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System Parameters’ Identification and Optimal Tracking Control for Nonlinear Systems
Proceedings, 2nd IFAC Conference on Proceedings, 2nd IFAC Conference on Modelling, Identification and Controlon of Nonlinear Systems Proceedings, 2nd IFAC Conference Modelling, Identification and Control of Nonlinear Systems Available online at www.sciencedirect.com Proceedings, 2nd IFAC Conference on Guadalajara, Mexico, June 20-22, 2018 Modelling, Identification and Control of Nonlinear Systems Guadalajara, Mexico, June 20-22, 2018 Modelling, Identification and Control of Nonlinear Systems Guadalajara, Mexico, June 20-22, 2018 Guadalajara, Mexico, June 20-22, 2018
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IFAC PapersOnLine 51-13 (2018) 431–436
System System System Optimal System Optimal Optimal Optimal
Parameters’ Identification and Parameters’ Identification and Parameters’ Identification and Tracking Control for Nonlinear Parameters’ Identification and Tracking Control for Nonlinear Tracking Control for Nonlinear Systems Tracking Control for Nonlinear Systems Systems Systems David Cortes Vega ∗∗ , Serafin Ramos Paz, ∗∗
David Cortes Vega ∗∗∗ , Serafin Ramos Paz, ∗∗ ∗ Fernando Ornelas-Tellez J. Rico-Melgoza David Cortes Vega ∗∗ and , Serafin Ramos Paz, ∗ ∗ Fernando Ornelas-Tellez J. Jesus Jesus Rico-Melgoza ∗ and ∗ ∗ David Cortes Vega , Serafin Ramos Paz, Fernando Ornelas-Tellez ∗ and J. Jesus Rico-Melgoza ∗∗ Fernando Ornelas-Tellez and J. Jesus Rico-Melgoza ∗ ∗ The authors are with the School of Electrical Engineering, ∗ The authors are with the School of Electrical Engineering, ∗ The authors are with the School of de Electrical Engineering, Universidad Michoacana de San Nicolas Hidalgo, Morelia, Universidad Michoacana de the SanSchool Nicolas Hidalgo,Engineering, Morelia, 58030, 58030, ∗ The authors are with of de Electrical Mexico (e-mail: [email protected], Universidad Michoacana de San Nicolas de Hidalgo, Morelia, 58030, Mexico (e-mail: [email protected], Universidad Michoacana de San Nicolas de Hidalgo, Morelia, 58030, [email protected], [email protected]) Mexico (e-mail: [email protected], [email protected], [email protected], [email protected], [email protected]) Mexico (e-mail: [email protected], [email protected], [email protected], [email protected]) [email protected], [email protected], [email protected]) Abstract: This This paper paper proposes proposes aa parameters’ parameters’ identification identification methodology methodology via via the the continuouscontinuousAbstract: time least-squares algorithm for those nonlinear systems which are linear with respect to their Abstract: This paper proposes a parameters’ identification methodology via the continuoustime least-squares algorithm for nonlinear systems which are linear with respect to Abstract: This paper proposes a parameters’ identification methodology via the continuoustime least-squares algorithm for those those nonlinear systems which are linear with respect toalltheir their parameters. The parameter identification method can be used for easily determining the parameters. The parameter identification method can bewhich used are for linear easilywith determining the time least-squares algorithm foronly those nonlinear systems respect toall their parameters in real plants from input-output data measurements, such that a posteriori, parameters. The parameter identification method can be used for easily determining all theaa parameters in real plants from only input-output data measurements, such that a posteriori, parameters. identification methoddata be used for easily determining all thea model-basedinThe control strategy can beinput-output synthesized. Itcan is measurements, worth mentioning that when control parameters realparameter plants from onlybe suchthat thatwhen a posteriori, model-based control strategy can synthesized. It is worth mentioning aaa control parameters in real plants from only input-output data measurements, such that a posteriori, a model-based control strategy can be synthesized. It is worth mentioning that when control engineer wants wants to to design aa modern modern and and sophisticated sophisticated controller, controller, usually usually the the system system model model is is used engineer model-based control strategy can on be its synthesized. Itcontroller, is worth in mentioning that when a control engineer wants to design design adepends modern and sophisticated usually the system model is used used for such purposes, which parameters, however, general the system parameters for such purposes, which adepends on parameters, controller, however, inusually generalthe the system parameters engineer wants design modern andits sophisticated system model is used are such not easy easy to to determine. Also, this this paper uses the the parameters’ parameters’ identification methodology for for purposes, which depends on its parameters, however, in general the system parameters are not to determine. Also, paper uses identification methodology for for such purposes, which depends on its parameters, however, in general the system parameters the design of an optimal tracking controller for state-dependent coefficient factorized (SDCF) are not easy to determine. Also, this paper uses the parameters’ identification methodology for the design of optimal tracking controller for state-dependent coefficient factorized (SDCF) are not easy Also, this paper uses parameters’ identification methodology for the design oftoan andetermine. optimal tracking controller for the state-dependent coefficient factorized (SDCF) nonlinear systems. Both, the the parameters’ identification scheme and and the optimal optimal control strategy strategy nonlinear systems. Both, parameters’ identification scheme the control the designsystems. of an optimal tracking controller for state-dependent coefficient factorized (SDCF) nonlinear Both, the parameters’ identification scheme andSynchronous the optimal control(PMSM), strategy are applied via simulations for the control of a Permanent Magnet Motor are appliedsystems. via simulations for the control identification of a Permanent Magnet Synchronous Motor (PMSM), nonlinear Both,machine. the scheme andSynchronous the optimal Motor control(PMSM), strategy a three-phase three-phase nonlinear are applied vianonlinear simulations forparameters’ the control of a Permanent Magnet a machine. are applied via simulations for the control of a Permanent Magnet Synchronous Motor (PMSM), a three-phase nonlinear machine. 2018, IFAC (International Federation of Automatic Control) Hosting by Elsevier Ltd. All rights reserved. a©three-phase nonlinear machine. Keywords: Keywords: Parameter Parameter identification, identification, optimal optimal nonlinear nonlinear control, control, PMSM. PMSM. Keywords: Parameter identification, optimal nonlinear control, PMSM. Keywords: Parameter identification, optimal nonlinear control, PMSM. 1. INTRODUCTION INTRODUCTION control signals signals that that will will cause cause aa process process to to satisfy satisfy the the physphys1. control ical constraints and at the same time minimize or maxi1. INTRODUCTION control signals that will cause a process to satisfy the physical constraints and at the same time minimize or maxi1. INTRODUCTION control signals that cause a process to satisfy physmizeconstraints performance index (Kirk (2004)). Linearthe optimal andwill at the (Kirk same time minimize oroptimal maxiModern control control techniques techniques are are based based on on aa good good underunder- ical mize aa performance index (2004)). Linear ical constraints and at the same time minimize or maxiModern control is special sort sort of optimal control, where where the plant a is performance index (Kirk (2004)). Linearthe optimal Modern techniques based on goodphysical under- mize standing control of the the plant to be be are controlled, by ausing control aaa special of control, plant mize a is performance index (Kirk (2004)). Linearthe optimal standing of to controlled, by control special sort ofasoptimal optimal control, where plant is assumed to be linear well as the controller, which is Modern control techniques based on goodphysical understanding ofpossible the plant plant to bea are controlled, by ausing using physical laws it is to get mathematical model of the is assumed to be linear asoptimal well as the controller, which is control is by a special sort ofas where the plant laws it isofpossible to to getbea controlled, mathematical modelphysical of the is obtained with quadratic indices. assumed toworking be linear well as control, theperformance controller, which is standing the plant byare using system, although many system parameters unknown or laws it is possible to get a mathematical model of the obtained by working with quadratic performance indices. assumed be achieves linear asthe well as the controller, which is system, although many system parameters are unknown or is The method that linear optimal control design bytothat working with quadratic performance indices. laws it although is possible to get a mathematical model of why the even difficult difficult to be bemany computed, that is the the main main reason system, system parameters are unknown or obtained The method achieves the linear optimal control design obtained by working with quadratic performance indices. even to computed, that is reason why is named named thethat linear quadratic regulator (LQR) (Andermethod achieves the linear optimal control(Anderdesign system, although system parameters are in unknown or The even difficult to bemany computed, is the main reason why system identification takes an that important role combinais the linear quadratic regulator (LQR) method achieves the linear optimal control(Anderdesign system identification takes an important role combinais named thethat linear quadratic regulator (LQR) son (1990)). A solution to the optimal nonlinear control even difficult tomodern be computed, is the main reason why The system identification takes an that important role in in combination with the control schemes. Different system sonnamed (1990)). Alinear solution to the optimal nonlinear control tion with the moderntakes control schemes. Different system is the quadratic regulator (LQR) (Anderproblem is obtained by using the method of dynamic proson (1990)). A solution to the optimal nonlinear control system identification an important role in combinaidentification techniques have been been reported with successsuccesstion with the techniques modern control schemes. Different system son problem is obtained by using theoptimal methodnonlinear of dynamic pro(1990)). A solution to the control identification have reported with gramming, developed by Bellman (Bellman (1957)), which problem is obtained by using the method of dynamic protion with the modern control schemes. Different system fully results as seen in (Rico and Heydt (2001)),(Laroche identification techniques have been reported with successgramming, developed by Bellman (Bellman (1957)), which problem obtained by the method of(1957)), dynamic profully results as seen in (Rico Heydt (2001)),(Laroche leads to to aaisnonlinear nonlinear partial differential equation named the developed byusing Bellman (Bellman which identification haveand been reported with successfully results(2008)) astechniques seenand in (Stinga (Rico and (2001)),(Laroche and Sedda Sedda andHeydt Soimu (2015)), applied gramming, leads partial differential equation named the gramming, developed by Bellman (Bellman (1957)), which and (2008)) and (Stinga and Soimu (2015)), applied leads to a nonlinear partial differential equation named the Hamilton-Jacobi-Bellman equation (HJB) (Kirk (2004)), fully results asparameter seenand in (Stinga (Rico and Heydt (2001)),(Laroche for the online estimation and control of inducand Sedda (2008)) and Soimu (2015)), applied Hamilton-Jacobi-Bellman equation (HJB) (Kirk (2004)), for the online parameter estimation and control ofapplied induc- leads to a nonlinear partial equation differential equation named the whose complicated for nonlinear sys(HJB) (2004)), and Sedda (2008)) andthe (Stinga Soimu (2015)),of tionthe motors. However leastand square algorithm deserves for online parameter estimation and control induc- Hamilton-Jacobi-Bellman whose solution solution in in general general is isequation complicated for (Kirk nonlinear sysHamilton-Jacobi-Bellman (HJB) (Kirk (2004)), tion motors. However the least square algorithm deserves tems. The application of this technique is well established whose solution in general is complicated for nonlinear sysfor the online parameter estimation and control of inducspecial mention amongthe theleast parameter identification tech- whose tion motors. However square identification algorithm deserves tems. The application of this technique is established solution in general is complicated nonlinear sysspecial mention among the parameter techtems. The application of this technique isforwell well established in solving solving the optimal control problem for for linear systems, tion motors. However square identification algorithm deserves special mention amongthe theleast parameter techniques: the previously mention approach is recursive that in the optimal control problem linear systems, tems. The application of this technique is well established niques: the previously mention approach is recursive that in solving the optimal control problem for linear systems, where its formulation results in the solution given by the special mention among the parameter identification techallows computer implementation. The niques: theefficient previously mention approach is recursive that in where its formulation results in the solution givensystems, by the allows for for efficient computer implementation. The reprerepresolving the optimalRiccati controlequation; problem for linear differential/algebraic hence, optimal conwhere its formulation results in the solution given by the niques: the previously mention approach is recursive that sentation ofefficient nonlinear systemsimplementation. in aa polynomial polynomialThe formrepremay where allows for of computer differential/algebraic Riccati equation; hence, given optimal conformulation in thetosolution byconthe sentation nonlinear systems in form may trol for forits linear systemsRiccati isresults considered be a solved solved problem; differential/algebraic equation; hence, optimal allows for ofefficient computer implementation. The reprebe possible by using different polynomial models, which sentation nonlinear systems in a polynomial form may trol linear systems is considered to be a problem; differential/algebraic Riccati equation; hence, optimal conbe possible by using different polynomial models, which trol for linear systems is considered to be a solved problem; however, the the nonlinear optimal optimal control control is is an open open issue. issue. sentation of approximate nonlinear systems in system a polynomial formwhich may however, be by using different polynomial are possible used to to the real real asmodels, polynomial trol for linear systems isoptimal considered to beisa an solved are used approximate the system as aa polynomial however, the nonlinear nonlinear control an openproblem; issue. be possible by using different polynomial models, which ones. The identification of dynamical models obtained ofare used to approximate the real system as a polynomial In this this sense, sense, the state-dependent Riccatiisequation equation (SDRE) ones.used The identification ofthe dynamical models obtained of- however, the the nonlinear optimal control an open(SDRE) issue. Riccati are approximateof real system as ais polynomial fline from from measurements or dynamical experimental data motivated ones. Theto identification models obtained of- In In this sense, the state-dependent state-dependent Riccati equation (SDRE) (L. Lin and Liang. (2015)) extends the LQR approach to fline measurements or experimental data is motivated (L. Lin and Liang. (2015)) extends the LQR approach to ones. The identification of dynamical models obtained ofthis sense, the state-dependent Riccati equation (SDRE) by using using the results for for orcontrol control purposes. On the other other In fline from the measurements experimental dataOn is motivated (L. Lin and Liang. (2015)) extends the LQR approach to the nonlinear case by allowing the matrices involved by results purposes. the the nonlinear case by allowing the matrices involved to fline from measurements experimental dataOn is controllers motivated by using results for orcontrol purposes. the other (L. Lin and Liang. (2015)) extends LQR approach to hand, it is isthe always of interest interest to design design efficient be functions functions ofcase state variables andthe transforming inputthe nonlinear by variables allowing the matrices involved to hand, it always of to efficient controllers be of state and transforming inputby using the results for control purposes. Onthe thesystem other the nonlinear case by allowing the matrices involved to in order to obtain a satisfactory behavior of hand, it is always of interest to design efficient controllers affine nonlinear systems into the so-called so-called state-dependent functions of systems state variables and transforming inputin order toalways obtainofa interest satisfactory behavior of the system be affine nonlinear into state-dependent hand, it isto design efficient controllers functions of systems state(SDCF) variables and transforming inputvariables and at the the same to time reach anofoptimization optimization in order obtain a satisfactory behavior the system be affine nonlinear into the the so-called state-dependent coefficient factorized nonlinear systems (Cloutier variables and at same time reach an coefficient factorized (SDCF) nonlinear systems (Cloutier in order Ato obtain a satisfactory behavior ofoptimization the system affine variables and at the same with time respect reach an nonlinear systems into the so-calledsystems state-dependent criteria. feasible approach these objectives (1997)), (Ornelas-Tellez et al. (2013)),(Cimen (2010)), a coefficient factorized (SDCF) nonlinear (Cloutier criteria. A feasible approach with respect these objectives (1997)), (Ornelas-Tellez et al.nonlinear (2013)),(Cimen (2010)), a variables atoptimal the same timetheory. reach an optimization criteria. Aand feasible approach with respect these objectives coefficient factorized (SDCF) systems (Cloutier is the use of the control linear like representation for nonlinear systems used to (1997)), (Ornelas-Tellez et al. (2013)),(Cimen (2010)), a is the use of the optimal control theory. linear like representation for nonlinear systems used to criteria. approach withtheory. respect these objectives (1997)), (Ornelas-Tellez et al. (2013)),(Cimen (2010)), a is the useAoffeasible the optimal control solve the the associated HJB equation. In essence, theused SDCF likeassociated representation for nonlinear systems to The optimal control is one one particular branch of of modern linear solve HJB In the SDCF is theoptimal use of the optimal control theory.branch linear likeassociated representation foris nonlinear systems used to The control is particular solve the HJB equation. equation. In essence, essence, thefor SDCF nonlinear control technique a systematic way synThe optimal control is one particular branch of modern modern control theory, which has the objective of determining the nonlinear control technique is a systematic way forSDCF synthe associated HJB equation. In essence, thefor control theory,control which is hasone theparticular objective of determining the solve nonlinear control technique is a systematic way synThe optimal branch of modern control theory, which has the objective of determining the nonlinear control technique is a systematic way for syncontrol has the objective of determining the Hosting by Elsevier Ltd. All rights reserved. 2405-8963theory, © 2018,which IFAC (International Federation of Automatic Control)
Proceedings, 2nd IFAC Conference on 431 Proceedings, 2nd IFAC Conference on 431 Control. Peer reviewIdentification under responsibility of International Federation of Automatic Modelling, and Control of Nonlinear Proceedings, 2nd IFAC Conference 431 Modelling, Identification and Controlon of Nonlinear 10.1016/j.ifacol.2018.07.324 Proceedings, 2nd IFAC Conference on 431 Systems Modelling, Identification and Control of Nonlinear Systems Modelling, Identification and Control of Nonlinear Guadalajara, Mexico, June 20-22, 2018
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thesizing optimal nonlinear feedback controllers, which mimics the controller synthesis as done for the linear case, by exploiting the nature of the nonlinear behavior throughout the state space. Notice that by using the SDCF technique the system is not being linearized around an equilibrium point, therefore the nonlinear system dynamic is completely considered; thus, the controller has a larger operation range over the system in comparison with the linear controllers; in addition, the nonlinear inherit nature of the system is exploited. In this work the least-square algorithm is used in order to identify the unknown parameters of a known nonlinear model of a dynamic system, which is achieved by obtaining the optimal values of a parameter vector θ by minimizing the identification error. The contribution of this paper is the introduction of an easy method for identifying the parameters in linear and nonlinear systems, which are linear with respect to the unknown parameters. A posteriori, the identified-parameterbased system is used for the synthesis of a nonlinear optimal controller. Simulations validate the effectiveness of the proposed methodology in identifying the parameters and for the control of a PMSM. The paper is organized as follows. Section 2 describes the parameter identification algorithm based on a recursive least-square method. Section 3 presents the robust nonlinear optimal control scheme. In section 4, the parameter identification scheme is applied to a permanent magnet synchronous motor in order to determine its unknown parameters, then based on the identified parameters, the nonlinear optimal control is applied to the system. Simulation results prove the identification and control scheme. Finally, Section 5 concludes the paper. 2. PARAMETER IDENTIFICATION ALGORITHM 2.1 Uncertain nonlinear systems Consider the uncertain in the parameters nonlinear system given as X˙ = W (X , u, d)θ∗ (1) Y = h(X ) where X ∈ Rn is the state vector, Y ∈ Rp is system output and u ∈ Rm is the system input; matrix W (X , u, d) is a nonlinear map containing states (X ), inputs (u) and bounded external disturbances (d); h is the system output map. Term θ∗ represent the parameters’ vector of the system. Note that (1) is linear with respect to θ∗ , which is a key feature in the parameters’ identification algorithm. In order to design a model-based control scheme for system (1), it is necessary to know the parameters of the system θ∗ , nonetheless such parameters are generally difficult to determine for physical systems. This work is focused in proposing an easy parameters’ identification scheme to determine θ∗ such that a model-based controller can be designed. For explanation purposes, vector θ∗ will be assumed as one containing the real parameters of a physical system, named for simplicity as the system ideal parameter vector. It is important pointing out that for the parameters’ identification algorithm, this paper considers 432
to (1) as the physical system and the corresponding methodology will be evaluated via simulations. Currently, work is progressing to have experimental results. 2.2 Parameter Identification Algorithm For the determination of the system parameters, consider an identifier system given as x˙ = W (x, u, d)θ (2) y = h(x) where x ∈ Rn is the identifier state vector, θ is the vector to be on-line adapted such that it converges toward θ∗ by means a recursive least-square algorithm (RLSA) (Sastry and Bodson (2011)), which optimally estimate the parameters’ vector θ by minimizing the identification error given as = x − X. (3) For the RLSA design it is assumed that there exist the unknown ideal parameters’ vector θ∗ , whose dynamics can be described by θ˙∗ = 0. (4) with the output (5) y = W T (x, u, d)θ∗ Assuming the right-hand sides of (4) and (5) are perturbed by zero mean white gaussian noises with spectral intensities Υ ∈ R2n×2n and g ∈ R, respectively, the RLSA results in θ˙ = −gΨW (x, u, d) (6) ˙ Υ, g > 0 Ψ = Υ − gΨW (x, u, d)W T (x, u, d)Ψ
which is also referred as the Kalman-Bucy filter (Anderson and Moore (1971)), where θ is the estimated value for θ∗ , vector W contains the selected polynomial basis to approximate the vector fields in (1), Υ and g are fixed design parameters used to ensure the convergence of the identification error. Matrix Ψ is of appropriate dimension and is the so-called covariance matrix and it is involved in the update law for θ; the initial conditions for Ψ are arbitrarily selected, usually chosen to reflect the confidence in the initial estimate value for θ(0), being Ψ(0) a symmetric and positive definite matrix. It is important to notice that the adaptation law (6) must be implemented for each state variable. 2.3 SDCF Nonlinear Systems Extended linearization (Friedland. (1996)), also known as apparent linearization (Wernli and Cook (1975)) or state-dependent coefficient factorization SDCF (L. Lin and Liang. (2015)),(Ornelas-Tellez et al. (2013)), is the process of factorizing a nonlinear system into a linearlike structure, which contains state-dependent matrices. To obtain such factorization, consider that system (2) can be decomposed and presented as x˙ = f (x, θ) + B(x, θ)u + Γ(θ) (7) y = h(x) where Γ is considered an external input containing possible disturbances; functions f (x, θ), B(x, θ) and h(x) are smooth of appropriate dimensions. Under the assumptions that f (·) ∈ C 1 , h(·) ∈ C 1 , f (0) = 0 and h(0) = 0, a
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continuous nonlinear matrix-valued functions exist such that f (x, θ) = A(x, θ)x and h(x) = C(x)x (Shamma and Cloutier (2003)); then, system (7) can be rewritten as x˙ = A(x, θ)x + B(x, θ)u + Γ(θ) (8) y = C(x) x. It is worth remarking that the factorizations A(x)x and C(x)x are not unique (Banks et al. (2007)). To obtain well-defined control schemes, appropriate factorizations for this representation should be determined such that the controllability and observability properties are fulfilled for system (8), as established in (Banks et al. (2007)) and (Ornelas-Tellez et al. (2014)). Remark. For the parameters’ identification, an arbitrary control input u must be used to excite the dynamics of systems and (2) to achieve the convergence of θ toward to θ∗ . By considering that θ has been determined, for easy of notation in the controller design in the next section such parameter dependence in (8) will be omitted in the system description. 3. NONLINEAR OPTIMAL TRACKING CONTROL
433
T
with xa = [Φ, x] , then system (8) is rewritten as an augmented system given as x˙ a =Aa (xa ) + Ba (xa )u + D (12) ya =Ca (xa )xa where 0 C 0 , Ba = , Aa (xa ) = 0 A(x) B (13) −r Da = . Ca = [ 0 C ] , Γ
For system (12), the control scheme proposed is considered to be optimal in the sense of minimizing the performance index 1 ∞ t Φ QI Φ + eT Qe + uT Ru dt (14) J= 2 t0 where matrix Q is matrix positive definite weighting the performance of the state vector, while R also a positive definite matrix weighting the control effort expenditure, if more importance is given to the state performance, a greater value for Q could be selected or reducing the value of R. The controller design is based on the following assumptions: H1. The pair [A(x), B(x)] is controllable and the pair [A(x), C(x)] is observable.
Nonlinear optimal tracking control (Ornelas-Tellez et al. (2014)), extends the nonlinear optimal control methodology by using the SDCF in order to track a time-varying reference signals. The problem considered is the infinitehorizon tracking control (Carlson et al. (1987)) of nonlinear systems, derived from the state-regulator problem (Athans and Falb (2007)), which has the objective of “keeping the state near the desired value”, then the minimum principle is used in order to obtain the necessary conditions for the optimal control. In order to give more robustness to the controller, an integral action is added to the control law in terms of the tracking error e, in such a way that constant parametric uncertainties as well as external disturbances may be rejected, and therefore, enables this control scheme for real-time applications. 3.1 Optimal Tracking Controller Design For system (8), let us consider a trajectory tracking error defined as e =r − y (9) =r − C(x)x where r is the desired reference to be tracked by the system output y. The nonlinear optimal control problem is to find a nonlinear optimal control law, such that the output of system (8) tracks a desired reference signal, in the optimal sense of minimizing the tracking error and the control effort. In order to enhance the controller robustness properties, an integral term is considered in the design, defined as ˙ = −e Φ (10) p where Φ ∈ R is a vector of integrators for a system with p outputs. Therefore, an augmented system can be stated, which includes the integrator as ˙ −e Φ x˙ a = = (11) A(x)x + B(x)u + Γ x˙ 433
H2. The state vector x is available for feedback. Assumption H1 is related to the existence and uniqueness of the Riccati equation solution (Anderson and Moore (1990); Sepulchre et al. (1997); Kalman (1960)). Assumption H2 implies the full state vector is available from measurements, but when it is unavailable, an observer could be used for estimating x, however, this issue is beyond the scope of this paper. Under assumptions H1 and H2, the nonlinear optimal controller (15) u∗ (xa ) = −R−1 BaT (P (xa ) − z(xa )) achieves robust trajectory tracking for system (8) along the desired trajectory r, as established in (Ornelas-Tellez et al. (2017)), where P (xa ) is the solution of the SDRE and z(xa ) is the solution of a vector differential equation, respectively where P˙ (xa ) = − Qa + P (xa )Ba R−1 BaT P (xa ) (16) − ATa (xa )P (xa ) − P (xa )Aa (xa ) with QI 0 (17) Qa = 0 C T QC z(x ˙ a ) = − [Aa (xa ) − Ba R−1 BaT P (xa )]T z(xa )
(18) + P Da − CaT Qr with boundary conditions P (x(∞)) = 0 and z(x(∞)) = 0. Fig 1 depicts the flow chart of the proposed parameter identification process for dynamic systems which are linear with respect to their parameters. The parameters’ identification process is done off-line, and once the system parameters have been identified, the state-feedback optimal control law is applied. It is important mentioning that this paper considers the unknown system parameters to be determined are constant, which is in general common in physical systems; for the case when the parameters are
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1 R where θ1 = fjr , θ2 = 3λ 2j , θ3 = j , θ4 = Ld , θ5 = 1, θ6 = R λ 1 θ7 = Lq , θ8 = 1, θ9 = Lq and θ10 = Lq .
Nonlinear identifier x˙ = W (x, u, d)θ
1 Ld ,
T
with the state vector x = [ x1 x2 x3 ] identifies to the actual state vector X . Which supposes that the actual system parameters are unknown and only the state variables are available for measurements. The vector θ for each state variable is identified by using (6) with the identification error = x − X .
Recursive least squares to identify θ∗ θ˙ = −gΨW (x, u, d) ˙ = Υ − gΨW (x, u, d)W T (x, u, d)Ψ Ψ Parameter convergence θ → θ∗ Nonlinear optimal control u∗ (xa ) = −R−1 BaT (P (xa ) − z(xa )) Fig. 1. Flow chart of the parameter identification procedure for the model-based controller design. piece-wise constant or slowly time-varying, the RLSA can be used, however, a faster convergence of the RLSA must be guaranteed. 4. APPLICATION PERMANENT MAGNET SYNCHRONOUS MOTOR The parameters’ identification methodology and controller design is illustrated in a nonlinear system, the PMSM, which is linear with respect to its parameters. A commonly used reference frame for PMSM models is the synchronous reference frame, also known as dq-frame. One of the main features of this reference frame is the fact that the required calculations for analysis and control tasks are simplified since the systems values are constant and not time-varying as in the abc-frame. Therefore, the chosen dynamic model for the PMSM (Yaramasu and Wu (2017)) is described in dq-frame as fr 3 Tm x˙ 1 = − x1 + P (λx3 + (Ld − Lq )x2 x3 ) − j 2j j R Lq Vd x˙ 2 = − x2 + P x1 x3 + (19) Ld Ld Ld λ Ld R Vq x˙ 3 = − P x1 − P x1 x2 − x3 + Lq Lq Lq Lq where x1 , x2 , x3 are the rotational speed, d -axis current and q-axis current respectively, R is the resistance of the stator winding, Ld , Lq are the stator dq-axis inductances, λ is the magnetic flux produced by the permanent magnets, P is the number of pole pairs, Vd , Vq are the dq-axis stator voltages; f r, J, Tm are the viscous friction, moment of inertia and mechanical torque of the shaft, respectively. 4.1 Polynomial parameter identification for the PMSM Based on the nonlinear model (19), the following polynomial identifier model structure is proposed x˙ 1 = −θ1 x1 + P θ2 x3 − Tm θ3 x˙ 2 = −θ4 x2 + P θ5 x1 x3 + θ6 Vd (20) x˙ 3 = −θ7 x3 − P θ8 x1 x2 − P θ9 x1 + Vq θ10 434
The parameters used in the simulation are g1 = 100, g2 = 0.1 and g3 = 0.1, with the selected bases T T W1 = [ x1 x3 1 ] , W2 = [ x2 x1 x3 1 ] and W3 = T [ x3 x1 x2 x1 ] . The identifier parameters to ensure the identification convergence are Υ1 = diag [ 10 10 10 ], Υ2 = diag [ 10 1 1 ] and Υ3 = diag [ 10 1 1 1 ]. Thus it is possible to estimate the convergence of vector θ, towards the ideal (real) physical parameters of system (19) which a priori where considered to be unknown. Table 1 depicts the identified parameters of the PMSM model (20) while Table 2 displays the real physical values of the system (19). Figures 4, 5 and 6 display the convergence of the vector θ toward the real values of the system. Table 1. PMSM identified parameters Friction coefficient Pole pairs Magnetic flux Moment of Inertia Stator Resistance d-axis inductance q-axis inductance Mechanical torque
0.00185798 N.m.s 4 0.1929967 W b 0.00250013 kg.m2 2.4802995 Ω 0.11177605 H 0.11201381 H 4 N.m
Table 2. PMSM parameters Friction coefficient Pole pairs Magnetic flux Moment of Inertia Stator Resistance d-axis inductance q-axis inductance Mechanical torque
0.00186 N.m.s 4 0.193 W b 0.0025 kg.m2 2.48 Ω 0.113 H 0.113 H 4 N.m
4.2 Nonlinear Optimal Control Based on the scheme proposed in (8)–(11), the augmented system including an integral term of the error is defined as y−r x˙ a = (21) A(x)x + B(x)u + Γ and the system matrices can be rewritten as presented in (13), with 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 fr 3pλ , Ba = (22) Aa = 0 0 0 0 0 0 − J 2J 1 ωLq R 0 0 0 0 − L d L d Ld 0 R d 0 L1q 0 0 0 − Lλ1 − ωL − Lq Ld 0 0 0 1 0 0 T Ca = 0 0 0 0 1 0 , Γa = −r TJm 0 0 0 0 0 0 0 1
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4.3 Nonlinear robust optimal control To evaluate the performance of the designed controller a simulation of the PMSM model is developed in Wolfram/Mathematica environment. The parameters used in the simulation are shown in Tables 1 and 2. The rotational speed reference value ωref is selected at 100rad/s while the d -axis current is controlled at 8A. The initial states are x(0) = 0 and the controller parameters QI , Q and R are as follows 20 0 0 10 0 0 0.01 0 QI = 0 100 0 Q = 0 10 0 R = . 0 0.05 0 0 1 0 0 1
400
Vq 200
Vd
0 300
200 0
300
400 600 0.0
Figure 2 illustrates the machine rotational speed response, the solid line corresponds to the actual rotational speed, and the dashed line to the desired reference value. The state converges to the reference without steady-state error due to the integral term added to the control design.
0.1
0.2
600
800 0
1
2
3
5
4
Time s
Fig. 4. Control actions applied to the PMSM
120
400
80
101
Θ i Identified Real
Ω 300
Ωref Identified value
Rotational speed rad s
Figure 4 shows the control actions used to drive the system to the desired reference values. It can be seen that a quick convergence is achieved with small overshoot. These results prove the efficiency of the designed controller based on the identified parameters to control the machine modeled with ideal values.
Voltage V
where the system parameters correspond to the ones obtained by means of the proposed identifier.
435
100
40 99 0
1
2
3
4
5
200
100
0
0
1
2
3
5
4
0
Time s
0
500
Fig. 2. Rotational speed of the PMSM Figure 3 shows the d -axis current. At t = 3s the desired current is reached and the error is almost zero.
1000
Time (s)
1500
2000
Fig. 5. Convergence of θ1 , θ2 , θ3 respectively toward the ideal physical values
12
Id Identified value
Current A
9 8.1
6
8.0
7.9
3
Θ i Identified Real
20
Idref 15 10 5
7.8 0
1
2
3
4
5
0 0
0 0
1
2
3
4
5
Time s
500
1000
Time (s)
1500
2000
Fig. 6. Convergence of θ4 , θ5 , θ6 respectively toward the ideal physical values
Fig. 3. d -axis current of the PMSM 435
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25
Θ i Identified Real
Identified value
20 15 10 5 0 0
500
1000
Time (s)
1500
2000
Fig. 7. Convergence of θ7 , θ8 , θ9 , θ10 respectively toward the ideal physical values 5. CONCLUSIONS In this paper, the least square algorithm is proposed in order to identify the physical parameters for systems which are linear with respect to their parameters. Subsequently based on this offline parameters identification an optimal robust controller was designed for a permanent magnet synchronous motor. Simulation studies demonstrated that the designed controller can effectively achieve the desired tracking objectives by means of the identified parameters of the PMSM instead of the ideal ones. REFERENCES Anderson, B. (1990). Optimal Control Linear Quadratic Methods. Dover Publications, Englewood Cliffs, New Jersey, USA. Anderson, B. and Moore, J. (1990). Optimal Control: Linear Quadratic Methods. Prentice-Hall, Englewood Cliffs, NJ, USA. Anderson, D. and Moore, J.B. (1971). The Kalman-Bucy Filter as a True Time-Varying Wiener Filter, volume SMC-1. IEEE Transactions on Systems, Man, and Cybernetics. Athans, M. and Falb, P.L. (2007). Optimal Control an Introduction to the Theory and its Applications. Dover, New York, USA. Banks, H.T., Lewis, B.M., and Tan, H.T. (2007). Nonlinear feedback controllers and compensators: a statedependent Riccati equation approach. Computational Optimization and Applications, 37(2), 177–218. Bellman, R.E. (1957). Dynamic Programming. Princeton University Press, Princeton, USA. Carlson, D., Haurie, A., and Leizarowitz, A. (1987). Infinite Horizon Optimal Control. Springer-Verlag, Berlin, Germany. Cimen, T. (2010). Systematic and effective design of nonlinear feedback controllers via the state-dependent riccati equation (sdre) method. Annual Reviews in Control, 34(1), 32 – 51. Cloutier, J. (1997). State-dependent riccati equation techniques: an overview. In Proceedings of the 1997 American Control Conference (Cat. No.97CH36041), volume 2, 932–936 vol.2. Friedland., B. (1996). Advanced Control Systems Design. Prentice Hall, USA. 436
Kalman, R.E. (1960). Contributions to the theory of optimal control. Bol. Soc. Mat. Mexicana, 5, 102–119. Kirk, D.E. (2004). Optimal Control Theory an Introduction. Dover Publications, New York, USA. L. Lin, J.V. and Liang., Y. (2015). Analytical representation of the state-dependent coefficients in the sdre/sddre scheme for multivariable systems. In Automatica, volume 59, 106–111. Laroche, E. and Sedda, E. (2008). Methodological insights for online estimation of induction motor paramteres. Transactions on Control Systems Technology, 16(5). Ornelas-Tellez, F., Rico, J.J., Espinosa-Juarez, E., and Sanchez, E.N. (2017). Optimal and robust control in DC microgrids. IEEE Transactions on Smart Grid, 1– 12. Ornelas-Tellez, F., Rico-Melgoza, J.J., and Sanchez, E.N. (2013). Optimal tracking for a class of nonlinear systems based on the state-dependent riccati equation. In 2013 10th International Conference on Electrical Engineering, Computing Science and Automatic Control (CCE), 42–47. Ornelas-Tellez, F., Rico, J.J., and Ruiz-Cruz, R. (2014). Optimal tracking for state-dependent coefficient factorized nonlinear systems. Asian Journal of Control, 16(3), 890–903. Rico, J. and Heydt, G. (2001). An algebraic approach for identifying operating point dependent parameters of synchronous machines using orthogonal series expansions. IRE Transactions on Energy Conversion, 16(1). Sastry, S. and Bodson, M. (2011). Adaptive Control, Stability, Convergence and Robustness. Dover, New York, United States. Sepulchre, R., Jankovic, M., and Kokotovi´c, P.V. (1997). Constructive Nonlinear Control. Springer-Verlag, Berlin, Germany. Shamma, J. and Cloutier, J. (2003). Existence of sdre stabilizing feedback. IEEE Transactions on Automatic Control, 48(3), 513–517. Stinga, F. and Soimu, A. (2015). Online estimation and control of an induction motor. 19th Conference on System Theory and Computing. Wernli, A. and Cook, G. (1975). Suboptimal control for the nonlinear quadratic regulator problem. Automatica, 11. Yaramasu, V. and Wu, B. (2017). Model Predictive Control of Wind Energy Conversion Systems. John Wiley and Sons, New Jersey, United States.
ScienceDirect
IFAC PapersOnLine 51-13 (2018) 431–436
System System System Optimal System Optimal Optimal Optimal
Parameters’ Identification and Parameters’ Identification and Parameters’ Identification and Tracking Control for Nonlinear Parameters’ Identification and Tracking Control for Nonlinear Tracking Control for Nonlinear Systems Tracking Control for Nonlinear Systems Systems Systems David Cortes Vega ∗∗ , Serafin Ramos Paz, ∗∗
David Cortes Vega ∗∗∗ , Serafin Ramos Paz, ∗∗ ∗ Fernando Ornelas-Tellez J. Rico-Melgoza David Cortes Vega ∗∗ and , Serafin Ramos Paz, ∗ ∗ Fernando Ornelas-Tellez J. Jesus Jesus Rico-Melgoza ∗ and ∗ ∗ David Cortes Vega , Serafin Ramos Paz, Fernando Ornelas-Tellez ∗ and J. Jesus Rico-Melgoza ∗∗ Fernando Ornelas-Tellez and J. Jesus Rico-Melgoza ∗ ∗ The authors are with the School of Electrical Engineering, ∗ The authors are with the School of Electrical Engineering, ∗ The authors are with the School of de Electrical Engineering, Universidad Michoacana de San Nicolas Hidalgo, Morelia, Universidad Michoacana de the SanSchool Nicolas Hidalgo,Engineering, Morelia, 58030, 58030, ∗ The authors are with of de Electrical Mexico (e-mail: [email protected], Universidad Michoacana de San Nicolas de Hidalgo, Morelia, 58030, Mexico (e-mail: [email protected], Universidad Michoacana de San Nicolas de Hidalgo, Morelia, 58030, [email protected], [email protected]) Mexico (e-mail: [email protected], [email protected], [email protected], [email protected], [email protected]) Mexico (e-mail: [email protected], [email protected], [email protected], [email protected]) [email protected], [email protected], [email protected]) Abstract: This This paper paper proposes proposes aa parameters’ parameters’ identification identification methodology methodology via via the the continuouscontinuousAbstract: time least-squares algorithm for those nonlinear systems which are linear with respect to their Abstract: This paper proposes a parameters’ identification methodology via the continuoustime least-squares algorithm for nonlinear systems which are linear with respect to Abstract: This paper proposes a parameters’ identification methodology via the continuoustime least-squares algorithm for those those nonlinear systems which are linear with respect toalltheir their parameters. The parameter identification method can be used for easily determining the parameters. The parameter identification method can bewhich used are for linear easilywith determining the time least-squares algorithm foronly those nonlinear systems respect toall their parameters in real plants from input-output data measurements, such that a posteriori, parameters. The parameter identification method can be used for easily determining all theaa parameters in real plants from only input-output data measurements, such that a posteriori, parameters. identification methoddata be used for easily determining all thea model-basedinThe control strategy can beinput-output synthesized. Itcan is measurements, worth mentioning that when control parameters realparameter plants from onlybe suchthat thatwhen a posteriori, model-based control strategy can synthesized. It is worth mentioning aaa control parameters in real plants from only input-output data measurements, such that a posteriori, a model-based control strategy can be synthesized. It is worth mentioning that when control engineer wants wants to to design aa modern modern and and sophisticated sophisticated controller, controller, usually usually the the system system model model is is used engineer model-based control strategy can on be its synthesized. Itcontroller, is worth in mentioning that when a control engineer wants to design design adepends modern and sophisticated usually the system model is used used for such purposes, which parameters, however, general the system parameters for such purposes, which adepends on parameters, controller, however, inusually generalthe the system parameters engineer wants design modern andits sophisticated system model is used are such not easy easy to to determine. Also, this this paper uses the the parameters’ parameters’ identification methodology for for purposes, which depends on its parameters, however, in general the system parameters are not to determine. Also, paper uses identification methodology for for such purposes, which depends on its parameters, however, in general the system parameters the design of an optimal tracking controller for state-dependent coefficient factorized (SDCF) are not easy to determine. Also, this paper uses the parameters’ identification methodology for the design of optimal tracking controller for state-dependent coefficient factorized (SDCF) are not easy Also, this paper uses parameters’ identification methodology for the design oftoan andetermine. optimal tracking controller for the state-dependent coefficient factorized (SDCF) nonlinear systems. Both, the the parameters’ identification scheme and and the optimal optimal control strategy strategy nonlinear systems. Both, parameters’ identification scheme the control the designsystems. of an optimal tracking controller for state-dependent coefficient factorized (SDCF) nonlinear Both, the parameters’ identification scheme andSynchronous the optimal control(PMSM), strategy are applied via simulations for the control of a Permanent Magnet Motor are appliedsystems. via simulations for the control identification of a Permanent Magnet Synchronous Motor (PMSM), nonlinear Both,machine. the scheme andSynchronous the optimal Motor control(PMSM), strategy a three-phase three-phase nonlinear are applied vianonlinear simulations forparameters’ the control of a Permanent Magnet a machine. are applied via simulations for the control of a Permanent Magnet Synchronous Motor (PMSM), a three-phase nonlinear machine. 2018, IFAC (International Federation of Automatic Control) Hosting by Elsevier Ltd. All rights reserved. a©three-phase nonlinear machine. Keywords: Keywords: Parameter Parameter identification, identification, optimal optimal nonlinear nonlinear control, control, PMSM. PMSM. Keywords: Parameter identification, optimal nonlinear control, PMSM. Keywords: Parameter identification, optimal nonlinear control, PMSM. 1. INTRODUCTION INTRODUCTION control signals signals that that will will cause cause aa process process to to satisfy satisfy the the physphys1. control ical constraints and at the same time minimize or maxi1. INTRODUCTION control signals that will cause a process to satisfy the physical constraints and at the same time minimize or maxi1. INTRODUCTION control signals that cause a process to satisfy physmizeconstraints performance index (Kirk (2004)). Linearthe optimal andwill at the (Kirk same time minimize oroptimal maxiModern control control techniques techniques are are based based on on aa good good underunder- ical mize aa performance index (2004)). Linear ical constraints and at the same time minimize or maxiModern control is special sort sort of optimal control, where where the plant a is performance index (Kirk (2004)). Linearthe optimal Modern techniques based on goodphysical under- mize standing control of the the plant to be be are controlled, by ausing control aaa special of control, plant mize a is performance index (Kirk (2004)). Linearthe optimal standing of to controlled, by control special sort ofasoptimal optimal control, where plant is assumed to be linear well as the controller, which is Modern control techniques based on goodphysical understanding ofpossible the plant plant to bea are controlled, by ausing using physical laws it is to get mathematical model of the is assumed to be linear asoptimal well as the controller, which is control is by a special sort ofas where the plant laws it isofpossible to to getbea controlled, mathematical modelphysical of the is obtained with quadratic indices. assumed toworking be linear well as control, theperformance controller, which is standing the plant byare using system, although many system parameters unknown or laws it is possible to get a mathematical model of the obtained by working with quadratic performance indices. assumed be achieves linear asthe well as the controller, which is system, although many system parameters are unknown or is The method that linear optimal control design bytothat working with quadratic performance indices. laws it although is possible to get a mathematical model of why the even difficult difficult to be bemany computed, that is the the main main reason system, system parameters are unknown or obtained The method achieves the linear optimal control design obtained by working with quadratic performance indices. even to computed, that is reason why is named named thethat linear quadratic regulator (LQR) (Andermethod achieves the linear optimal control(Anderdesign system, although system parameters are in unknown or The even difficult to bemany computed, is the main reason why system identification takes an that important role combinais the linear quadratic regulator (LQR) method achieves the linear optimal control(Anderdesign system identification takes an important role combinais named thethat linear quadratic regulator (LQR) son (1990)). A solution to the optimal nonlinear control even difficult tomodern be computed, is the main reason why The system identification takes an that important role in in combination with the control schemes. Different system sonnamed (1990)). Alinear solution to the optimal nonlinear control tion with the moderntakes control schemes. Different system is the quadratic regulator (LQR) (Anderproblem is obtained by using the method of dynamic proson (1990)). A solution to the optimal nonlinear control system identification an important role in combinaidentification techniques have been been reported with successsuccesstion with the techniques modern control schemes. Different system son problem is obtained by using theoptimal methodnonlinear of dynamic pro(1990)). A solution to the control identification have reported with gramming, developed by Bellman (Bellman (1957)), which problem is obtained by using the method of dynamic protion with the modern control schemes. Different system fully results as seen in (Rico and Heydt (2001)),(Laroche identification techniques have been reported with successgramming, developed by Bellman (Bellman (1957)), which problem obtained by the method of(1957)), dynamic profully results as seen in (Rico Heydt (2001)),(Laroche leads to to aaisnonlinear nonlinear partial differential equation named the developed byusing Bellman (Bellman which identification haveand been reported with successfully results(2008)) astechniques seenand in (Stinga (Rico and (2001)),(Laroche and Sedda Sedda andHeydt Soimu (2015)), applied gramming, leads partial differential equation named the gramming, developed by Bellman (Bellman (1957)), which and (2008)) and (Stinga and Soimu (2015)), applied leads to a nonlinear partial differential equation named the Hamilton-Jacobi-Bellman equation (HJB) (Kirk (2004)), fully results asparameter seenand in (Stinga (Rico and Heydt (2001)),(Laroche for the online estimation and control of inducand Sedda (2008)) and Soimu (2015)), applied Hamilton-Jacobi-Bellman equation (HJB) (Kirk (2004)), for the online parameter estimation and control ofapplied induc- leads to a nonlinear partial equation differential equation named the whose complicated for nonlinear sys(HJB) (2004)), and Sedda (2008)) andthe (Stinga Soimu (2015)),of tionthe motors. However leastand square algorithm deserves for online parameter estimation and control induc- Hamilton-Jacobi-Bellman whose solution solution in in general general is isequation complicated for (Kirk nonlinear sysHamilton-Jacobi-Bellman (HJB) (Kirk (2004)), tion motors. However the least square algorithm deserves tems. The application of this technique is well established whose solution in general is complicated for nonlinear sysfor the online parameter estimation and control of inducspecial mention amongthe theleast parameter identification tech- whose tion motors. However square identification algorithm deserves tems. The application of this technique is established solution in general is complicated nonlinear sysspecial mention among the parameter techtems. The application of this technique isforwell well established in solving solving the optimal control problem for for linear systems, tion motors. However square identification algorithm deserves special mention amongthe theleast parameter techniques: the previously mention approach is recursive that in the optimal control problem linear systems, tems. The application of this technique is well established niques: the previously mention approach is recursive that in solving the optimal control problem for linear systems, where its formulation results in the solution given by the special mention among the parameter identification techallows computer implementation. The niques: theefficient previously mention approach is recursive that in where its formulation results in the solution givensystems, by the allows for for efficient computer implementation. The reprerepresolving the optimalRiccati controlequation; problem for linear differential/algebraic hence, optimal conwhere its formulation results in the solution given by the niques: the previously mention approach is recursive that sentation ofefficient nonlinear systemsimplementation. in aa polynomial polynomialThe formrepremay where allows for of computer differential/algebraic Riccati equation; hence, given optimal conformulation in thetosolution byconthe sentation nonlinear systems in form may trol for forits linear systemsRiccati isresults considered be a solved solved problem; differential/algebraic equation; hence, optimal allows for ofefficient computer implementation. The reprebe possible by using different polynomial models, which sentation nonlinear systems in a polynomial form may trol linear systems is considered to be a problem; differential/algebraic Riccati equation; hence, optimal conbe possible by using different polynomial models, which trol for linear systems is considered to be a solved problem; however, the the nonlinear optimal optimal control control is is an open open issue. issue. sentation of approximate nonlinear systems in system a polynomial formwhich may however, be by using different polynomial are possible used to to the real real asmodels, polynomial trol for linear systems isoptimal considered to beisa an solved are used approximate the system as aa polynomial however, the nonlinear nonlinear control an openproblem; issue. be possible by using different polynomial models, which ones. The identification of dynamical models obtained ofare used to approximate the real system as a polynomial In this this sense, sense, the state-dependent Riccatiisequation equation (SDRE) ones.used The identification ofthe dynamical models obtained of- however, the the nonlinear optimal control an open(SDRE) issue. Riccati are approximateof real system as ais polynomial fline from from measurements or dynamical experimental data motivated ones. Theto identification models obtained of- In In this sense, the state-dependent state-dependent Riccati equation (SDRE) (L. Lin and Liang. (2015)) extends the LQR approach to fline measurements or experimental data is motivated (L. Lin and Liang. (2015)) extends the LQR approach to ones. The identification of dynamical models obtained ofthis sense, the state-dependent Riccati equation (SDRE) by using using the results for for orcontrol control purposes. On the other other In fline from the measurements experimental dataOn is motivated (L. Lin and Liang. (2015)) extends the LQR approach to the nonlinear case by allowing the matrices involved by results purposes. the the nonlinear case by allowing the matrices involved to fline from measurements experimental dataOn is controllers motivated by using results for orcontrol purposes. the other (L. Lin and Liang. (2015)) extends LQR approach to hand, it is isthe always of interest interest to design design efficient be functions functions ofcase state variables andthe transforming inputthe nonlinear by variables allowing the matrices involved to hand, it always of to efficient controllers be of state and transforming inputby using the results for control purposes. Onthe thesystem other the nonlinear case by allowing the matrices involved to in order to obtain a satisfactory behavior of hand, it is always of interest to design efficient controllers affine nonlinear systems into the so-called so-called state-dependent functions of systems state variables and transforming inputin order toalways obtainofa interest satisfactory behavior of the system be affine nonlinear into state-dependent hand, it isto design efficient controllers functions of systems state(SDCF) variables and transforming inputvariables and at the the same to time reach anofoptimization optimization in order obtain a satisfactory behavior the system be affine nonlinear into the the so-called state-dependent coefficient factorized nonlinear systems (Cloutier variables and at same time reach an coefficient factorized (SDCF) nonlinear systems (Cloutier in order Ato obtain a satisfactory behavior ofoptimization the system affine variables and at the same with time respect reach an nonlinear systems into the so-calledsystems state-dependent criteria. feasible approach these objectives (1997)), (Ornelas-Tellez et al. (2013)),(Cimen (2010)), a coefficient factorized (SDCF) nonlinear (Cloutier criteria. A feasible approach with respect these objectives (1997)), (Ornelas-Tellez et al.nonlinear (2013)),(Cimen (2010)), a variables atoptimal the same timetheory. reach an optimization criteria. Aand feasible approach with respect these objectives coefficient factorized (SDCF) systems (Cloutier is the use of the control linear like representation for nonlinear systems used to (1997)), (Ornelas-Tellez et al. (2013)),(Cimen (2010)), a is the use of the optimal control theory. linear like representation for nonlinear systems used to criteria. approach withtheory. respect these objectives (1997)), (Ornelas-Tellez et al. (2013)),(Cimen (2010)), a is the useAoffeasible the optimal control solve the the associated HJB equation. In essence, theused SDCF likeassociated representation for nonlinear systems to The optimal control is one one particular branch of of modern linear solve HJB In the SDCF is theoptimal use of the optimal control theory.branch linear likeassociated representation foris nonlinear systems used to The control is particular solve the HJB equation. equation. In essence, essence, thefor SDCF nonlinear control technique a systematic way synThe optimal control is one particular branch of modern modern control theory, which has the objective of determining the nonlinear control technique is a systematic way forSDCF synthe associated HJB equation. In essence, thefor control theory,control which is hasone theparticular objective of determining the solve nonlinear control technique is a systematic way synThe optimal branch of modern control theory, which has the objective of determining the nonlinear control technique is a systematic way for syncontrol has the objective of determining the Hosting by Elsevier Ltd. All rights reserved. 2405-8963theory, © 2018,which IFAC (International Federation of Automatic Control)
Proceedings, 2nd IFAC Conference on 431 Proceedings, 2nd IFAC Conference on 431 Control. Peer reviewIdentification under responsibility of International Federation of Automatic Modelling, and Control of Nonlinear Proceedings, 2nd IFAC Conference 431 Modelling, Identification and Controlon of Nonlinear 10.1016/j.ifacol.2018.07.324 Proceedings, 2nd IFAC Conference on 431 Systems Modelling, Identification and Control of Nonlinear Systems Modelling, Identification and Control of Nonlinear Guadalajara, Mexico, June 20-22, 2018
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thesizing optimal nonlinear feedback controllers, which mimics the controller synthesis as done for the linear case, by exploiting the nature of the nonlinear behavior throughout the state space. Notice that by using the SDCF technique the system is not being linearized around an equilibrium point, therefore the nonlinear system dynamic is completely considered; thus, the controller has a larger operation range over the system in comparison with the linear controllers; in addition, the nonlinear inherit nature of the system is exploited. In this work the least-square algorithm is used in order to identify the unknown parameters of a known nonlinear model of a dynamic system, which is achieved by obtaining the optimal values of a parameter vector θ by minimizing the identification error. The contribution of this paper is the introduction of an easy method for identifying the parameters in linear and nonlinear systems, which are linear with respect to the unknown parameters. A posteriori, the identified-parameterbased system is used for the synthesis of a nonlinear optimal controller. Simulations validate the effectiveness of the proposed methodology in identifying the parameters and for the control of a PMSM. The paper is organized as follows. Section 2 describes the parameter identification algorithm based on a recursive least-square method. Section 3 presents the robust nonlinear optimal control scheme. In section 4, the parameter identification scheme is applied to a permanent magnet synchronous motor in order to determine its unknown parameters, then based on the identified parameters, the nonlinear optimal control is applied to the system. Simulation results prove the identification and control scheme. Finally, Section 5 concludes the paper. 2. PARAMETER IDENTIFICATION ALGORITHM 2.1 Uncertain nonlinear systems Consider the uncertain in the parameters nonlinear system given as X˙ = W (X , u, d)θ∗ (1) Y = h(X ) where X ∈ Rn is the state vector, Y ∈ Rp is system output and u ∈ Rm is the system input; matrix W (X , u, d) is a nonlinear map containing states (X ), inputs (u) and bounded external disturbances (d); h is the system output map. Term θ∗ represent the parameters’ vector of the system. Note that (1) is linear with respect to θ∗ , which is a key feature in the parameters’ identification algorithm. In order to design a model-based control scheme for system (1), it is necessary to know the parameters of the system θ∗ , nonetheless such parameters are generally difficult to determine for physical systems. This work is focused in proposing an easy parameters’ identification scheme to determine θ∗ such that a model-based controller can be designed. For explanation purposes, vector θ∗ will be assumed as one containing the real parameters of a physical system, named for simplicity as the system ideal parameter vector. It is important pointing out that for the parameters’ identification algorithm, this paper considers 432
to (1) as the physical system and the corresponding methodology will be evaluated via simulations. Currently, work is progressing to have experimental results. 2.2 Parameter Identification Algorithm For the determination of the system parameters, consider an identifier system given as x˙ = W (x, u, d)θ (2) y = h(x) where x ∈ Rn is the identifier state vector, θ is the vector to be on-line adapted such that it converges toward θ∗ by means a recursive least-square algorithm (RLSA) (Sastry and Bodson (2011)), which optimally estimate the parameters’ vector θ by minimizing the identification error given as = x − X. (3) For the RLSA design it is assumed that there exist the unknown ideal parameters’ vector θ∗ , whose dynamics can be described by θ˙∗ = 0. (4) with the output (5) y = W T (x, u, d)θ∗ Assuming the right-hand sides of (4) and (5) are perturbed by zero mean white gaussian noises with spectral intensities Υ ∈ R2n×2n and g ∈ R, respectively, the RLSA results in θ˙ = −gΨW (x, u, d) (6) ˙ Υ, g > 0 Ψ = Υ − gΨW (x, u, d)W T (x, u, d)Ψ
which is also referred as the Kalman-Bucy filter (Anderson and Moore (1971)), where θ is the estimated value for θ∗ , vector W contains the selected polynomial basis to approximate the vector fields in (1), Υ and g are fixed design parameters used to ensure the convergence of the identification error. Matrix Ψ is of appropriate dimension and is the so-called covariance matrix and it is involved in the update law for θ; the initial conditions for Ψ are arbitrarily selected, usually chosen to reflect the confidence in the initial estimate value for θ(0), being Ψ(0) a symmetric and positive definite matrix. It is important to notice that the adaptation law (6) must be implemented for each state variable. 2.3 SDCF Nonlinear Systems Extended linearization (Friedland. (1996)), also known as apparent linearization (Wernli and Cook (1975)) or state-dependent coefficient factorization SDCF (L. Lin and Liang. (2015)),(Ornelas-Tellez et al. (2013)), is the process of factorizing a nonlinear system into a linearlike structure, which contains state-dependent matrices. To obtain such factorization, consider that system (2) can be decomposed and presented as x˙ = f (x, θ) + B(x, θ)u + Γ(θ) (7) y = h(x) where Γ is considered an external input containing possible disturbances; functions f (x, θ), B(x, θ) and h(x) are smooth of appropriate dimensions. Under the assumptions that f (·) ∈ C 1 , h(·) ∈ C 1 , f (0) = 0 and h(0) = 0, a
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continuous nonlinear matrix-valued functions exist such that f (x, θ) = A(x, θ)x and h(x) = C(x)x (Shamma and Cloutier (2003)); then, system (7) can be rewritten as x˙ = A(x, θ)x + B(x, θ)u + Γ(θ) (8) y = C(x) x. It is worth remarking that the factorizations A(x)x and C(x)x are not unique (Banks et al. (2007)). To obtain well-defined control schemes, appropriate factorizations for this representation should be determined such that the controllability and observability properties are fulfilled for system (8), as established in (Banks et al. (2007)) and (Ornelas-Tellez et al. (2014)). Remark. For the parameters’ identification, an arbitrary control input u must be used to excite the dynamics of systems and (2) to achieve the convergence of θ toward to θ∗ . By considering that θ has been determined, for easy of notation in the controller design in the next section such parameter dependence in (8) will be omitted in the system description. 3. NONLINEAR OPTIMAL TRACKING CONTROL
433
T
with xa = [Φ, x] , then system (8) is rewritten as an augmented system given as x˙ a =Aa (xa ) + Ba (xa )u + D (12) ya =Ca (xa )xa where 0 C 0 , Ba = , Aa (xa ) = 0 A(x) B (13) −r Da = . Ca = [ 0 C ] , Γ
For system (12), the control scheme proposed is considered to be optimal in the sense of minimizing the performance index 1 ∞ t Φ QI Φ + eT Qe + uT Ru dt (14) J= 2 t0 where matrix Q is matrix positive definite weighting the performance of the state vector, while R also a positive definite matrix weighting the control effort expenditure, if more importance is given to the state performance, a greater value for Q could be selected or reducing the value of R. The controller design is based on the following assumptions: H1. The pair [A(x), B(x)] is controllable and the pair [A(x), C(x)] is observable.
Nonlinear optimal tracking control (Ornelas-Tellez et al. (2014)), extends the nonlinear optimal control methodology by using the SDCF in order to track a time-varying reference signals. The problem considered is the infinitehorizon tracking control (Carlson et al. (1987)) of nonlinear systems, derived from the state-regulator problem (Athans and Falb (2007)), which has the objective of “keeping the state near the desired value”, then the minimum principle is used in order to obtain the necessary conditions for the optimal control. In order to give more robustness to the controller, an integral action is added to the control law in terms of the tracking error e, in such a way that constant parametric uncertainties as well as external disturbances may be rejected, and therefore, enables this control scheme for real-time applications. 3.1 Optimal Tracking Controller Design For system (8), let us consider a trajectory tracking error defined as e =r − y (9) =r − C(x)x where r is the desired reference to be tracked by the system output y. The nonlinear optimal control problem is to find a nonlinear optimal control law, such that the output of system (8) tracks a desired reference signal, in the optimal sense of minimizing the tracking error and the control effort. In order to enhance the controller robustness properties, an integral term is considered in the design, defined as ˙ = −e Φ (10) p where Φ ∈ R is a vector of integrators for a system with p outputs. Therefore, an augmented system can be stated, which includes the integrator as ˙ −e Φ x˙ a = = (11) A(x)x + B(x)u + Γ x˙ 433
H2. The state vector x is available for feedback. Assumption H1 is related to the existence and uniqueness of the Riccati equation solution (Anderson and Moore (1990); Sepulchre et al. (1997); Kalman (1960)). Assumption H2 implies the full state vector is available from measurements, but when it is unavailable, an observer could be used for estimating x, however, this issue is beyond the scope of this paper. Under assumptions H1 and H2, the nonlinear optimal controller (15) u∗ (xa ) = −R−1 BaT (P (xa ) − z(xa )) achieves robust trajectory tracking for system (8) along the desired trajectory r, as established in (Ornelas-Tellez et al. (2017)), where P (xa ) is the solution of the SDRE and z(xa ) is the solution of a vector differential equation, respectively where P˙ (xa ) = − Qa + P (xa )Ba R−1 BaT P (xa ) (16) − ATa (xa )P (xa ) − P (xa )Aa (xa ) with QI 0 (17) Qa = 0 C T QC z(x ˙ a ) = − [Aa (xa ) − Ba R−1 BaT P (xa )]T z(xa )
(18) + P Da − CaT Qr with boundary conditions P (x(∞)) = 0 and z(x(∞)) = 0. Fig 1 depicts the flow chart of the proposed parameter identification process for dynamic systems which are linear with respect to their parameters. The parameters’ identification process is done off-line, and once the system parameters have been identified, the state-feedback optimal control law is applied. It is important mentioning that this paper considers the unknown system parameters to be determined are constant, which is in general common in physical systems; for the case when the parameters are
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1 R where θ1 = fjr , θ2 = 3λ 2j , θ3 = j , θ4 = Ld , θ5 = 1, θ6 = R λ 1 θ7 = Lq , θ8 = 1, θ9 = Lq and θ10 = Lq .
Nonlinear identifier x˙ = W (x, u, d)θ
1 Ld ,
T
with the state vector x = [ x1 x2 x3 ] identifies to the actual state vector X . Which supposes that the actual system parameters are unknown and only the state variables are available for measurements. The vector θ for each state variable is identified by using (6) with the identification error = x − X .
Recursive least squares to identify θ∗ θ˙ = −gΨW (x, u, d) ˙ = Υ − gΨW (x, u, d)W T (x, u, d)Ψ Ψ Parameter convergence θ → θ∗ Nonlinear optimal control u∗ (xa ) = −R−1 BaT (P (xa ) − z(xa )) Fig. 1. Flow chart of the parameter identification procedure for the model-based controller design. piece-wise constant or slowly time-varying, the RLSA can be used, however, a faster convergence of the RLSA must be guaranteed. 4. APPLICATION PERMANENT MAGNET SYNCHRONOUS MOTOR The parameters’ identification methodology and controller design is illustrated in a nonlinear system, the PMSM, which is linear with respect to its parameters. A commonly used reference frame for PMSM models is the synchronous reference frame, also known as dq-frame. One of the main features of this reference frame is the fact that the required calculations for analysis and control tasks are simplified since the systems values are constant and not time-varying as in the abc-frame. Therefore, the chosen dynamic model for the PMSM (Yaramasu and Wu (2017)) is described in dq-frame as fr 3 Tm x˙ 1 = − x1 + P (λx3 + (Ld − Lq )x2 x3 ) − j 2j j R Lq Vd x˙ 2 = − x2 + P x1 x3 + (19) Ld Ld Ld λ Ld R Vq x˙ 3 = − P x1 − P x1 x2 − x3 + Lq Lq Lq Lq where x1 , x2 , x3 are the rotational speed, d -axis current and q-axis current respectively, R is the resistance of the stator winding, Ld , Lq are the stator dq-axis inductances, λ is the magnetic flux produced by the permanent magnets, P is the number of pole pairs, Vd , Vq are the dq-axis stator voltages; f r, J, Tm are the viscous friction, moment of inertia and mechanical torque of the shaft, respectively. 4.1 Polynomial parameter identification for the PMSM Based on the nonlinear model (19), the following polynomial identifier model structure is proposed x˙ 1 = −θ1 x1 + P θ2 x3 − Tm θ3 x˙ 2 = −θ4 x2 + P θ5 x1 x3 + θ6 Vd (20) x˙ 3 = −θ7 x3 − P θ8 x1 x2 − P θ9 x1 + Vq θ10 434
The parameters used in the simulation are g1 = 100, g2 = 0.1 and g3 = 0.1, with the selected bases T T W1 = [ x1 x3 1 ] , W2 = [ x2 x1 x3 1 ] and W3 = T [ x3 x1 x2 x1 ] . The identifier parameters to ensure the identification convergence are Υ1 = diag [ 10 10 10 ], Υ2 = diag [ 10 1 1 ] and Υ3 = diag [ 10 1 1 1 ]. Thus it is possible to estimate the convergence of vector θ, towards the ideal (real) physical parameters of system (19) which a priori where considered to be unknown. Table 1 depicts the identified parameters of the PMSM model (20) while Table 2 displays the real physical values of the system (19). Figures 4, 5 and 6 display the convergence of the vector θ toward the real values of the system. Table 1. PMSM identified parameters Friction coefficient Pole pairs Magnetic flux Moment of Inertia Stator Resistance d-axis inductance q-axis inductance Mechanical torque
0.00185798 N.m.s 4 0.1929967 W b 0.00250013 kg.m2 2.4802995 Ω 0.11177605 H 0.11201381 H 4 N.m
Table 2. PMSM parameters Friction coefficient Pole pairs Magnetic flux Moment of Inertia Stator Resistance d-axis inductance q-axis inductance Mechanical torque
0.00186 N.m.s 4 0.193 W b 0.0025 kg.m2 2.48 Ω 0.113 H 0.113 H 4 N.m
4.2 Nonlinear Optimal Control Based on the scheme proposed in (8)–(11), the augmented system including an integral term of the error is defined as y−r x˙ a = (21) A(x)x + B(x)u + Γ and the system matrices can be rewritten as presented in (13), with 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 fr 3pλ , Ba = (22) Aa = 0 0 0 0 0 0 − J 2J 1 ωLq R 0 0 0 0 − L d L d Ld 0 R d 0 L1q 0 0 0 − Lλ1 − ωL − Lq Ld 0 0 0 1 0 0 T Ca = 0 0 0 0 1 0 , Γa = −r TJm 0 0 0 0 0 0 0 1
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4.3 Nonlinear robust optimal control To evaluate the performance of the designed controller a simulation of the PMSM model is developed in Wolfram/Mathematica environment. The parameters used in the simulation are shown in Tables 1 and 2. The rotational speed reference value ωref is selected at 100rad/s while the d -axis current is controlled at 8A. The initial states are x(0) = 0 and the controller parameters QI , Q and R are as follows 20 0 0 10 0 0 0.01 0 QI = 0 100 0 Q = 0 10 0 R = . 0 0.05 0 0 1 0 0 1
400
Vq 200
Vd
0 300
200 0
300
400 600 0.0
Figure 2 illustrates the machine rotational speed response, the solid line corresponds to the actual rotational speed, and the dashed line to the desired reference value. The state converges to the reference without steady-state error due to the integral term added to the control design.
0.1
0.2
600
800 0
1
2
3
5
4
Time s
Fig. 4. Control actions applied to the PMSM
120
400
80
101
Θ i Identified Real
Ω 300
Ωref Identified value
Rotational speed rad s
Figure 4 shows the control actions used to drive the system to the desired reference values. It can be seen that a quick convergence is achieved with small overshoot. These results prove the efficiency of the designed controller based on the identified parameters to control the machine modeled with ideal values.
Voltage V
where the system parameters correspond to the ones obtained by means of the proposed identifier.
435
100
40 99 0
1
2
3
4
5
200
100
0
0
1
2
3
5
4
0
Time s
0
500
Fig. 2. Rotational speed of the PMSM Figure 3 shows the d -axis current. At t = 3s the desired current is reached and the error is almost zero.
1000
Time (s)
1500
2000
Fig. 5. Convergence of θ1 , θ2 , θ3 respectively toward the ideal physical values
12
Id Identified value
Current A
9 8.1
6
8.0
7.9
3
Θ i Identified Real
20
Idref 15 10 5
7.8 0
1
2
3
4
5
0 0
0 0
1
2
3
4
5
Time s
500
1000
Time (s)
1500
2000
Fig. 6. Convergence of θ4 , θ5 , θ6 respectively toward the ideal physical values
Fig. 3. d -axis current of the PMSM 435
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25
Θ i Identified Real
Identified value
20 15 10 5 0 0
500
1000
Time (s)
1500
2000
Fig. 7. Convergence of θ7 , θ8 , θ9 , θ10 respectively toward the ideal physical values 5. CONCLUSIONS In this paper, the least square algorithm is proposed in order to identify the physical parameters for systems which are linear with respect to their parameters. Subsequently based on this offline parameters identification an optimal robust controller was designed for a permanent magnet synchronous motor. Simulation studies demonstrated that the designed controller can effectively achieve the desired tracking objectives by means of the identified parameters of the PMSM instead of the ideal ones. REFERENCES Anderson, B. (1990). Optimal Control Linear Quadratic Methods. Dover Publications, Englewood Cliffs, New Jersey, USA. Anderson, B. and Moore, J. (1990). Optimal Control: Linear Quadratic Methods. Prentice-Hall, Englewood Cliffs, NJ, USA. Anderson, D. and Moore, J.B. (1971). The Kalman-Bucy Filter as a True Time-Varying Wiener Filter, volume SMC-1. IEEE Transactions on Systems, Man, and Cybernetics. Athans, M. and Falb, P.L. (2007). Optimal Control an Introduction to the Theory and its Applications. Dover, New York, USA. Banks, H.T., Lewis, B.M., and Tan, H.T. (2007). Nonlinear feedback controllers and compensators: a statedependent Riccati equation approach. Computational Optimization and Applications, 37(2), 177–218. Bellman, R.E. (1957). Dynamic Programming. Princeton University Press, Princeton, USA. Carlson, D., Haurie, A., and Leizarowitz, A. (1987). Infinite Horizon Optimal Control. Springer-Verlag, Berlin, Germany. Cimen, T. (2010). Systematic and effective design of nonlinear feedback controllers via the state-dependent riccati equation (sdre) method. Annual Reviews in Control, 34(1), 32 – 51. Cloutier, J. (1997). State-dependent riccati equation techniques: an overview. In Proceedings of the 1997 American Control Conference (Cat. No.97CH36041), volume 2, 932–936 vol.2. Friedland., B. (1996). Advanced Control Systems Design. Prentice Hall, USA. 436
Kalman, R.E. (1960). Contributions to the theory of optimal control. Bol. Soc. Mat. Mexicana, 5, 102–119. Kirk, D.E. (2004). Optimal Control Theory an Introduction. Dover Publications, New York, USA. L. Lin, J.V. and Liang., Y. (2015). Analytical representation of the state-dependent coefficients in the sdre/sddre scheme for multivariable systems. In Automatica, volume 59, 106–111. Laroche, E. and Sedda, E. (2008). Methodological insights for online estimation of induction motor paramteres. Transactions on Control Systems Technology, 16(5). Ornelas-Tellez, F., Rico, J.J., Espinosa-Juarez, E., and Sanchez, E.N. (2017). Optimal and robust control in DC microgrids. IEEE Transactions on Smart Grid, 1– 12. Ornelas-Tellez, F., Rico-Melgoza, J.J., and Sanchez, E.N. (2013). Optimal tracking for a class of nonlinear systems based on the state-dependent riccati equation. In 2013 10th International Conference on Electrical Engineering, Computing Science and Automatic Control (CCE), 42–47. Ornelas-Tellez, F., Rico, J.J., and Ruiz-Cruz, R. (2014). Optimal tracking for state-dependent coefficient factorized nonlinear systems. Asian Journal of Control, 16(3), 890–903. Rico, J. and Heydt, G. (2001). An algebraic approach for identifying operating point dependent parameters of synchronous machines using orthogonal series expansions. IRE Transactions on Energy Conversion, 16(1). Sastry, S. and Bodson, M. (2011). Adaptive Control, Stability, Convergence and Robustness. Dover, New York, United States. Sepulchre, R., Jankovic, M., and Kokotovi´c, P.V. (1997). Constructive Nonlinear Control. Springer-Verlag, Berlin, Germany. Shamma, J. and Cloutier, J. (2003). Existence of sdre stabilizing feedback. IEEE Transactions on Automatic Control, 48(3), 513–517. Stinga, F. and Soimu, A. (2015). Online estimation and control of an induction motor. 19th Conference on System Theory and Computing. Wernli, A. and Cook, G. (1975). Suboptimal control for the nonlinear quadratic regulator problem. Automatica, 11. Yaramasu, V. and Wu, B. (2017). Model Predictive Control of Wind Energy Conversion Systems. John Wiley and Sons, New Jersey, United States.