upon the ^ 2 ðsÞ), reference dynamics and uncertainty weighting (W magnitude (jG R ðsÞ), actuator saturation (k), plant dynamics (Po ðsÞ), and a scaling constant (L). Note that if the control effort specification is infinite, i.e. k ¼ 1; ^ 2 ðjoÞj and therefore W 2 ðsÞ would then jW 2 ðjoÞjXjW only be contingent upon the uncertainty weighting for all frequencies. The weighting function, W 1 ðsÞ; emphasizes the most important frequencies generated by the reference command. Therefore the shape of W 1 ðsÞ is similar to the reference dynamics, G R ðsÞ: This requires the H 1 controller design to minimize the energy between the reference command and the tracking deviations at those frequencies where the reference command dynamics contain significant energy. The magnitude of the reference step, j; and the tracking deviation constraint,

1417

d; also play a role in the minimizing the energy between the reference command and the tracking deviations. However, this modifies only the static gain of W 1 ðsÞ: 3.2.2. Tracking H 1 controller synthesis Two procedures are now proposed for the H 1 controller design. First, a procedure for controller synthesis such that the allowable reference step size is maximized will be given. The idea is to maximize the reference command step size, j; until the result violates one of the time domain specifications. The optimization ^ until one of the is actually performed by maximizing j conditions set forth in Theorems 1 or 2 is violated. The values of j and L are unknown through the design. After the controller is synthesized, the actual largest reference step command size (j ) is determined through ^ is an iterative simulation. The maximization of j ^ is process and an initial guess for j ^ l oe; e  1: 0oj

(14)

^ is The design is optimal in the sense that the maximum j found for the weighting functions selected (based on the enforcement of time domain specifications). A second option for controller synthesis is a procedure that maximizes the system response to a step change in reference. Essentially, the idea driving this process is to follow the same procedure outlined above, however, relax the tracking deviation constraint while maximizing the reference command step size (j). This forces the controller synthesis to maximize the actuator effort. Again, the optimization is actually performed by ^ relaxing the tracking deviation constraint maximizing j; (i.e. allowing d to be large), until one of the conditions ^ is set forth in Theorems 1 or 2 is violated. The value of j found as described above in an iterative process using an initial guess for j^ as provided in Eq. (14). 3.3. Stability analysis of the closed-loop system using the circle criterion Once the H 1 controller design process is complete, the closed-loop stability of the entire system may be quantified. The modeling and nonlinear inversion errors contained within pðt; aÞ are assumed to be a memoryless time-varying nonlinearity, fðs; tÞ; which belongs to the sector [a,b] (Fig. 2). To quantify closed-loop robustness, the Lur’e problem of Fig. 3 is analyzed using the wellknown circle criterion stated in Theorem 3. The block ∧ e(t)

∧ u(t) +

k(t)

φ (σ,t)

∧ p(t,α)

∧ y(t)

-

Fig. 6. Feedback interconnection of controller, modeling and nonlinear inversion errors, and plant.

ARTICLE IN PRESS G.A. Ingram et al. / Control Engineering Practice 13 (2005) 1413–1423

1418

diagram of the controller, modeling and nonlinear inversion errors, and plant to be analyzed is shown in Fig. 6 and is manipulated into the Lur’e problem. The circle criterion theorem given in Sastry (1999) is stated below.

the largest sector bound of the memoryless time-varying nonlinearity fðs; tÞ:

Theorem 3. Consider the Lur’e system in Fig. 3, where L½gðtÞ ¼ GðsÞ 2 A where A represents the algebra of transfer functions (Sastry, 1999) and fðs; tÞ is a memoryless, possibly time-varying, nonlinearity in the sector ½ r; r: Then, the closed-loop system is finite gain L2 stable if

The problem of determining the largest sector bounds of fðs; tÞ becomes one of optimizing two variables, a and b. Instead of solving an optimization problem of two variables, a constrained optimization problem using one variable may be developed. Consider the following, ^ 1 and nðÞ shown in Fig. if no errors exist between ½nðÞ 2, fðs; tÞ can be represented by a mapping of a line with unity slope and zero intercept. Since the upper and lower bounds of the sector [a,b] represent limits on the total modeling and nonlinear inversion errors that may be tolerated by the system, the sector [a,b] represents the maximum and minimum gains about unity, i.e. a line with unity slope and zero intercept. Therefore, given this case, the optimization problem of two variables may be rewritten as a constrained optimization problem of one variable, y (Fig. 8). In this case, y is maximized and a and b are constrained by p  a ¼ tan y (20) 4 and  p b ¼ tan y þ : (21) 4 The maximum sector bounds are found when y is maximized while satisfying the provisions of Theorem 3,   1 ^ : (22) ymax ¼ maxfyg sup jGðjoÞjor

sup jGðjoÞjor 1 ;

(15)

o2R

where R is the field of real numbers. For the case where fðs; tÞ belongs to the sector [a,b] for arbitrary a, b, define c¼

aþb ; 2

(16)

r¼

b a 2

(17)

and ^ ¼ GðsÞ

GðsÞ : 1 þ cGðsÞ

(18)

^ 2A Then the closed-loop system is L2 stable if GðsÞ (Sastry, 1999) and 1 ^ sup jGðjoÞjor :

(19)

o2R

Proof. See Sastry (1999).

&

The allowable modeling and nonlinear inversion errors, fðs; tÞ; may be quantified by applying Theorem 3 to the system described in Fig. 3. In many cases, a free integrator is added to the controller to ensure zero steady-state error. Therefore Eq. (19) must be satisfied for stability according to the circle criterion. In this case, the Lur’e problem may be rewritten into the form shown in Fig. 7 and Theorem 3 applied. The modeling and nonlinear inversion errors will be quantified by finding

3.4. Quantifying allowable modeling and nonlinear inversion errors

o2R

φ (σ,t)

slope=b slope=1

2θ ∧ u(t) u1(t) +

e1(t) + -

-

xe(t)

∧ y(t) ∧ p(t,α)

yc(t)

+∧ e(t)

y1(t)

k(t)

slope=a

c σ c -

y2(t) +

φ (σ,t)

e2(t)

+ u (t) 2 +

Fig. 7. System feedback interconnection problem with loop transformation.

Fig. 8. Sector bounds of fðs; tÞ:

ARTICLE IN PRESS G.A. Ingram et al. / Control Engineering Practice 13 (2005) 1413–1423

A simple bisection method may be implemented to determine ymax and the resulting a and b determined by Eqs. (20) and (21).

3.5. Allowable input signal size for BIBO stability Once the maximum sector bounds of the modeling and nonlinear inversion errors are determined, Corollary 1 provides additional information regarding the stability of the system with respect to the actual system ^ and yðtÞ; ^ input and output, uðtÞ respectively (Figs. 6 and 7). Corollary 1 follows a proof in Vidyasagar (1978) and provides a result of finite-gain L2 stability between the actual system input and output under certain conditions.

1419

4. Spark ignition engine mass air flow control application The proposed robust SISO controller design methodology is applied to synthesize a robust feedback controller to regulate the MAF of a 4.6 L V8 spark ignition engine equipped with an electronic throttle. The associated controller design challenges are three-fold. First, the system contains a time delay, which must be incorporated in the design. Second, the throttle has a nonlinear gain, which varies as a function of throttle angle and engine speed. Third, the controller must provide adequate performance from idle to high speed/ load engine conditions. 4.1. Identification of the dynamic model

Corollary 1. Consider the system in Fig. 7, where KðsÞ is the controller (with a free integrator), fðs; tÞ and c are ^ aÞ is the plant as described in Theorem 3, and Pðs; ^ aÞ as ^ aÞ; and Gðs; described by Eq. (5). Define F^ ðs; aÞ; Hðs; F^ ðs; aÞ ¼

^ aÞ ¼ Hðs;

^ aÞ Pðs; ; ^ aÞKðsÞc 1 Pðs; KðsÞ

(23)

;

(24)

^ ^ aÞ ¼ Pðs; aÞKðsÞ : Gðs; ^ aÞKðsÞc 1 Pðs;

(25)

^ aÞKðsÞc 1 Pðs;

^ aÞ 2 A 8a and satisfy ^ aÞ; Gðs; Suppose F^ ðs; aÞ; Hðs; sup jF^ ðjo; aÞj9g2 o1

8a;

(26)

^ aÞj9g1 o1 sup jHðjo;

8a;

(27)

^ sup jGðjo; aÞj9go or 1

8a;

(28)

o2R

o2R

o2R

The proposed two-part system identification process is employed. First, a static (steady-state) map is empirically determined to capture the majority of the nonlinearities associated with the engine throttle characteristics. This map is then inverted to serve as a precompensator to the system. Next, a transfer function estimation of the combined system, i.e. the inverted steady-state throttle map and engine, is calculated to develop the dynamic system model with uncertainty. The two-part system identification is detailed in the following subsections. 4.1.1. Development of the mass air flow static map The flow characteristics of a throttle are inherently nonlinear. The relationship between throttle position and MAF has been addressed by Stefanopoulou, Grizzle, and Freudenberg (1994), Benninger and Plapp (1991), Bidan, Boverie, and Chaumerliac (1995), as well as others. The desired mapping proposed in this work is an experimental process involving a series of throttle sweeps. The engine was held at a constant speed by the dynamometer while throttle position was slowly swept.

where r is defined in Theorem 3. Then the closed-loop ^ is finite-gain system, including the mapping from u^ to y; ^ 2 L2 : L2 stable provided u1 ðtÞ; u2 ðtÞ; uðtÞ

A methodology has been presented to quantify the modeling and nonlinear inversion errors by determining the largest sector bounds of a memoryless time varying nonlinearity while guaranteeing L2 stability. Furthermore, Corollary 1 provides an extension of Theorem 3 to guarantee finite-gain L2 stability between the actual ^ ^ system input and output, uðtÞ and yðtÞ; respectively. Using this information, the closed-loop system may be evaluated.

Throttle Cmd (normalized)

Proof. Similar form of proof found in Vidyasagar (1978). &

0.5 0.4 0.3 0.2 0.1 0 4 3 MA F

0 1000

2 (vo lts

1 )

0

2000 ) PM d (R pee

3000 4000

S

Fig. 9. Inverted mass air flow static map.

ARTICLE IN PRESS

Desired MAF Engine Speed

Inverted Static Map

Throttle Command

MAF (volts) Fig. 10. Experimental setup for system identification of mass air flow system.

0 -10 Nominal Plant -20 100 101 Frequency (rad/sec)

Phase (degrees)

4.1.2. System identification of the inverted mass air flow static map and engine A model of the engine dynamics related to engine MAF is required for controller design. To this point, a static map has been determined experimentally and inverted to remove the majority of the nonlinearities associated with the throttle flow characteristics. Now, the frequency response estimation of the combined system, i.e. the steady-state throttle map and engine, may commence. A block diagram of the dynamic system to be identified is shown in Fig. 10. White noise was sent as a frequency-rich command input to the system. The white noise perturbs the desired MAF while current engine speed is also provided to the static map. The output of the inverted static map commands the electronic throttle position. Desired and actual engine MAF were measured for several engine operating conditions: 40 kph (1000 RPM, 105:8 N m), 76 kph (1260 RPM, 149:1 N m), 93 kph (1550 RPM, 210:2 N mÞ; and 119 kph (2000 RPM, 192:5 N m). The ^ aÞ was identified implementing a standard model Pðs; FRF calculation (Eq. (4)) for each operating condition resulting in the FRFs presented in Fig. 11. It is clear from the FRFs of the system that a time delay exists. This delay is associated with the breathing dynamics of the engine. A restriction of H 1 controller design synthesis is that the nominal plant may not contain a time delay term. The delay may be incorporated into the nominal plant by a Pade´ approximation. However, the Pade´ approximation would increase the order of the nominal plant and therefore a higher-order H 1 controller would be synthesized. The higher-order controller may be more difficult to reduce. In this case, the delay will be incorporated into the uncertainty weighting as discussed by Doyle et al. (1992). The

10

-30 102

0 -100 -200

Nominal Plant

-300 -400 100 101 Frequency (rad/sec)

102

Fig. 11. System identification of mass air flow system.

Low Coherence

20 10 Magnitude (dB)

The experiment was repeated for several engine speeds between 600 and 4000 RPM. To mitigate the nonlinear throttle behavior, this static map (look-up table) must be inverted to provide the relationship between MAF and engine speed to throttle commanded position (Fig. 9). By inverting the engine map, a desired MAF and current engine speed may be used to predict a throttle command.

Magnitude (dB)

G.A. Ingram et al. / Control Engineering Practice 13 (2005) 1413–1423

1420

0 -10 -20 -30 -40 100

101 Frequency (rad/sec)

102

^ 2 ðsÞ Bounding the uncertainty of the nominal plant. Fig. 12. W

selected nominal plant transfer function is Po ðsÞ ¼

1:2589ðs=16 þ 1Þ ðs=20 þ 1Þððs=50Þ2 þ ð2ð0:707Þ=50Þs þ 1Þ

(29)

(Fig. 11). This transfer function, Po ðsÞ; was chosen such that the magnitude of its frequency response is approximately equal to the mean of the experimental frequency response magnitudes for each frequency. Now that a nominal plant has been determined, the nonlinearities and uncertainty must be incorporated into the model. For this design, multiplicative uncertainty was chosen to capture the system nonlinearities and system delay. Multiplicative uncertainty may be written as shown in Eq. (5) with the condition stated in Eq. (6). The uncertainty weighting, incorporating the time delay, may be determined by the expression given in Eq. (7)

ARTICLE IN PRESS G.A. Ingram et al. / Control Engineering Practice 13 (2005) 1413–1423

(Doyle et al., 1992). The experimental multiplicative uncertainty for this system is shown in Fig. 12 and in equation form as

^ was found to be 0.41. This The maximum value of j resulted in the final weighting functions of W 1 ðsÞ ¼

^ 2 ðsÞ ¼ 0:45ðs=9 þ 1Þ : W s=130 þ 1

(30)

s=83 þ 1


2


0:92 s=14:5 þ 1 W 2 ðsÞ ¼ : s=119 þ 1

Now that the inverted MAF static map and MAF path of the engine have been identified, the H 1 MAF controller may be synthesized. The closed-loop MAF block diagram may be written in the form shown in Fig. 4 and with the proper selection of weighting functions, transformed to the form shown in Fig. 5.

(32)

4.2.1. Weighting function selection The control orientated goal is to design a tracking controller which maximizes the system responsiveness to a step change in reference. Therefore, Theorems 1 and 2 are implemented to determine the proper weighting functions while the time domain tracking deviation specification (d) is relaxed and set to a relatively large value. The time domain control effort specification (k) is dependent on the electronic throttle actuator and is 0.7 V. To implement Theorems 1 and 2, the reference dynamics, G R ðsÞ; are required. The reference dynamics for this application were chosen to capture the reference command dynamics in a slightly larger frequency range than the plant would respond, i.e. (31)

One of the system requirements of this controller design is the closed-loop system must have zero steadystate error, requiring the controller to have one free integrator. To incorporate a free integrator into the design, the nominal plant transfer function is augmented with a free integrator. Now Theorems 1 and 2 may be used to aid in the controller synthesis. 4.2.2. Robust mass air flow controller design The H 1 controller design follows the procedure, outlined in a previous section, that maximizes system responsiveness to a step change in reference. The ^ optimization is actually performed by maximizing j while relaxing the tracking deviation constraint (i.e. allowing d to be large) until one of the conditions set forth in Theorems 1 or 2 is violated. This forces the controller synthesis to maximize the actuator effort. In ^ is found this case, d was set to equal 100. The value of j as described above in an iterative process using an initial ^ as provided in Eq. (14). guess for j

(33)

The H 1 MAF controller was determined using the MATLABs 1 command hinflmi. After the design was complete, a balanced truncation method was employed to reduce the order of the controller. The reduced order controller, with the free integrator added, is KðsÞ ¼

1 : ðs=83 þ 1Þ2

0:0041

and

4.2. H 1 mass air flow controller design

G R ðsÞ ¼

1421

419:4s2 þ ð2:675e4Þs þ 1:105e6 : s3 þ 424:8s2 þ ð9:78e4Þs

(34)

The values of j and L are unknown through the design. Now that the controller is synthesized, the actual largest reference command step size (j ) is determined via simulation and found to be 0.88. 4.3. Analysis of the closed-loop system Before the experimental evaluation of the controller, the closed-loop stability of the entire system may be quantified and assessed. To this end, Theorem 3 is used to determine the largest sector bounds of a memoryless time varying nonlinearity representing the amount of modeling and nonlinear inversion errors (Fig. 2) that can be tolerated by the controller and plant. Once the sector bounds have been obtained, Corollary 1 provides conditions under which the actual input and output of ^ and yðtÞ ^ respectively) will be finite-gain the system (uðtÞ L2 stable. To apply Theorem 3 and Corollary 1, the plant will be described using multiplicative uncertainty. As mentioned previously, time delays may not be directly included in the nominal plant for H 1 design synthesis. However, Theorem 3 does not contain such a restriction. Therefore, to provide a plant model with less uncertainty, a time delay is added to the nominal plant description and a new weighting function associated with the multiplicative uncertainty of the system is determined using Eq. (7). The nominal plant considered for stability analysis is Pstab ðsÞ ¼ Po ðsÞ e 0:027s

(35)

and the associated multiplicative uncertainty weighting function is W stab ðsÞ ¼

0:39ðs=11:5 þ 1Þ : s=30 þ 1

(36)

1 MATLAB is a registered trademark of MathWorks, Inc. of Natick Massachusetts.

ARTICLE IN PRESS G.A. Ingram et al. / Control Engineering Practice 13 (2005) 1413–1423

1422

Theorem 3 is applied using the plant description given by Eqs. (5), (6), (35), and (36) as well as the controller given by Eq. (34). The result of a bisection routine implemented to determine the largest sector bounds, constrained by Eqs. (20) and (21), is fðs; tÞ 2 ½0:229; 4:35:

(37)

This sector represents the allowable modeling and nonlinear inversion errors within the system. Specifically, under the conditions provided by the circle criterion, the plant gain may increase or decrease by a factor of 4.35 or 0.229 respectively without the system becoming unstable. Furthermore, the conditions of Corollary 1 are satisfied and therefore the actual input/output relationship between the requested MAF and the actual MAF is finite-gain L2 stable. Requested + MAF

Desired MAF MAF Controller Engine Speed

-

Inverted Static Map

Fig. 13. Block diagram of mass air flow controller.

14 13 12 11

40 kph

10 0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

22 21 20 19 MAF (g/s)

Once the feedback controller design was finalized, the H 1 MAF controller was implemented in real time via dSpace and an engine dynamometer. A block diagram of the implementation is provided in Fig. 13. Step changes of MAF were accomplished in approximately 1.2, 1.9, 1.4, and 2.7 engine cycles for the 40, 76, 93, 119 kph operating conditions respectively (Fig. 14). At lower engine loads there exists some overshoot of the actual MAF. This can be explained by the nonlinear throttle behavior. At smaller throttle openings the system gain is large. Therefore an overshoot may be anticipated if the static map is not exact for all engine conditions. An advantage of the robust controller methodology employed for this design is the fact the system, in general, responds well over a large operating range despite the throttle nonlinearities.

Throttle Command

Actual MAF

0

4.4. Experimental validation of the mass air flow controller

5. Conclusions The methodology for robust SISO controller synthesis advanced in this manuscript can be applied to a class of nonlinear systems described by uncertain Hammerstein models. The methodology is applicable for maximizing the allowable reference step size or maximizing the tracking response given time domain constraints. Furthermore, once the design is complete, the maximum allowable modeling and nonlinear inversion errors can be quantified. A guarantee of L2 BIBO stability may be given provided the system input belongs to L2 and certain frequency domain conditions are satisfied. Finally, the controller design methodology has been successfully applied to MAF tracking control of a spark ignition engine.

76 kph

18 0

0.1

0.2

0.3

0.4

0.5

0.6

35 34 33 32 31 30 29

0.7

0.8

93 kph 0

0.1

0.2

0.3

0.4

0.5

0.6

41 40 39 38 37 36 35

0.7

0.8

119 kph 0

0.1

0.2

0.3

0.4 0.5 Time (seconds)

0.6

0.7

0.8

Fig. 14. System performance at various speed/load conditions, reference command (solid), actual MAF (dashed).

References Al-Duwaish, H., & Bettayeb, M. (1997). Control of nonlinear dynamical systems using genetic algorithms. International symposium on intelligent control (pp. 421–424). Beaven, R., Wright, M., & Seaward, D. (1996). Weighting function selection in the H 1 design process. Control Engineering Practice, 4(5), 625–633. Bendat, J., & Piersol, A. (1993). Engineering applications of correlation and spectral analysis (2nd ed.). New York: Wiley. Benninger, N., & Plapp, G. (1991). Requirements and performance of engine management systems under transient conditions. Technical Report 910083, SAE. Bidan, P., Boverie, S., & Chaumerliac, V. (1995). Nonlinear control of a spark-ignition engine. IEEE Transactions on Control Systems Technology, 3(1), 4–13. Billings, S., Chen, S., & Korenberg, M. (1989). Identification of MIMO non-linear systems using a forward-regression orthogonal estimator. International Journal of Control, 49(6), 2157–2189. Chun, Y., & Hori, Y. (1996). On general weighting functions for the application of H 1 optimization method. Proceedings of the fourth

ARTICLE IN PRESS G.A. Ingram et al. / Control Engineering Practice 13 (2005) 1413–1423 international workshop on advanced motion control, vol. 2 (pp. 571–576). Donha, D., Desanj, D., & Katebi, M. (1997). Automatic weight selection for H 1 control design. International conference on genetic algorithms in engineering systems: innovations and applications (pp. 50–55). Doyle, J., Francis, B., & Tannenbaum, A. (1992). Feedback control theory. New York: Macmillian. El-Farra, N., & Christofides, P. (2001). Integrating robustness, optimality and constraints in control of nonlinear processes. Chemical Engineering Science, 56(5), 1841–1868. Franchek, M. (1996). Selecting the performance weights for mu and Hinfinity synthesis methods for SISO regulating systems. Transactions of the ASME, 118, 126–131. Grimble, M., & Biss, D. (1988). Selection of optimal control weighting functions to achieve good H 1 robust designs. International conference on control (pp. 683–688). Hedrick, J. (1998). Nonlinear controller design for automated vehicle applications. International conference on control, vol. 1 (pp. 23–32). Hu, J., Bohn, C., & Wu, H. (1999). Practical H 1 weighting functions and their application to real-time control of a pilot plant. American control conference, vol. 2 (pp. 920–924). Hu, J., Bohn, C., & Wu, H. (2000). Systematic H 1 weighting function selection and its application to the real-time control of a vertical take-off aircraft. Control Engineering Practice, 8(3), 241–252.

1423

Hu, J., Unbehauen, H., & Bohn, C. (1996). A practical approach to selecting weighting functions for H 1 control and its application to a pilot plant. UKACC International conference on control, vol. 2 (pp. 998–1003). Ljung, L. (1999). System identification: theory for the user (2nd ed.). Englewood Cliffs, NJ: Prentice-Hall. Meghani, A., & Latchman, H. (1992). H 1 vs. classical methods in the design of feedback control systems. Proceedings of Southeastcon, vol. 1 (pp. 59–62). Postlethwaite, I., Tsai, M., & Gu, D. (1990). Weighting function selection in H 1 design. International federation of automatic control world congress (pp. 127–132). Sastry, S. (1999). Nonlinear systems analysis stability and control. Berlin: Springer. Stefanopoulou, A., Grizzle, J., & Freudenberg, J. (1994). Engine airfuel ratio and torque control using secondary throttles. Proceedings of the IEEE conference on decision and control, vol. 3 (pp. 2748–2753). Vidyasagar, M. (1978). Nonlinear systems analysis. Englewood Cliffs, NJ: Prentice-Hall Inc. Yang, C., Ju, H., & Liu, S. (1994). Experimental design of H 1 weighting functions for flight control systems. American control conference, vol. 3 (pp. 2516–2520). Zhuang, M., & Atherton, D. (1996). CAD of nonlinear controllers for nonlinear systems. International conference on control (pp. 545–550).