Object-oriented components for simulation of adaptive controllers

The35th IEEE Conference on Decision and Control CDC'96,Kobe,Japan,December11-13,1996.Object-Oriented Componentsfor Simulation of Adaptive ControllersMattias Grundelius Sven Erik Mattsson Karl J.ÅströmDepartment of Automatic Control,Lund Institute of TechnologyBox118,S-22100LUND,SwedenE-mail:mattias@control.lth.se svenerik@control.lth.se kja@control.lth.seAbstractThe ideas of object orientation are convenient to describe complex technical systems.This paper de-scribes how an adaptive controller is conveniently structured in this way.This approach isflexible and safe.It also permits reuse and easy modification of models through the mechanisms of inheritance and specifications.The particular implementation is based on the language Omola and the interactive simulation environment OmSim.1.IntroductionA basic aim of object-oriented modeling is to support reuse so a model component can be used as a part in different applications to solve a variety of problems.A user can make the model simply by combining com-ponents,if there are good model libraries available. This paper presents an object-oriented approach to definition of adaptive controllers.The modeling language used is Omola[1,8,7].It supports object-oriented structuring concepts.Mod-els can be decomposed hierarchically with well-defined interfaces that describe interaction.All model components are represented as classes.Inher-itance and specialization support easy modification. Omola supports behavioural descriptions in terms of differential-algebraic equations(DAE),differential equations and difference equations.The behavioural descriptions may also contain discrete event ele-ments.The primitives for describing discrete events allow implementation of high level descriptions as Petri nets and Grafcet.An interactive environment called OmSim supports modeling and simulation[2]. Omola and OmSim has been used in applications to model and simulate power networks,power genera-tion,process systems,towed sonar arrays etc.The paper givesfirst an introduction to Omola.A modularized approach to adaptive control is then presented and the internals of the model components are described.An application is given in[6].2.Introduction to OmolaTo introduce Omola,we will discuss definition of model components whose behaviour is given as a continuous time transfer functionG(s)b0s n+b1s n−1+⋅⋅⋅+b nA transfer function can be parameterized in severalways.Another form isf1s n−1+⋅⋅⋅+f nG(s) d+(2)q n+a1q n−1+⋅⋅⋅+a nwhere q is the forward shift operator.We can reuse SISOShell and add a behaviour:DSISO ISA SISOshell WITHCLOCK ISAN InputEvent;x TYPE DISCRETE Column[n];y0TYPE DISCRETE Real;WHEN CLOCK DOnew(x):=-trans(A[2..n+1)*x[1]+[x[2..n];zeros(min(n,1),1)]+trans(F)*u;new(y0):=x[1]+d*new(u);END;y=y0;END;The input terminal CLOCK defines the sampling in-stants.It allows synchronization with other compo-nents.The event body‘WHEN CLOCK DO actions END’defines what happens each time the event CLOCK istriggered.The construct‘new(x)’refers to the valueof x immediately after the event,while‘x’refers tothe value immediately before the event.It is alsonecessary to define the values of x and y0betweenthe sampling instances.Here it is natural to let themkeep their values constant between the sampling in-stances,i.e.,to assume zero-order hold.This is de-clared by the keyword DISCRETE.The values of dis-crete variables can only be changed as an effect ofafired discrete event.Note that different events areallowed to set same variables.An event may trigger many events.When this hap-pens the actions are merged and sorted.An eventdefinition need not explicitly define every single ele-ment of the state update.A continuity assumption isused to deduce equations such as‘new(v):=v’forall variables that are not assigned explicitly in theevent definitions,and that cannot be deduced fromthe continuous time part equations.This means thatmost event actions can be defined in terms of vari-ables that are local in the same model component asthe event definition.The global effect of the event is deduced automatically from the connections of the structured model.Continuous time relations such as‘y=y0’are as-sumed to give constraints on continuous time vari-ables.This means that‘y=y0’actually implies ‘y:=y0’and that y can be deduced to be a discrete time variable.OmSim handles this automatically in an efficient way.We could have avoided the introduc-tion of y0by redefining y as a discrete time variable.3.ModularizationIn the object-oriented modeling approach wefirst identify the model components and define their inter-faces.The behaviour of each component type is then described.The behavior of a component can either be given as equations or as interconnected components. Our modeling approach is thus hierarchical.An adaptive controller can be built in many ways. See e.g.the textbook[3].Here we will focus on an indirect self-tuning regulator.The object diagram in Fig.1identifies the key components of an adaptive controller and their interactions.3.1Process ModelThe modules Estimation and Design must agree on model structure.To be specific,the following single-input,single-output model is usedA(q)y(t) B(q)u(t)(3)where y is the output and u the input of the process. Estimation estimates process parameters and trans-fers them to Design.A terminal class defined asModelPars ISA RecordTerminal WITHA,B ISA DiscrRowTerm;Update ISAN EventTerminal;END;Figure1An indirect self-tuning regulator.supports a mailbox scheme for passing parameters. When Estimation has new estimates available it puts them in A and B and triggers Update.A connection between two simple terminals T1and T2mean that their values should be equal,T1=T2. The model compilation procedure deduces the com-putational causality automatically.There is also a class ZeroSumTerminal which is used for terminals that should sum to zero.For a composite terminal, i.e.,a terminal that is a subclass of RecordTerminal, a connection means that the components are con-nected.A connection also propagates information on dimen-sionality of simple terminals.For example,assume that the model parameter terminals of Estimation and Design are called MP.It is sufficient to define the dimensions of A andB of Estimation.MP ex-plicitly.A connection between Estimation.MP and Design.MP then implicitly defines the dimensions of Design.MP.A and Design.MP.B.OmSim will auto-matically deduce the regulator order,if Design is provided with relations between model order and regulator order.3.2Control structureDesign and Controller must agree on control struc-ture and parameters used.A general linear,discrete time controller can be described byR(q)u(t) T(q)u c(t)−S(q)y(t)(4) where R,S and T are polynomials.To allow passing of controller parameters,we intro-duce the terminal classControllerPars ISA RecordTerminal WITHR,S,T ISA DiscrRowTerm;Update ISAN EventTerminal;END;Controller includes the sampling clock.It is sup-posed to make measured values and calculated con-trol value available to Estimation: ProcessDataTerminal ISA RecordTerminal WITH u,y ISA SimpleTerminal;Update ISAN EventTerminal;END;3.3The modulesThe controller(4)is a MISO system which can be implemented as a simple extension of the SISO sys-tem described in Section2.Multiple inputs are read-ily handled by an observability canonical realization. We will here focus on the design and estimation mod-ules.4.The Design ModuleAs an example of a design module let us discuss minimum-degree pole placement[3,pp.92–102]. 4.1Minimum-degree pole placementLet the process model be(3),where the polynomial B is factored as B+B−with B+being monic.Let the desired closed loop response be specified byA m(q)y(t)B m(q)B−(q)u(t)(5) where deg A m deg A and deg B m deg B+.The controller is(4)where R R′B+and S is the solution to the Diophantine equationAR ′+B−S AoA m(6)where A0is a given polynomial with deg A0 deg A−deg B+−1.Form T A0B m.There are some interesting special cases.•All zeros cancelled.It is natural to choose B mA m(1)q d0where d0 deg A−deg B.ThenB− b0,B+ B/b0and T A m(1)q d0/b0.The closed-loop polynomial becomes B+A0A m.This approach requires that all process zerosare stable and well damped.•No zeros cancelled.Take B+ 1,B− B andB m βB,whereβ A m(1)/B(1).The closed-loop characteristic polynomial is A0A m.4.2ImplementationThe interface of the design module consists of one terminal MP for obtaining estimated parameters and one terminal CP to pass on calculated controller parameters:DesignShell ISA Model WITHMP ISA ModelPars;CP ISA ControllerPars;END;An implementation of a minimum degree controller with no cancellation of zeros is given byMDNZC_Design ISA DesignShell WITHdegAo TYPE Integer:=MP.A.n-2;CP.R.n:=MP.A.n-1;CP.S.n:=MP.A.n-1; CP.T.n:=degAo+1;Ao ISA RowPar WITH n:=degAo+1;END;Am ISA RowPar WITH n:=MP.A.n;END;nRS TYPE Integer:=CP.R.n+CP.S.n;RS TYPE DISCRETE Row[nRS];Am1,B1TYPE DISCRETE Real;WHEN MP.Update DOnew(RS):=diophant(MP.A,MP.B,mult(Am,Ao));new(CP.R):=new(RS[1..CP.R.n]);new(CP.S):=new(RS[CP.R.n+1..nRS]);new(Am1):=Am*ones(MP.A.n,1)new(B1):=B*ones(MP.B.n,1);new(CP.T):=new(Am1/B1)*Ao;schedule(CP.Update,0);END;END;The function‘diophant(MP.A,MP.B,mult(Am,Ao))’returns a solution to(6).The numerical routine is based on[4].The call schedule(E,delay_time) implies that the event E should be triggered at a time delay_time after this event.In this case the delay time is zero,which mean that CP.Update is triggered immediately after the controller parameters have been calculated.To obtain a synchronization with the receiver of the parameters,the code should look like %...WHEN MP.Update CAUSE CP.Update DO%as above,but no schedule statement END;%...However,in this case Design needs no synchroniza-tion,since the informationflow is unidirectional.The design procedure does not refer to any quantities in the receiving controller.5.The Estimation Module Recursive least square is a simple estimation proce-dure(see e.g.,[3,p.53]).5.1Recursive least square estimation Recursive least squares estimation with exponential forgetting is given by the equationsK(t) P(t−1)ϕ(t)/(λ+ϕT(t)P(t−1)ϕ(t)θ(t) θ(t−1)+K(t)(η(t)−ϕT(t)θ(t−1))P(t) (I−K(t)ϕT(t))P(t−1)/λFor a model model of the form(3)we haveϕT(t) [−y(t−1)...−y(t−n a)u(t−d)...u(t−d−n b)]η(t) y(t)θT(t) [a1...a n a b0...b n b] where d deg A−deg B.The variablesη,ϕandθhave other interpretations for other models.How-ever,the same estimation algorithm can be used.It is thus useful to decompose the module into two parts,one that implements the estimation algorithm and another part that forms the regression vector,super-vises the estimation and handles starts and stops.It is then possible to obtain more sophisticated al-gorithms by adding regression filters and other fea-tures.5.2ImplementationThe interface of the estimation module consists of one terminal P for the process values and one termi-nal MP to pass on the estimated model parameters.It should be easy to exchange the modules.Inheri-tance and redefinition support this requirement.For this purpose we first define a shell model,which specifies terminals,components and their connec-tions.Let us call it EstimationShell .This is con-veniently done using a graphical editor in OmSim to draw an object diagram.See Fig.2.The component EstCont is of the class EstContShell and the compo-nent EstAlg is of the class EstAlgShell .These two shell models define terminals.The terminals con-necting EstCont and EstAlg is of the classEstComTerminal ISA RecordTerminal WITH eta,e ISA DiscreteTerminal;phi,theta ISA DiscrVectorTerm;Newphi,Newtheta ISA EventTerminal;END;It handles communication between the estimation al-gorithm and the control module,which are assumed to collaborate in the following way1.When EstimationShell.P.Update is fired ex-ternally this indicates that new measurements are available in EstimationShell.P .2.EstCont sets phi (ϕ)and eta (η)and triggers Newphi .3.EstAlg calculates a new parameter estimate theta (θ)and the prediction error e and trig-gers Newtheta .4.EstCont sets MP.A and MP.B and triggers MP.Update to indicate that new parameter es-timates areavailable.Figure 2An object diagram for the estimation module.A recursive least squares module given byRLS_Algorithm ISA EstAlgShell WITH lambda ISA Parameter;nPar TYPE Integer :=EC.theta.n:th0ISA VectorPar WITH n :=nPar;END;P0ISA SqMatrixPar WITH n :=nPar;END;P TYPE DISCRETE Matrix[nPar,nPar];K TYPE DISCRETE Column[nPar];WHEN Init DOnew(EC.theta):=th0;new(P):=P0;schedule(EC.Newtheta,0);END;WHEN EC.Newphi DO new(K):=P*EC.phi/(lambda +trans(EC.phi)*P*EC.phi);new(EC.e):=EC.eta -trans(EC.phi)*EC.theta;new(EC.theta):=EC.theta +new(K*EC.e);new(P):=(P -new(K)*trans(EC.phi)*P)/lambda;schedule(EC.Newtheta,0);END;END;A simple estimation controller is given bySISO_EstCont ISA EstContShell WITH degA,degB TYPE Integer;nPar TYPE Integer :=degA +degB +1;MP.A.n :=degA+1;MP.B.n :=degB+1;EC.theta.n :=nPar;EC.phi.n :=nPar;WHEN P.Update DOnew(EC.eta):=P.y;schedule(EC.Newphi,0);END;WHEN EC.Newtheta DOnew(EC.phi):=[-P.y;EC.phi[1..degA-1];P.u;EC.phi[degA+1..nPar-1]];new(MP.A):=[1,trans(EC.theta[1..degA])];new(MP.B):=trans(EC.theta[degA+1..nPar]);schedule(MP.Update,0);END;END;A recursive least square estimator is given bySISO_RLS_Estimation ISA EstimationShell WITH EstCont ISA SISO_EstCont;EstAlg ISA RLS_Algorithm;END;The class specifications for EstCont and EstAlg over-writes those inherited from EstimationShell ,but all other attributes such as the connection remain unchanged.Inheritance can be viewed as a kind of parameterization where every attribute can be ex-changed by a part with a compatible interface.6.The Self-Tuning RegulatorIt is convenient to define a shell model,which spec-ifies components and their connections.See Fig.3.Estimation is of class EstimationShell and Design is a DesignShell .The complete implementation is used for RST_Controller ,since this class in itself is general.For example,an indirect self-tuning regulator based on minimum-degree pole placement with no zero can-cellation,and and recursive least square algorithm for parameter estimation is defined asISTR_MDNZC_RLS ISA ISTR_Shell WITH Design ISA MDNZC_Design;Estimation ISA SISO_RLS_Estimation;END;A self-tuning regulator has a number of parameters,such as model order and polynomials A m and A 0.To improve the interface and facilitate use,we may de-fine them as parameters of the self-tuning regulator class and express the corresponding parameters of the Design and Estimation in terms of these.7.ConclusionsThis paper has discussed use of object-oriented mod-eling for definition of components of adaptive con-trollers.The approach makes it easy to test and evaluate new designs and applications of adaptive controllers.Omola allows specification of components in a natural way.The compilation procedure trans-forms the description to a representation which is ef-ficient for numerical solution.The description is also a sound basis for generating code for real controllers.However,this is not supported by OmSim today.In-heritance and specialization make it easy to modify or exchange a eful shells are easily developed.The components have been used to sim-ulate adaptive controllers for systems with backlash [5,6].It is also used in the course Adaptive Control,see URL:http://www.control.lth.se/~kursar .OmSim is available via anonymous FTPfromFigure 3An object diagram for an indirect self-tuning controller.URL:ftp://ftp.control.lth.se/pub/cace .Cur-rently there are versions for Sun workstations and HP workstations as well as for PC’s running the Linux operating system For information on Omola and access to OmSim see WWW at URL:http://www.control.lth.se/~cace .AcknowledgementsThe work has been supported by the Swedish Na-tional Board for Industrial and Technical Develop-ment.under contract 9304688-3and the Swedish Re-search Council for Engineering Sciences under con-tract 95-956.8.References[1]M.A NDERSSON .Object-Oriented Modeling andSimulation of Hybrid Systems .PhD thesis ISRN LUTFD2/TFRT--1043--SE,December 1994.[2]M.A NDERSSON ,S.E.M ATTSSON ,D.B R ¨UCK,AND T.S CHÖNTHAL .“OmSim—An integrated environ-ment for object-oriented modelling and simula-tion.”In Proceedings of the IEEE/IFAC Joint Symposium on Computer-Aided Control System 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