Jtam-A4.dvi JOURNAL OF THEORETICAL AND APPLIED MECHANICS 53, 3, pp. 711-722, Warsaw 2015 DOI: 10.15632/jtam-pl.53.3.711 REALIZATION OF COORDINATION TECHNOLOGY OF HIERARCHICAL SYSTEMS IN DESIGN OF ACTIVE MAGNETIC BEARINGS SYSTEM Kanstantsin Miatliuk, Arkadiusz Mystkowski Bialystok University of Technology, Department of Automatic Control and Robotics, Białystok, Poland e-mail: k.miatliuk@pb.edu.pl; a.mystkowski@pb.edu.pl A cybernetic technology of mechatronic design of active magnetic bearings systems (AMB) originated from theory of systems is suggested in the paper. Traditional models of artificial intelligence and mathematics do not allow describing mechatronic systems being designed on all its levels in one common formal basis. They do not describe the systems structure (the set of dynamic subsystems with their interactions), their control units, and do not treat them as dynamic objects operating in some environment. They do not describe the environment structure either.Therefore, the coordination technology of hierarchical systems has been chosen as a theoreticalmeans for realization of design and control. The theoretical basis of the given coordination technology is briefly considered. An example of technology realization in conceptual and detailed design of AMB system is also presented. Keywords: hierarchical systems, design, coordination, mechatronic, magnetic bearings 1. Introduction In the design process of active magnetic bearings (AMB) we deal with mechatronic objects which contain connected mechanical, electromechanical, electronic and computer subsystems. Various methods andmodels which are used for each system coordination (design and control) cannot describe all subsystems in common theoretical basis and, at the same time, describe the mechanism with all interactions in the structure of a higher level and the system as a unit in its environment. It is important to define the common theoretical means which will describe all subsystems of a mechatronic object being designed (AMB systems) and its coordination (design and control) system in a common formal basis. This task is topical for the systems of computer aided design (CAD). Besides, theoretical means of the coordination technology must allow performing the design and control tasks under condition of any information uncertainty, i.e. (1) to create and changemechatronic system construction and technology by selecting units of lower levels and settling their interactions to make the state and activity of the system in higher levels (environment) best coordinatedwith environmental aims (selection stratum); (2) to change the ways (strategies) of the design task performingwhen the designed unit is multiplied and theknowledgeuncertainty is removed (learning stratum); (3) to change the abovementioned strata when new knowledge is created (self-coordination stratum). The coordination technology must also cohere with traditional forms of information repre- sentation in mechatronics, i.e. numerical and geometrical systems. The theoretical basis of the design process in agreement with these requirements must be a hierarchical construction con- necting any level unit with its lower and higher levels. Mathematical and cybernetic theories based on the set theory are incoherent with the above design requirements since the set theory describes one-level world outlook. In this paper, the coordination technology ofHierarchical SystembyMesarovich et al. (1970) with its standard block aed (ancientGreekword) byNovikava et al. (1990, 1995, 1997)Miatliuk 712 K.Miatliuk, A. Mystkowski (2003), Novikava andMiatliuk (2007) has been chosen as the theoretical basis for performing a mechatronic design task. In comparisonwith traditional methods, aed technology allows presen- tation of the designed object structure, its dynamic representation as a unit in the environment, the environment itself and the control system in common formal basis together with easy for- malization of the design process. In the paper, the aed formal basis and coordination technology of hierarchical systems are described.AMB system construction and the system conceptual and detailed design are presented as practical examples of the proposed technology. Finally, the developed technology for the design of exemplary AMB mechatronic systems is analysed. 2. Formal basis of design technology The aed model Sℓ considered below unites the codes of the two level system (Measarovic et al., 1970) and general systems theory byMesarovic and Takahara (1990), the number codeLS, geometry and cybernetics methods. The dynamic representation (ρ,ϕ) is themainmeans of the description of the named codes.Aed is a standard element of hierarchical systems (Novikava et al., 1990, 1995, 1997; Miatliuk, 2003; Novikava and Miatliuk, 2007), which realizes the general laws of systems organization on each level and the inter-level connections. AedSℓ contains ωℓ and σℓ models which are connected by the coordinator Sℓ0 Sℓ ↔{ω,S0,σ} ℓ (2.1) where ωℓ is a dynamic representation of any level ℓ ∈ LS system in its environment, σℓ is the system structure, Sℓ0 is coordinator. The structure diagram of aedS ℓ is presented in Fig. 1. Fig. 1. Structure diagram of aed – standard block of Hierarchical Systems. S0 is the coordinator, Sω is the environment, Si are subsystems, Pi are subprocesses,P l is the process of level ℓ,Xl and Y l are the input and output of the system Sl;mi, zi, γ, wi, ui, yi are interactions Aggregated dynamic representations ωℓ of all aed connected elements, i.e. the object oS ℓ, processes oP ℓ, ωP ℓ and environment ωS ℓ are presented in form of the dynamic system (ρ,ϕ)ℓ ρℓ = {ρt : Ct×Xt →Yt ∧ t∈T} ℓ ϕℓ = {φtt′ : Ct×Xtt′ →Ct′ ∧ t,t ′ ∈T ∧ t′ >t}ℓ (2.2) whereCℓ is the state,Xℓ – input, Y ℓ – output, Tℓ – time of level ℓ, ρℓ andϕℓ are the reactions and state transition functions, respectively. Dynamic representations ωℓ of the object oS ℓ, the processes oP ℓ, ωP ℓ and the environment ωS ℓ are connected by their states, inputs and outputs. Themodel of the system structure is defined as follows σℓ = {Sℓ0,{ω ℓ−1,σU ℓ}}= {Sℓ0, σ̃ ℓ} (2.3) Realization of coordination technology of hierarchical systems... 713 where Sℓ0 is the coordinator, ω ℓ−1 are aggregated dynamic models of the subsystems S ℓ−1 = {Sℓ−1i : i∈ I ℓ} of the lower level ℓ−1, σU ℓ are structural connections σU ℓ ⊃ ωU ℓ−1 = {ωU ℓ−1 i : i∈ I ℓ} of the subsystemsS ℓ−1 . σ̃ℓ is the connection of the dynamic systemsωℓ−1 and their structural interactions σU ℓ coordinated with the external ones ωU ℓ = σU ℓ+1|Sℓ. The coordinator Sℓ0 is the main element of hierarchical systems which realizes the processes of systems design and control (Novikava et al., 1995; Miatliuk, 2003). It is defined according to aed presentation of Eq. (2.1) in the following form Sℓ0 = {ω ℓ 0,S ℓ 00,σ ℓ 0} (2.4) where ωℓ0 is the aggregated dynamic realization of S ℓ 0, σ ℓ 0 is the structure of S ℓ 0, S ℓ 00 is the coor- dinator control element. Sℓ0 is defined recursively. The coordinator S ℓ 0 constructs its aggregated dynamic realizationωℓ0 and the structureσ ℓ 0 by itself.S ℓ 0 performs thedesignand control tasks on its selection, learning and self-organization strata (Miatliuk, 2003). All metric characteristics µ of systems being coordinated (designed and controlled) and the most significant geometry si- gns are determined in the frames of aed informational basis in the codes of numeric positional system LS (Miatliuk, 2003; Novikava andMiatliuk, 2007). The external connections ωU ℓ of ωℓ with other objects are its coordinates in the environ- ment ωS ℓ. The structures have two basic characteristics: ξℓ (connection defect) and δℓ (con- structive dimension); µℓ, ξℓ and δℓ are connected and described in the positional code of theLS system (Miatliuk, 2003; Novikava andMiatliuk, 2007). For instance, the numeric characteristic (constructive dimension) δℓ ∈∆ℓ of the system Sℓ is presented in theLS code as follows δ̃ℓ =(n3, . . . ,n0)δ δ̃ ℓ ∈{δℓσ,δ ℓ ω} (ni)δ =(n3−i)ξ (ni)δ ∈N i=0,1,2,3 (2.5) where δℓω and δ ℓ σ are constructive dimensions of σ ℓ and ωℓ, respectively. This representation of geometrical information allows execution of all operations with geometric images on the computer as operations with numeric codes. The aed technology briefly described above presents a theoretical basis for AMB systems design and control. In comparisonwith the two-level systemproposedbyMesarovic et al. (1970), the presented informationalmodel ofaedSℓ has newpositive characteristic features (Novikava et al., 1990, 1995, 1997; Miatliuk, 2003; Novikava and Miatliuk, 2007). Formalization, availability of the environment block ωS ℓ, description of the inter-level relations, coordination technology and information aggregation make the aed technology more efficient in the design tasks. 3. Coordination technology realization in the design of AMB system 3.1. Conceptual formal model of an AMB system Formal description of the Active Magnetic Bearing (AMB) system in aed form is an exam- ple of the Hierarchical System (HS) (aed) coordination technology realization in the conceptual design of amechatronic system. TheAMBs systems are usually used in rotatingmachinery, fly- wheels, industrial turbomachinery, etc. (Schweitzer andMaslen, 2009). In this paperwe focus on anAMBsystemwhich is a part of the experimental stand of a suspension system (Fig. 2) develo- ped at Automation andRobotics Department, Bialystok University of Technology (Mystkowski and Gosiewski, 2007, Gosiewski andMystkowski, 2006, 2008). The AMB system is presented in aed form as follows MS ℓ ↔M{ω,S0,σ} ℓ (3.1) 714 K.Miatliuk, A. Mystkowski Fig. 2. AMB-beam test rig where Mω ℓ is an aggregated dynamic representation of the AMB system MS ℓ, see Eq. (2.2), Mσ ℓ is the system structure, MS ℓ 0 is coordinator, i.e. design and control system, ℓ is the index of level. TheAMBsystemconstructionMσ ℓ contains the set of sub-systemsωℓ−1 and their structural connections σU ℓ. Thus, according toEq. (2.3), the structural subsystemspresented in aggregated dynamic form ωℓ−1 are: • front AMB – Mω ℓ−1 1 • rear AMB – Mω ℓ−1 2 • thrust passive magnetic bearing (PMB) – Mω ℓ−1 3 • shaft – Mω ℓ−1 4 . In their turn, each subsystem has its own structural elements – the lower level ℓ−1 subsys- tems. In the AMB subsystem Mω ℓ−1 1 , these are eight i=8 electromagnetic coils Mω ℓ−2 1i and the displacement sensors assembly Mω ℓ−2 1,9 which creates the external part of theAMB.The internal part is the magnetic core Mω ℓ−2 1,10 attached to the shaft. The subsystems Mω ℓ−1 are connected by their common parts – the structural connections σU ℓ−1 that are elements of lower levels. For instance, the shaft Mω ℓ−1 4 and the front AMB Mω ℓ−1 1 are connected by their common element – the magnetic core σU ℓ−1 1,4 ↔ Mω ℓ−2 1,10 ↔ Mω ℓ−2 4,1 , where Mω ℓ−2 1,10 is aggregated dynamic realization of the magnetic core being the subsystem of the front AMB Mω ℓ−1 1 , and Mω ℓ−2 4,1 the realization of the magnetic core being the subsystem of the shaft Mω ℓ−1 4 . Aggregated dynamic realizations Mω ℓ−1, i.e. dynamic models i(ρ,ϕ) ℓ−1, Eq. (2.2), of the subsystems MS ℓ−1 , are formed after definition of their inputs-outputs concerning each concrete sub-process they execute. Thus, for the shaft Mω ℓ−1 4 concerning its rotation process, the in- put MX ℓ−1 4 is the torqueM obtained from the loading system (motor), and the output MY ℓ−1 4 is the angular velocity Ω of the shaft (Fig. 2). The shaft dynamic model Mω ℓ−1 4 in this case is presented at the detailed design stage in formof the differential equation described byGosiewski andMystkowski (2006, 2008). The environment ωS ℓ of the AMB system has its own structure and contains: ωℓ1 – measuring and signal conditioning system (electronic), ωℓ2 – loading system –motor/generator (electromechanical), ωℓ3 – control systems in feedback loop of the general control AMB system (computer system). Thus, the object being controlled MS ℓ (AMB system), environment subsystems, i.e. measu- ring ωS ℓ 1 (sensors, filters, estimators), loading ωS ℓ 2 (electromotor, generator, clutch) and control systems ωS ℓ 3 in the feedback loop (computer, processor, converters DAC and ADC) create the general control AMB system. The immediate input MX ℓ for the AMB system (which is at the Realization of coordination technology of hierarchical systems... 715 same time the output ωY ℓ M = MX ℓ of the environment of the AMB system) are signals from the loading system – themotor torque and control signal, i.e. the voltage/current or fluxwhich come from internal or external controllers of the control system. The output of theAMB system is the axial displacement of the shaft in the plane orthogonal to the shaft symmetry axis, me- asured currents, flux, rotor angular speed, coil temperature, etc. The output MY ℓ of the AMB system MS ℓ, i.e. the displacement of the shaft, is at the same time the input ωX ℓ M = MY ℓ of the environment which is measured by eddy-current sensors or optical (laser) sensors. The states MC ℓ i of the AMB system MS ℓ are: Mc ℓ 1 – displacements, Mc ℓ 2 – velocities, Mc ℓ 3 – accelerations, Mc ℓ 4 – magnetic forces. The dynamic representation Mω ℓ of the AMB system is constructed in form of Eq. (2.2) by the inputs MX ℓ, states MC ℓ and outputs MY ℓ mentioned above. The dynamic representation at the conceptual stage can be given in (ρ,ϕ), which is transformed into the state-space matrix form at the detailed design stage ẋ=Ax+Bu y=Cx (3.2) The first state equation in Eq. (3.2) corresponds to the state transition function ϕ in Eq. (2.2), and the secondoutput equation corresponds to the reaction ρ.Vectorsx,y,uandmatrices A,B,C of the equations are defined byGosiewski andMystkowski (2006). Therefore, Eq. (2.2) is the dynamic representation Mω ℓ of the AMB system at the stage of conceptual design, and Eq. (3.2) is the AMB model which is used at the detailed design stage of the AMB system life circle (Ulman, 1992). TheAMBsystemprocessPℓ is apartof thehigher-level processPℓ+1 in the environmentωS ℓ, i.e. the general control AMB system. This process contains: Pℓ1 – control of the shaft displacement, vibration damping and machine diagnostics (by the AMB system MS ℓ), Pℓ2 – measuring of output values of the AMB system by themeasuring and signal conditioning system, Pℓ3 – reading of measured values and converting by the Digital Signal Processor (DSP) or any other real-time digital processor, Pℓ4 – processing and estimating, Pℓ5 – creation of the simulation model and sending it to DSPmemory, Pℓ6 – sending control signals to the AMB system in real time, Pℓ7 – AMB system loading realized by the electromotor or generator that causes rotation of the shaft or convertion of the kinetic energy. Pℓ8 – shaft rotation. Pℓ1 and P ℓ 7 are realized by electromechanical subsystems of the general mechatronic system (general control AMB system), Pℓ2-P ℓ 6 are realized by the computer subsystem, and P ℓ 8 by the mechanical one. The general process is composed of sub-processes P ℓ executed by the general control AMB system, which includes the ABM system MS ℓ and its environment ωS ℓ. So, all the subsystems of the general control AMB system, i.e. mechanical (shaft Sℓ−14 ), electromechanical (AMB system MS ℓ andmotor ωS ℓ 2), computer-electronic (measuring ωS ℓ 1 and control system ωS ℓ 3) have their aggregated dynamic ω ℓ and structural σℓ descriptions. All the 716 K.Miatliuk, A. Mystkowski connected descriptions of the subsystemsS ℓ and processesP ℓ are presented in the informational resources (data bases) of the coordinator which realizes the design process connecting in this way the structure Mσ ℓ and the functional dynamic realization Mω ℓ of the AMB system being designed. The coordinator MS ℓ 0 in our case is realized in form of an automated design and control system of the AMB, which maintains its functional modes by the control system and realizes the design process by a higher level computer aided design (CAD) system (general supervisor) if necessary. The AMB control system is designed according to the hierarchical concept and contains low-level and high-level controllers (Fig. 4). All metrical characteristics of the subsystems and processes described above are presented in form of numeric positional systems LS (Novikava et al., 1990, 1995, 1997; Miatliuk, 2003; Novikava andMiatliuk, 2007). 3.2. System architecture The hierarchical system coordination technology allows one to describe active magnetic be- arings (AMBs) coupled architecture and its coordination, i.e. design and control (Schweitzer and Maslen, 2009; Miatliuk et al., 2010a). This technology enables one to allocate the inter- subsystems in theAMBstructure. In this case, by using a novel approach, the conceptual design of theAMBsystem is considered as amultilevel model which enables introduction of further ne- cessary changes into AMB construction and technology. This approach supports the design and assembling of AMB parts and can be considered as a self-optimization process. Themain AMB model layers reflects AMB mechatronic subsystems, i.e. the mechanical subsystem, electrical subsystem and control software (supervisory intelligence), see Fig. 3. These subsystems can be Fig. 3. Structure diagram of the AMB hierarchical system constructed due tomachine demands by selecting partsωℓ−1 and setting their interactions σU ℓ, seeEq. (2.3). Thus, thewhole design process can bedivided into engineering departments accor- ding to due knowledge. For example, high dynamics of the electrical AMB subsystem (at a low level) is faster than themechanical one and requires different controller/actuators/sensors with a suitable bandwidth.Thus, these subsystems should be designedwith taking into account their specified performances according to thewhole system functional requirements. According to the Realization of coordination technology of hierarchical systems... 717 hierarchical control structure (see Fig. 1), the design technology realization steps are as follows. First, the low level (inner) closed-loop sub-system is designed in which the inner controller pro- vides a fast response of the control loop with respect to the model of the electrical part of the AMBsystem(Schweitzer andMaslen, 2009).Here, since the electrical subsystemdynamics of the AMBmodel has uncertainties and consists of nonlinearities, the nonlinear control low is realized with robust controller (Gosiewski andMystkowski, 2006, 2008). The robust controller overcomes control plant uncertainties and provides a fast response due to variations of the desired signals from the high level controller. Second, the high level control sub-system is designed based on the outer measured signals in the AMB mechanical sub-system. This high level control loop works slower than the inner controller since the dynamics of the AMB mechanical part refers to the significant inertia of AMB position control. The design process is formally presented in form of coordination strategies realized on the selection layer of the coordinator and described by the output functions λ of the coordinator canonical model (ϕ,λ) (Miatliuk, 2003). The change of coordination strategies in the coordinator learning and self-organization layers is described by the state transition functions ϕ. 3.3. Control structure The hierarchical structure of the AMB control system consists of (at least) three layers. The first one (high level) consist of a complex AMB dynamic model (nonlinear) which refers to the concrete plant system. This plant model after simplification is used for controller synthesis and refers to the abstract system Sℓ, Eq. (3.1). The second layer consists of the low level controller presented in form of the coordinator Sℓ0, Eq. (2.4), responding to the low level control task by direct impact onAMBdynamics and it is strongly nonlinear. The low level ℓ control subsystems represent a decentralized (local) control loop based on command signals from the high level ℓ+1 control system. The last layer represents a high level controller (global) given in from of Sℓ+10 coordinator which performs high order tasks. The main advantage of such approaches is the decoupling of control laws for simpler evaluation by the designing engineers. For such a control structure, the high level controller is not dependent on the nonlinearities located in the low level layer. This enables designing a linear high level controller. However, the refinement of inter- couplings due to the nonlinear nature of this dynamic system is the main challenge. Referring to the two-level control architecture as shown in Fig. 4, the plantSℓ behaviour is assumed to be described by the Mω ℓ model built on the relation of AMB inputsXℓ, outputs Y ℓ and statesCℓ, see Eq. (2.2). Cℓ is defined by the control inputs Gℓ−1 from the low level controller, i.e. the coordinatorSℓ0. Themeasured plant outputsW ℓ−1 are the feedback from the plantSℓ to the low level controllerSℓ0. The low level controllerS ℓ 0 is directly connected by its inputX ℓ 0 = {G l,W l−1} and outputY ℓ0 = {G l−1,W1}with the plantmodel andwith the high level controllerSℓ+10 where {Gl−1,W l−1} and {Gl,W l} are low level and high level signals, respectively. Similarly, the high level controller Sℓ+10 has its inputsX ℓ+1 0 = {G l+1,W l} and outputs Y ℓ+10 = {G l,W1+1} as well. Ccontrol signals of the controllers are presented in form of coordinator strategies described by the output functions λ̂ℓ0 of the coordinator canonical models (ϕ̂, λ̂) ℓ 0 (Miatliuk, 2003) built on its inputs, outputs and states as follows λ̂ℓ0t : C ℓ 0× ̂̃ X ℓ 0 → ̂̃ Y ℓ 0 (3.3) For instance, the control signal from the low-level ℓ/(ℓ−1) controllerSℓ0 to the plant is presented in form of the coordinator Sℓ0 output function λ̂ ℓ/(ℓ−1) 0t λ̂ ℓ/(ℓ−1) 0 = { λ̂ ℓ/(ℓ−1) 0t : ̂̃ C ℓ 0×W̃ ℓ−1 → ̂̃ G ℓ−1} (3.4) where ̂̃ C ℓ 0 is the controller (coordinator) states space. 718 K.Miatliuk, A. Mystkowski Fig. 4. Hierarchical AMB control architecture The change of controller states is described by the state transition function ϕ̂ ℓ 0 of the coor- dinator canonic model (Miatliuk, 2003) ϕ̂ ℓ 0 = {ϕ̂ ℓ 0tt′ : C ℓ 0×X ℓ 0tt′ →C ℓ 0} (3.5) For the current (or flux) controlled AMB, the high level controller provides the vector of 4 control currents which after biasing the vector of 8 reference currents (reference forces) are presented by the signals Gℓ (Fig. 4). The reference forces are provided to the low level control loops. The referenced voltages Gℓ−1 are input to the drives and actuators of the AMB system. The rotor displacements in the bearing planes (Wℓ−1) are estimated based on the measured rotor displacements in the sensor planes (Wℓ−1). They are provided to the low level controller. The desired rotor position is the reference signal of the high level (rotor position) controller and the desired electromagnetic force is the reference signal of the low level (current/flux) controller, respectively. In order to simplify the design of the control system, the one-degree-of-freedom (1 DOF) AMBdynamic control model (Fig. 4) is considered as the hierarchical system. Its control model is considered as a cascade of two simple systems consisting of high level (electrical) and low level (mechanical) mechatronic subsystems with their coordinators. In this case, the AMB controller structure is coupled to the position and flux feedback, which refers to global and local control loops, respectively. The given conceptualmodel of theAMBsystem is concretized at its detailed design stage. 4. Exemplary detailed design of an AMB system 4.1. Simplified AMB model At the detailed design stage which follows the conceptual one in the AMB system life circle (Ullman, 1992) the simplified 1 DOF (one degree of freedom) AMB model is used. The AMB consists of two opposite and identicalmagnetic actuators (electromagnets), which are generating the attractive forces F1 and F2, on the rotor (Schweitzer and Maslen, 2009). To control the position x of the rotor of mass m to the equilibrium state x = 0, the voltage inputs of the electromagnets V1 and V2 are used to design the control law, see Fig. 5. Realization of coordination technology of hierarchical systems... 719 Fig. 5. A simplified one-dimensional AMB (Schweitzer andMaslen, 2009) The simplifiedmechatronicmodel of theAMB is nonlinear and coupledwithmechanical and electrical dynamics. Referring to Fig. 5, neglecting gravity, the dynamic equation is given by Schweitzer andMaslen (2009) m d2x dt2 = Φ|Φ| µ0A =F(Φ) (4.1) whereΦ is the totalmagnetic flux through each active coil,A is the cross area of each electroma- gnet pole and µ0 is the permeability of vacuum (4π ·10 −7Vs/Am). Equation (4.1) corresponds to the dynamic representation (ρ,ϕ) given at the AMB conceptual design stage. The system nonlinearity in Eq. (4.1) is given by the function η(Φ) = Φ|Φ|, and it is non- decreasing. The total flux generated by the i-th electromagnet is Φi = Φ0 +φi. In the case of zero-bias operation, the bias fluxΦ0 equals zero and the total flux is equal to the control fluxφi. Then, we define the generalized flux which is given by φ :=φ1−φ2 = 1 N (∫ (V1−Ri1) dt− ∫ (V2−Ri2) dt ) i=1,2 (4.2) whereN is the number of turns of the coil of each electromagnet, V is applied control voltage, and i is current in the electromagnet with resistanceR. 4.2. Low level controller The fast inner controller (low level coordinator Sℓ0) generates the required fluxes in the AMB structure due to nonlinear characteristics of the controlled flux φ versus the generated for- ceF. Since themagnetic flux sensorsmay complicate significantly the electrical andmechanical structure of the AMB system, a low level flux observer can be applied. The low level observer estimates the flux φ based on current measurements in the electrical part of the AMB system. The low level control loop consists of the electrical dynamics of theAMB system.The governing equations for this dynamics are given by Schweitzer andMaslen (2009) d dt φ1 = 1 N (V1−Ri1) d dt φ2 = 1 N (V2−Ri2) (4.3) After neglecting the resistance in Eq. (4.3), the electrical dynamics is simplified φ̇i = Vi N i=1,2 (4.4) 720 K.Miatliuk, A. Mystkowski The low level controller works in the inner flux loop. The reference force signal fr for the low level fluxcontroller is providedby thehigh level position controller. Thus, the transform function for the low level control feedback rule in the s-domain Gl(s)= fc(s) fr(s) := φc(s) φr(s) (4.5) The control force fc depends on the control flux φc which fulfils the condition of switching scheme: —when φc ­ 0 φc =φ1 φ2 =0 —when φc < 0 φc =−φ2 φ1 =0 The low level control law uφ =−fφ(φr−φc), where fφ is a nonlinear control functionwhich also ensures the bounds of φi, i.e. limt→∞φi(t)=min{φ1(0),φ2(0)}. Equations (4.3)-(4.5) correspond to the dynamic representation (ϕ̂, λ̂)ℓ0 of the low level co- ordinator Sℓ0 given at the AMB conceptual design stage. 4.3. High level controller Now, with respect to the outer controller (high level coordinator Sℓ+10 ), since the AMB model from the force f to the position x is linear, no linearization is needed and, therefore, the position control law can be linear. Moreover, the high level controller is not coupled with the low level control loop. The high level control loop provides the reference force fr and consists the mechanical dynamics of the AMB system. The high level position feedback control rule in s-domain is based on themeasured rotor displacement xmat at the magnetic bearing plane and the referenced displacement xr Gh(s)= xm(s) xr(s) (4.6) where the displacement xm is estimated (by the linear high level position observer) based on the measured mass displacement x. In order to provide the equilibrium state of dynamics Eq. (4.1) the time derivatives in Eq. (4.1) go to zero d2x dt2 = Φ|Φ| µ0mA → 0 (4.7) If the static gain of the control loop ofGh is defined as the state feedback controller ( static gain matrix K), then lim s→0 Gh =K when d2x dt2 → 0 (4.8) Therefore, Eqs. (4.1)-(4.8) present detailed design models of the AMB system and its con- trollers. Equation (4.1) corresponds to the dynamicmodel (ρ,ϕ) of the AMBgiven at theAMB conceptual design stage, and Eqs. (4.3)-(4.5) and Eqs. (4.6)-(4.8) correspond to the dynamic models of the low-level and high-level controllers, respectively. Realization of coordination technology of hierarchical systems... 721 5. Conclusions The realization of the coordination technology for AMB mechatronic systems (design and con- trol) in the formal basis of hierarchical systems is briefly given in the paper. In comparison with traditional methods of mathematics and artificial intelligence, the proposed formal model contains connected descriptions of the designed object structure, its aggregated dynamic repre- sentation as a unit in its environment, the environment model and the control system. All the descriptions are connected by the coordinator which performs the design and control tasks on its strata. Besides, the proposed aed technology coheres with traditional systems of information presentation inmechatronics: numeric, graphic and natural language forms (Novikava andMia- tliuk, 2007). The technology is also coordinated with general requirements of the design and control systems (Novikava et al., 1990, 1995) as it considersmechatronic subsystems of different nature (mechanical, electromechanical, electronic, computer) in common theoretical basis. The presentation of the AMB system in the formal basis of HS allows creation of the AMB conceptual model necessary for its transition to concrete mathematical models used at the detaileddesign stage of theAMB.At thedetaileddesign stage, the low level andhigh level control loopsof theAMBcontrol structureare introduced.Each sub-systemconsists of the controller and observer structureswhich provide reference signals to each other. In this approach, the high level control loop is not dependent on the low level one. Thus, themagnetic force field nonlinearities in the low level sub-subsystem are not dependent on the high level position control loop. In the proposed approach, the electromagnetic nonlinearities are shifted from the high level control loop into the low level control loop. At the detailed design stage, theAMB (control) subsystems are described by traditional DE. At the conceptual design stage, the subsystems are presented in form of (ρ,ϕ) which are generalizations of DE and algebra systems. So, the transition from the conceptual to the detailed design stage in frames of the proposed technology is convenient and requires concretisation of the abstract dynamic system only. The given technology brings new informationalmeans for the conceptual and detailed design ofmechatronic systems andAMB systems in particular. 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