Control Engineering Practice

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Control Engineering Practice 46 (2016) 105–114Contents lists available at ScienceDirectControl Engineering Practicejournal homepage: www.elsevier.com/locate/conengpracNonlinear internal model controller design for wastegate control of aturbocharged gasoline engine Zeng Qiu a,n, Jing Sun b, Mrdjan Jankovic c, Mario Santillo caDepartment of Electrical Engineering and Computer Science, University of Michigan, Ann Arbor, MI, USADepartment of Navel Architecture & Marine Engineering, University of Michigan, Ann Arbor, MI, USAcFord Motor Company, Dearborn, MI, USAbart ic l e i nf oa b s t r a c tArticle history:Received 17 February 2015Received in revised form1 September 2015Accepted 19 October 2015Internal Model Control (IMC) has a great appeal for automotive powertrain control in reducing thecontrol design and calibration effort. Motivated by its success in several automotive applications, thiswork investigates the design of nonlinear IMC for wastegate control of a turbocharged gasoline engine.The IMC design for linear time-invariant (LTI) systems is extended to nonlinear systems. To leverage theavailable tools for LTI IMC design, the quasi-linear parameter-varying (quasi-LPV) models are explored.IMC design through transfer function inverse of the quasi-LPV model is ruled out due to parametervariability. A new approach for nonlinear inversion, referred to as the structured quasi-LPV model inverse, is developed and validated. A fourth-order nonlinear model which sufficiently describes the dynamic behavior of the turbocharged engine is used as the design model in the IMC structure. The controller based on structured quasi-LPV model inverse is designed to achieve boost-pressure tracking. Finally, simulations on a validated high-fidelity model are carried out to show the feasibility of the proposed IMC. Its closed-loop performances are compared with a well-tuned PI controller with extensivefeedforward and anti-windup built in. Robustness of the nonlinear IMC design is analyzed using simulations.& 2015 Elsevier Ltd. All rights reserved.Keywords:Internal model controlWastegate controlTurbocharged gasoline engineQuasi-linear parameter varying1. IntroductionInternal Model Control (IMC), whose diagram is shown in Fig. 1,is a well-established control design methodology with an intuitivecontrol structure (Morari & Zafiriou, 1989). It incorporates a system model as an explicit element in the controller so that thecontrol actions are determined based on the difference betweenthe model output and the plant output. It has several desiredfeatures and closed-loop properties as established in Morari andZafiriou (1989) and Garcia and Morari (1982), such as dual stabilitycriterion, zero offset, and perfect control. The design, analysis, andimplementation of IMC for linear systems have been well developed. Rivera, Morari, and Skogestad (1986) showed that processindustrial IMCs for many SISO models can lead to PID controllers,occasionally augmented with a first-order lag. They also demonstrated the superiority of using IMC for PID tuning in terms ofclosed-loop performance and robustness. This work is supported by Ford Motor Company.Corresponding author.E-mail addresses: connieqz@umich.edu (Z. Qiu), jingsun@umich.edu (J. Sun),mjankov1@ford.com (M. Jankovic), msantil3@ford.com (M. 2015.10.0120967-0661/& 2015 Elsevier Ltd. All rights reserved.The efficacy of IMC for nonlinear systems, however, has beeninvestigated with limited comprehensive results. Economou,Morari, and Palsson (1986) presented an important result ofnonlinear IMC, proving that the dual stability criterion, zero offset,and perfect control properties of LTI IMC would carry over tononlinear cases. The IMC was implemented by finding a nonlineardynamic inverse, which remained to be the key challenge in extending the IMC design to nonlinear systems. While the invertibility condition, inverse structure, and derivation for nonlineardynamic system inverse were studied (Hirschorn, 1979), the derivation of the nonlinear inverse involved higher-order derivativesand caused problems when noises and disturbances were presentin the system. In Economou et al. (1986), the nonlinear inverse wasderived by exploiting the Hirschorn nonlinear inverse structureand solving it numerically using the contraction principle methodor Newton's method. Stability of the IMC structure was discussedunder the ideal circumstance that the model was the same as theplant. Henson and Seborg (1991) also exploited the result ofHirschorn nonlinear inverse for nonlinear IMC design. Several assumptions were made to calculate the higher-order derivatives.Feedforward/feedback linearization approach was adopted byCalvet and Arkun (1988) to derive the model for the nonlinear

106Z. Qiu et al. / Control Engineering Practice 46 (2016) 105–114Fig. 1. Internal model control structure.plant in IMC. Their approach accounted for the disturbances andinput constraints.Nonlinear IMC was also investigated in the adaptive controlframework. Hunt and Sbarbaro (1991) used artificial neural networks for adaptive control of nonlinear IMC. Feasibility of identifying the nonlinear model and its inverse by a neural network wasexplored and demonstrated. Boukezzoula, Galichet, and Foulloy(2000) and Xie and Rad (2000) used fuzzy logic to estimate themodel dynamics. The inverse was derived from this fuzzy model.The black-box identifications of neural network and fuzzy logicmade it difficult to incorporate physical knowledge about the plantin the IMC design. In adaptive IMC scheme, using linear models torepresent the dynamics of the nonlinear plant though adaptationhas also been exploited (Brown, Lightbody, & Irwin, 1997; Datta,1998; Shafiq, 2005).Another possible avenue to exploit the linear IMC design toolsfor nonlinear systems would be through the linear parametervarying (LPV) model. Mohammapour, Sun, Karnik, and Jankovic(2013) applied IMC on a quasi-LPV model with two approaches. Inthe first approach, the IMC controller parameters were scheduledbased on the LPV model parameters which were assumed to beknown in real time and not vary rapidly. In the second approach,the design problem was formulated in the H framework as a setof linear matrix inequalities (LMI). Solving the associated LMIproblem, however, was computationally intensive. Toivonen,Sandström, and Nyström (2003) derived the LPV model based onvelocity-based linearization, then developed the IMC controllerbased on linear IMC theory. It was much less computationallydemanding, but it was only applicable when there were a smallnumber of scheduling parameters.This paper explores nonlinear IMC for turbocharged gasolineengines driven by the need for developing robust and easy-tocalibrate powertrain control solutions and motivated by severalsuccessful industrial applications. IMC was first applied to turbocharged diesel engines for automotive applications. Alfieri, Amstutz, and Guzzella (2009) applied IMC based on the classicalSmith predictor structure to air–fuel ratio control in turbochargeddiesel engines with exhaust gas recirculation. Schwarzmann,Nitsche, and Lunze (2006) treated boost-pressure control of aturbocharged diesel engine with a variable nozzle turbine withIMC. Their IMC utilized a flatness-based approach to design theinverse Q, in which flatness means that the system inputs can beexplicitly expressed in terms of internal system dynamics. In afollow up work, the same group also dealt with a two-stagedturbocharged diesel engine using IMC (Schwarzmann, Nitsche,Lunze, & Schanz, 2006). The inverse Q was designed based ongeometric nonlinear control design method. As turbocharged gasoline engines are becoming more popular, advanced control designs including IMC have been applied to turbocharged gasolineengines for improved performance. Thomasson, Eriksson, Leufven,and Andersson (2009) utilized IMC for PID tuning of wastegatecontrol in turbocharged gasoline engines. Karnik and Jankovic(2012) later applied IMC directly to wastegate control for a turbocharged gasoline engine, motivated by the successful applications on turbocharged diesel engines. They used a first-ordermodel which was simplified from a fourth-order nonlinear modelusing singular perturbation. While the simplicity of the first-ordermodel-based design was an advantage for implementation, itsperformance was limited by the linear approximation, as it is defined for a particular operating point.This work investigates the feasibility, performance, advantages,and limitations of a nonlinear IMC for automotive powertraincontrol design, using the fixed geometry turbocharged gasolineengine as a case study. While the nonlinear dynamics of the system can be sufficiently described by a fourth-order model, inverting the nonlinear model for the IMC design represents themajor challenge. To facilitate the IMC design, a quasi-LPV model(Rugh & Shamma, 2000) for the nonlinear model is developed.More importantly, the special quasi-LPV model structure is explored, and a structured quasi-LPV model is proposed, which leadsto a feasible nonlinear inverse, referred to as the structured quasiLPV inverse. The IMC based on the structured quasi-LPV inverse isdeveloped, and its performance is analyzed. Simulation results,using a validated “virtual” plant model, are presented to demonstrate the effectiveness of the proposed design. This work is applicable to IMC with SISO nonlinear models of higher-order and isnot limited by the number of scheduling parameters. The proposed IMC was originally presented as a conference paper (Qiu,Sun, Jankovic, & Santillo, 2014), whereas this paper represents anexpanded version. More specifically, the design procedure is discussed in more detail from the stability point of view and robustness analysis is included.The paper is organized as follows: Section 2 states the problem,presents two main tools used: IMC and LPV. Section 3 presents thenonlinear model for the turbocharged gasoline engine and exploitsquasi-LPV approach to derive its inverse. Section 4 analyzes theIMC implementation results. Section 5 summarizes the paper.2. Control problem and preliminariesGasoline engines have been aggressively downsized in an effortto reduce fuel consumption and CO2 emissions. However, thetorque provided by the engine is proportional to the air deliveredto the cylinders. To meet the consumer demands for performanceon the downsized engines, i.e., to maintain the engine outputtorque, turbochargers are widely adopted. They compress the intake air to increase the density of the engine airflow, thereby increasing the torque. The schematic of a turbocharged gasolineengine is shown in Fig. 2. The wastegate is the main actuator tocontrol boost pressure. It affects the engine operation by changingthe rotational speed of the turbine/compressor. The air is compressed by the compressor, and passes through an intercooler anda throttle before entering the engine intake port. The engine exhaust port is connected to the turbine, which is mechanicallyconnected to the compressor. An electric wastegate actuator controls the opening of the turbine bypass path in this application(Karnik & Jankovic, 2012), affecting the compressor speed andtherefore the boost pressure.The turbocharged gasoline engine is expected to produce thedesired engine torque, with higher fuel efficiency, power density,and lower emission (Guzzella & Onder, 2010). To achieve such goal,the desired engine torque is calculated from the driver pedal position. The desired engine torque is then mapped into desired intake manifold pressure and boost pressure considering the fueleconomy and emission. These two pressures are then trackedthrough throttle and wastegate. This two input two output controlproblem can often be tackled with a decentralized controller:

Z. Qiu et al. / Control Engineering Practice 46 (2016) 105–114107IMC controller Inverse Plant ModelFig. 3. Equivalent internal model control structure.Fig. 2. System schematic of a turbocharged gasoline engine (Buckland, 2009).using the throttle to track the intake manifold pressure and usingthe wastegate to track the boost pressure (Karnik, Buckland, &Freudenberg, 2005). In this paper, we will focus on using thewastegate to track the desired boost pressure, and the throttle isconsidered as an exogenous input. Boost pressure set-pointtracking is a critical enabling technology for achieving improvedfuel efficiency, power density, and emission reduction (Guzzella &Onder, 2010). However, operating conditions vary widely in automotive applications. This variation could result in inadequateboost at low speeds and loads, and over-boost situation at highspeeds and loads. The wide range in operating conditions addscomplexity to control design and calibration. IMC with a nonlinearmodel is adopted for its advantages in reducing the design andcalibration effort.A nonlinear IMC is developed in this paper for the controlproblem of boost-pressure tracking with the electric wastegate asthe actuator. Measurements for boost pressure Pb, temperatures Tb ,Ti , Te , throttle opening uth, and engine speed Nen are assumed to beavailable for feedback or feedforward control. The invertibility ofthe nonlinear model is assumed. To proceed with the designprocedure of nonlinear IMC with structured quasi-LPV inverse, thepreliminaries of IMC and quasi-LPV model are presented as below.the model inverse is the main challenge and focus in this work. Forboth linear and nonlinear systems, once the model and its validinverse are derived, IMC control design follows immediately. IMChave three main properties as established for LTI models (Garcia &Morari, 1982) and later extended to nonlinear cases (Economouet al., 1986):(1) Dual stability property: When the model M is exact, stabilityof both inverse Q and plant P is sufficient for overall systemstability.(2) Perfect control property: By changing the position of thesignal addition block in the IMC in Fig. 1, one can get an equivalentform of IMC as in Fig. 3, which has the same structure as theclassical control. With this structure, the IMC controller can bepresented by Q (1 MQ ) 1, therefore, the sensitivity function of theIMC control system is (1 MQ )(1 MQ PQ ) 1. When MQ ¼1, i.e.,Q M 1, the sensitivity function is 0, which means “perfect control” is achieved.(3) Zero offset property: A controller which satisfiesQ (0) M (0) 1 adds integral action to the controller, and yields zerooffset for a constant reference command r.2.2. Quasi linear parameter varying modelQuasi-LPV models are LPV models for nonlinear systems wherenonlinearities are hidden through state-dependent parameters, sothat a nonlinear model can be represented by an LPV model andtreated by LPV design techniques (Rugh & Shamma, 2000).In general, a nonlinear model in the form ofẋ f (x, u)2.1. Internal model control(2)can be expressed as an LPV model in the form ofThe schematic of a system with IMC is shown in Fig. 1, where P,M, and Q denote the plant, model, and inverse respectively. TheIMC controller contains two elements: the inverse Q and themodel M. Q takes the reference command and the difference between the outputs of the plant and the model as its inputs. If themodel is exact, i.e., M¼P, the IMC becomes a feedforward controller. Q is generally designed such that the norm of the differencee between the reference command r and the plant output y isminimized. Thus Q is chosen by solvingmin e 2 min (1 QM )r 2 ,QQ(1)subject to the constraint that Q is stable and casual. An absoluteminimum could be reached at Q M 1 if M is invertible. Howeverthis is not feasible for a non-minimum phase (NMP) model, whichis not stably invertible. An approximate inverse Q can be solvedalgebraically depending on the input r. Therefore, once the linearmodel is derived, the inverse of the model can be derived bysolving the minimization problem (1). For a nonlinear model,however these tools are inapplicable, and effective approaches forinverting nonlinear models are very limited. Therefore, derivingẋ A(p)x B(p)u(3)if the model (2) is affine in u and the time varying parametervector p in (3) is allowed to be state-dependent to disguise thenonlinearities (Rugh & Shamma, 2000). For example, the nonlinearmodelx1̇ x12 x1x2 ,x2̇ sin x1 ucan be expressed in quasi-LPV form as x1 x1 x 0 u,ẋ A(p)x Bu sin(x1) 0 x1 1 with p x1, sin(x1) x1 T, or x x 0 12 x 0 u,ẋ A(p)x Bu sin(x1) 0 1 x1 with p x1 x2, sin(x1) x1 T.

108Z. Qiu et al. / Control Engineering Practice 46 (2016) 105–114In the next section, a nonlinear model for the turbochargedgasoline engine is presented. The quasi-LPV approach is exploitedfor representing it in a linear structure to aid deriving an inversefor the nonlinear model in IMC implementation.compressor, throttle, intake, engine, exhaust, turbine, and wastegate respectively. Modeling of W (mass flow rate) and H (power)are described in detail in Karnik and Jankovic (2012) and Buckland(2009), and the resulting functional expressions are summarizedas follows: P Wc fc b , Nt , Pa 3. Quasi-LPV model and its inversion3.1. A nonlinear model for IMC designWth Control-oriented models serve the IMC design and implementation in two different ways: first, the IMC incorporates asystem model directly in its implementation as shown in Fig. 1;second, the standard IMC design procedure takes an inverse of theprocess model and augments it with a proper filter to avoid noncausal implementation to form the inverse Q.The nonlinear model for the boost-pressure dynamics of aturbocharged engine presented in this paper is based on the workof Buckland (2009). The nonlinear model has the following statesand one input:x [Pb, Pi, Pe, Nt ]T ,RTbdPbRT b (Wc Wth),dtVbdPiRT i (Wth Wen),dtVi dPeRT 1 A/F e Wen Wt Ww ,dtVe A/FdNt1 (Ht Hc ),dtItNt(4)where R is the ideal gas constant, A /F is the air to fuel ratio, T, V, W,I and H are temperature, volume, mass flow rate, inertia, andpower respectively. The subscript indicates the physical location ofthe variable as in Fig. 2, and b, c, th, i, en, e, t, and w are boost,Table 1Nomenclature for modeling of turbocharged gasoline engine.Variables P γPbϕ i , Pb V NWen Piηen en en ,RTi 2Ww sat(0, u w , 1)RTe P γPeϕ x , Pe P N PWt ft e , t e ,Te Te PxHt cp, eTeWtηt ψt ,u uw ,where Pb is the boost pressure, Pi is the intake pressure, Pe is theexhaust pressure, Nt is the turbocharger speed, and the input uw isthe wastegate, which is the fraction of the opening and takes values in the range of [0, 1]. The dynamics of the pressures Pb, Pi, andPe are derived using mass conservation along with isothermalmanifold assumptions, while the dynamics of the turbochargerspeed Nt are derived by a power balance between the turbine andthe compressor as described in Karnik and Jankovic (2012) andBuckland (2009). The equations are summarized as follows:Hc cp, aTaWc1ψ,ηc c P (γe 1) / γeψt 1 x , Pe P (γa 1) / γa 1,ψc b Pa (5)where Pa is the ambient pressure, Px is the turbine exit pressure,ϕ(·) is a function of pressure ratio across the component, ψ is amass flow parameter, γ is the specific heat ratio for air, cp,(·) is thespecific heat at constant pressure, η is the isentropic efficiency, uthis the throttle opening, Nen is the engine speed, and sat(0, u, 1)limits u to be in the range [0, 1]. The temperatures Tb, Ti, and Te areassumed to be measured. Typically the temperature sensors have aslow response time and delay, and the measurements are leadfiltered to improve the response time. Therefore, the measurementinaccuracy is not considered in this work. A /F is the stoichiometricratio of gasoline. uth and Nen are considered as exogenous inputs inthis work and they are measurable. All the variables and subscriptsfor the nonlinear model of turbocharged gasoline engine aresummarized in Table 1.The nonlinear model is evaluated by comparing its responseswith those of the virtual “plant”, which is a high fidelity Fordproprietary model that has been validated extensively. It includesthe intercooler, throttle, engine, wastegate, turbine, and compressor (Buckland, 2009). Responses to a step change in the wastegate setting from 0.25 to 0.75 at t 5 s for the nonlinear modeland the virtual “plant” are shown in Fig. 4, confirming that thecontrol-oriented nonlinear model and the “plant” have very similar dynamic responses.SubscriptsA/Fcp,(·)Air to fuel ratioSpecific heat at constant pressureabAmbientBoostf· (·)INPHRTuVWϕ(·)Compressor/turbine mapcCompressorMoment of inertiaRotational speedPressurePowerIdeal gas constantTemperatureFraction/degree of openingVolumeMass flow rateFunction of pressure ratioacross the componentIsentropic efficiencyMass flow parameterRatio of specific stegateTurbine exitηψγsat(0, uth , 1)3.2. Quasi-LPV turbocharged gasoline engine modelIMC control can be achieved by designing the inverse Q in Fig. 1as the inverse of the model M. For linear models, their inverses canbe achieved by inverting their transfer functions and appending aproper filter to assure causality. To extend this approach to nonlinear models, the quasi-linear parameter varying model approachis explored to represent the nonlinear model in a linear structure.Note that there are an infinite number of quasi-LPV models inthe form of (3) that can match (2), depending on the choice of thevarying parameter p. For the turbocharged gasoline engine system,the physical couplings of state variables are considered and thefollowing structure that leads to the most sparse A, B matrices ischosen:

200Intake Pres [kPa]Boost Pres [kPa]Z. Qiu et al. / Control Engineering Practice 46 (2016) 105–11415010005Time [sec]10200109NonlinearPlant15010005Time [sec]105Time [sec]10Turbo Speed [rad/sec]Exhaust Pres [kPa]420015010005Time [sec]101.5x 1010.50Fig. 4. Comparison of responses of the nonlinear model and the “plant” for a step change in wastegate actuation. a 11 a21A 0 000a220a32 a330a43a14 0 ,0 a44 x [Pb, Pi, Pe, Nt ]T , 0 0B , b3 0 u uw ,y x1.(6)The non-zero elements in (6) are defined as follows:RTb Wth Vb Pba11 a14RTbVb P sat(0, uth , 1)γϕ i , Pb Σ1: x1̇ a11x1 a14x4 x1 Σ1(x4 ), RT WRTb Pb b c f , Nt ,Vb NtVbNt c Pa Σ 2: x2̇ a22x2 a21x1 x2 Σ 2(x1),Σ 3: x3̇ a33x3 a32x2 b3u x3 Σ 3(x2 , u),RT WRT sat(0, uth , 1)A max Pi ϕ γ ,a21 i th iVi PbVi Pb RTbΣ4: x4̇ a44x4 a43x3 x4 Σ4(x3).Further expressingRTi WenRT V N i en en ,Vi PiVi 2RTia22 1 A/F RTe Wen1 A/F TeηenVenNen ,A/F Ve PiA/F2VeTiR Te Pe Nt RT W ,a33 e t f ,Ve PeVe t PxTe b3 HcItNt2 y x1 Σ1(Σ4(Σ 3(Σ 2(x1), u))) Σ1(Σ4(Σ 31(Σ 2(x1)) Σ 32(u))),RTe 1 1(Σ4 (Σ1 1(y)) Σ 31(Σ 2(y))),u Σ 32Ve P γPeϕ x . Pe (9)whose block diagram representation is shown in Fig. 5.Expressing the input u in terms of the output y based on (9),11cp, aTaWc ψc ,ηcItNt2RTeWw Ve sat(0, u w , 1)Σ3 to bewe can show that the input–output relation of the quasi-LPVmodel can be expressed as a composition of these sub-models:HtW1 cp, eTe t ηt ψt ,PeItNtPeItNta44 (8)Σ 3(x2 , u) Σ 31(x2) Σ 32(u),a32 a43 minimize the approximation error.(1) Quasi-LPV model inverse structure: Exploiting the sparsity ofthe A, B matrices of model (6), the quasi-LPV model is expressed asan integration of several first-order sub-models. With this veryspecial structure of the nonlinear model, the inverse can be pursued by deriving the inverse of multiple first order nonlinearmodels, which will involve limited approximation.Given the sparse matrices A, B in the form of (6), the followingfirst-order sub-models Σ1, Σ 2, Σ3, Σ4 are defined as(7)3.3. Structured quasi-LPV inverseGiven that the parameters defined by (7) are varying fast during transients, treating the parameters as frozen and deriving thetransfer function of (6) will not be effective for deriving the inverse. Indeed, by numerical simulations it is confirmed that thetransfer function inverse does not represent the nonlinear modelinverse. In this section, the special form of the quasi-LPV structureof (6) is explored to derive its inverse model in an effort to(10)which can be viewed as an inverse model of (9). Fig. 6 shows theblock diagram representation of the inverse of the quasi-LPVmodel through the integration of several inverse models of first 1order blocks as expressed in (10), in which Σ1 1, Σ4 1, and Σ32areapproximate inverses of Σ1, Σ4, and Σ32. The derivations of the 1are explained in the following section. x1– x4 inΣ1 1, Σ4 1, and Σ32Fig. 6 are approximations of x1–x4 in Fig. 5.(2) First-order quasi-LPV inverse: We now derive the inverse ofthe first-order quasi-LPV model and define its property in order toderive the representation for the inverse model (10).For a first-order quasi-LPV model:Σi: ẋ ax bu,define(11)

110Z. Qiu et al. / Control Engineering Practice 46 (2016) 105–1141x z,τb 1ḃa 1 z ̇ z 2 x . b τ b bτu (18)Remark 1. Treat x as the input, ū as the output, and a, b, and τ asthe parameters, the BIBO stability of the first order system (13) canbe easily established given that the system has a single frozenFig. 5. Interconnection of the first-order quasi-LPV sub-models for the fourth-orderturbocharged gasoline engine LPV model.1time pole at τ , and1 1 1, (τb τ bτ ḃb2a b ) are bounded.Remark 2. Note that since u u for small τ, one can treat (13) asan approximate inverse model of (11). Moreover, u u 6 (τ ),namely, the inverse model error can be made arbitrarily small witha properly chosen τ.Remark 3. Lemma 1 assumes a continuous-time implementationof the inverse of (11). When (13) is discretized for real engineimplementation, its BIBO stability remains due to Remark 1. If thedelay caused by discretization is small, its performance will not besubstantially affected.Fig. 6. Interconnection of first-order quasi-LPV sub-models for inverse of the LPVmodel shown in Fig. 5. 1 u ,u τs 1 (12)where { τs 1 1 } denotes the first-order filter with a transfer function1.τs 1Then, the following lemma gives the representation of ū.Lemma 1. Let a, b be time varying parameters with a, b 3 , b 0,and b *1. Then, for any u, ū given by (12) can be expressed in termsof the state x as1x z,τb 1ḃa 1 z ̇ z 2 x ,b τ b bτu (13)where x is given by (11).Proof. Note that 1 1 x ̇ax u u . τs 1 τs 1 bb (14)Sinced x ẋbẋ 2, dt bbb(15)we have, from (14), that ax 1 x1 d x bẋ 2 τs 1 u b τs 1 b τs 1 dt b τb 1 τs ḃa 1 x . 2b 1 b bτ (16)Note that the time invariant operator {τs 1} can now becanceled with { τs 1 1 } since there is no time varying signal in between. Let ḃa 1 1 2 x ,z b b τs 1 bτNow the approximate inverse model Σ i 1 is given by (13). Forsimplicity, one can drop the ḃ/b2 term if the parameter variation issubstantially slower than the system dynamics. However, this isnot the case in this application. ḃ/b2 as in (7) includes the states.Therefore, ḃ/b2 is not slower than the system dynamics. The following simulation also verified that ḃ/b2 should not be omitted: Tovalidate the first-order sub-model inverse, the two systems Σi andΣ i 1 are connected in cascade as shown in Fig. 7. According toRemark 2, the output v in Fig. 7 should be close to the input v̄ . Anumerical analysis of the inverse performance is shown in Fig. 8. Itis obvious that the inverse incorporating the ḃ/b2 term withsmaller time constant τ has better accuracy. Therefore, the inverseincorporating ḃ/b2 is adopted in the subsequent derivation. Thetime constant τ is the tuning parameter for the IMC design. Fig. 8indicates that smaller time constants lead to a better inverse, asexpected.(3) Structured quasi-LPV inverse: Representing each first-ordermodel inverse in (10) with (13), an inverse model for the nonlinearmodel is derived, which will be referred to as the structured quasiLPV inverse.To incorporate the quasi-LPV inverse model in the IMC structure, two implementable configurations are possible as shown inFigs. 9a and b. Firstly, one can use the states in the nonlinearmodel Σ to schedule the parameters in the inverse model Σ 1 (asin Fig. 9a). Since the states used for parameter scheduling areexternal to Σ 1, Fig. 9a is referred to as the externally scheduledquasi-LPV inverse. Secondly, since all the states in the originalquasi-LPV model are explicit in its inverse structure, one can derive parameters used in A and B (defined in (6)) for Σ 1 based onthe internal states in Σ 1 (as in Fig. 9b), which is referred to as theinternally scheduled quasi-LPV inverse.While the externally scheduled quasi-LPV inverse looks appealing at first, its utility is ruled out after more in-depth analysisand simulation. The dashed line in Fig. 9a which represents thegain scheduling signal forms a feedback loop, which causes instability. In this work, the proposed IMC controller uses the internally scheduled quasi-LPV inverse model for its implementation. It should be noted that the internally scheduled quasi-LPVinverse is not possible for inverse LPV model derived by general(17)then the dynamics from output x to ū can be represented by a firstorder LPV modelFig. 7. Structure for validation of first-order inverse.

Z. Qiu et al. / Control Engineering Practice 46 (2016) 105–114111Fig. 10. Parameter scheduling relationship in the internally scheduled quasi-LPVinverse (blue solid lines indicate that the states are actually used for scheduling;red dotted lines represent the use of the steady state value generated from steadystate mapping in scheduling). (For interpretation of the references to color in thisfig

The IMC was implemented by finding a nonlinear dynamic inverse, which remained to be the key challenge in ex- . works for adaptive control of nonlinear IMC. Feasibility of identi-fying the nonlinear model and its inverse by a neural network was . (2013) applied IMC on a quasi-LPV model with two approaches. In

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