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Self-Excited Systems of Two-Level Adaptive Control
Copyright © IFAC E\'aluation of Adaptive Con trol Strategies, T bilisi, USSR, 1989
SELF-EXCITED SYSTEMS OF TWO-LEVEL ADAPTIVE CONTROL Vu. V. Mitrishkin institute of CU llt ml SCifll CfS, M usculi', USSR
Abstract. The class of piecewise linear self-excited systems of twolevel adaptive control (SESTLAC) is presented. The paper considers the SESTLAC features; their field of application; issues of classification, synthesis and analysis. The paper deals with the dynamic objects control problems solved by SESTLAC, as well as the problems of constructing elementary models of SESTLAC objects and methods of estimating their parameters and external disturbance. A number of new technical designs used in SESTLAC are listed. Keywords. Adaptive control; adaptive systems; self-exicited oscillations; dual control; identification; feedback control.
In some cases SESTLAC are the systems of dual control (Feldbaum, 1972).
FEATURES AND FIELD OF APPLICATION The SESTLAC systems were developed to solve important and interesting scientific and technical problems, e.g., in the field of modern power - engineering. The SESTLAC class is technical systems whose characteristic features (attributes) were established during the development of c ontrol devices for feedbsck systems in controlling the dynamic objects with concentrated and distributed parameters. These features are: SESTLAC systems are hierarchical: they c onform to two control le vels. At the lower level the main control aim is achieved: maintaining within certain limits or varying some of the object coordinates by a certain Isw. At the upper level the problem of adaptation for disturbance, is solved (Feldbsum, 1965; Tsypkin, 1968). SESTLAC operate in the mode of self-exc i ted oscillations set up at the lower or upper (or both) control levels (Androno v , Vit t and Khaikin, 1959). Self-excited oscillations in SESTLAC are caused by the presence of nonlinesr links with discontinuous chara c teristics in the executive and (or) control devices. The SESTLAC discontinuous characteristics can be of a relay type, both with a static c onnection between the output and input, and with the presence of the insensitivity time zone, or they can change the systems structure, in particular, the feedback sign (Mitrishkin, 1985).
First, at the SESTLAC upper adaptive level the devices for determing the states and parameters of an object in the observation mode can be used. In particular, these devices can be adaptive state estimators where the object model parameters are automatically adjusted for the object psrameters (Gribov and co-workers, 1986). In SESTLAC with identification the input action at the self-excited oacillations frequencies tests the object and simultaneously controls it. Second, in case of optimization in SESTLAC by means of the automatic search through self-excited oscillations, a direction is determined and movement is simultaneously performed towards the extremum of the performance index. And the oscillations are combined test ' and operation movements (Mitrishkin, 1985) • Among the problems solved by means of SESTLAC in the field of power-engineering, above all are the problems of controlling a high-temperature plasma in open and closed (tokamaks) magnetic traps (Gvozdkov and others, 1985), a s well aa the problems of automatic optimization of energy transformation in independent power supply systems (Mitrishkin, 1978). During stabilization of the plasma groove instability in the open magnetic trap by means of the feedback linear system, relaxation oscillations of the plasma density are set up. In (Mitrishkin, 1977) the author considers the statement and aolution of the problem of automatic optimization of the plssma threshold density by means of various types of automatic optimizers adjusting the feedback system parameters.
An object in SESTLAC is linear, and it is affected both by coordinate and parametric disturbances. The nonlinear links with discontinuous characteristics impart the features of piecewise linear systems to SESTLAC, i.e., the systems with a state space consisting of domains where dynamic equations of movements are linear (Andronov, Vitt and Khaikin, 1959).
In order to contain plasma in tokamaks when the discharge periods are long, it is necessary to stabilize the position of 15 1
Yu. V. l\litrishkin
plasma column. The stabilization is performed by means of the external magnetic fields. Their generation in modern tokamaks requires large power provided by special actuators with the increased speed and efficiency - thyristor switches. The utilization of thyristor switches for the plasma equilibriwn of clcsed systems with discontinuous control, which leads to the excitement of oscillations at the SESTLAC lower level. And since the parameters of plasma vary over a considerablY wide range during its containment ill a toksmak, then adaptation methods are expedient to provide a high-quality stabilization (Gvozdkov and others, 1985; Mitrishkin, 1985). In independent power supply systems the automatic pulse optimizers with switch elements in the power section are used to coordinate the mode of operation of the primary energy converter and storage battery and load (Mitrishkin, 1978, 1985). Since the volt-ampere characteristic of primary converters is nonstationary, and the load is of a varying character, then in order to improve the accuracy of tracking the maximum power of a primary converter, it is possible to purposely vary the parameters of the pulse optimizer control section. Such adaptation can decrease, for example, the hunting losses. GENERALIZED BLOCK DIAGRAM Fig. 1 shows a generalized block diagram of a SESTLAC system. It is the general diagram for adaptive systems (Feldbaum, 1965; Gachinsky snd colleagues, 1985). The system lower level contains the main (closed) control loop including an object, actuator and section of the controller. The lower level can include additional loops as well, embracing, for example, the controller and actuator. In SESTLAC the controller is devided into two sections: the main section belongs to the lower control level, and adaptive section - to the upper level. Signals in SESTLAC can be both scalar and vector values depending on the object character and control problem statement. In Fig. 1 these signals are: x, u - output signal and input action of the object respectively; f - additive disturbance in the right side of the object model differential equations as a time function (or vector-function);x*- reference action; y - input signal of the actuator; z - output signal of the controller adaptive section affecting the controller main section. COMBINATION OF CONTROL PRINCIPLES Various control laws can be used at the lower and upper level of SESTLAC: regulation and search. Therefore, it is expedient to classify the SESTLAC systems by the mutual combination of these laws at different levels, similarly to the general classification of adaptive systems proposed in (Gachinsky and colleagues, 1985). Depending on the kind of control used at the upper level, SESTLAC can be devided into the searchless, search, combined, and those with the situation iden-
tification (Fig. 2). DYNAMIC OBJECTS CONTROL PROBLEMS The SESTLAC systems are designed to control non-stationary dynamic objects. SESTLAC with self-excited oscillations at the lower level are considered most interesting due to the peculiarities of solving regUlation and extremal control prpblems by the main loop. In the control ' problems the object output value x (Fig.1) must approach reference action x* as close as possible, oscillating with respect to it, no matter if x* is constant (stabilization), x* variation law is known (program) and value x* is arbitrary in time (tracking). In the search problems output value must oscillate with respect to the value corresponding to the performance index extremum. As a rule, the control problems in SESTLAC have two interrelated statements. First, it is required to transfer the dynamic object control from the initial conditions to the assigned state space domain (or the assigned state). Second, it is required to Keep the object in this domain. And the object transfer from one state to another can be performed by means of SESTLAC or in the forced operation mode (without oscillations in the main loop), or in the oscillations mode. In the required state space domain the object can be kept in the oscillations mode only, since the oscillations break-down causes the object moving beyond the required domain and can make the control process unacceptable. It is expedient to point out the problems of suppressing various kinds of instabilities of dynamic objects by SESTLAC. Such problems are more and more frequent as the theory and technology of automatic c ontrol de velop, particularly, in the controlled thermonuclear synthesis. The instabilities considerably complicate the object transfer and its keeping in the required state, since, in particular, the unstable object parameters fluctuation can cause the oscillations break-down in SESTLAC. Another important problem solved by SESTLAC is to counteract the disturbance effects (in some cases uncontrollable). The disturbance can cause the oscillations break-down, increase in statical and dynamic control errors. SYNTHESIS The SESTLAC synthesis is based upon the development of the lower level control loop. It is at the lower level that the control main goal must be achieved, and the upper level must favor the efficient achievement of this goal. To solve the lower level control problems the classical control theory methods can be used. They provide, for example, the development of regulators for the main section of a control unit with . P-, PD-, PID-laws. To increase the accuracy and speed in controlling nonstationary objects, it is necessary to vary the controllers parameters in accordance with the measured parameters of an object.
Self-Excited Systems
To solve the extreme problems of the lower level in SESTLAC the automatic optimization methods are used which lead to the development of one-channel and multichannel optimizers (Gachinsky and colleagues, 1985; Mitrishkin, 1985). The control upper level can vary the search step, when moving towards the extremum, and the frequency of test actions thus increasing the accuracy and speed of search. In some cases the upper level control laws can extend the stability domain in the lower level automatic search system parameters space, and even transfer the system from the unstable search domain to the stability domain (Merzlikin and Mitrishkin, 1978). In synthesizing the lower level control laws the invariant theory methods can be used. They provide for the development of the main control loop whose operation is invariant to the disturbance effect. As a result, a combined system is synthesized composed of a deviation control principle and disturbance control principle. Thus, for example, in (Gribov and co-workers, 1985) a SESTLAC system is developed where the invariance is achieved to the additive disturbance effect in the average component of the oscillation process. To synthesize the SESTLAC main loop the optimal control theory methods can be used. They provide for finding the best modes of operation of the main loop and creating the control laws for these modes. The control laws in SESTLAC (including optimal ones) under the conditions of the object nonstationary state are realized by means of adaptation methods (Tsypkin, 1968). By varying the lower level operation algorithm by means of the upper level, one can optimize the performance index (or indexes) of the main loop operation. In particular, an optimal system has been developed in (Mitrishkin, 1989), where by means of adaptive variation of the control unit main section parameters the oscillations amplitude is minimized. The SESTLAC synthesis necessarily provides for the utilization of the identification methods for constructing the models of objects under control and actuators, as well as for estimating their states and parameters. ANALYSIS Adequate models of SESTLAC objects with concentrated and distributed parameters are described by the systems of ordinary differential equations. Besides, in actuator and controller the analog elements are mainly used in which the processes are also rather accurately represented by ordinary differential equations. Therefore the theory of ordinary differential equations is a theoretical basis in the development of SESTLAC. This theory makes the basis for the known analytical methods of studying linear and nonlinear control systems. All these methods are to this or that extent applicable to SESTLAC (Mitrishkin, 1985). And the difficulties of the SESTLAC theoretical studying sometimes are unresolvable, since SESTLAC are generally described by complex nonlinear sets of differential equations with the coefficients variable in time. Therefore,
153
various assumptions have to be used which make it possible to justify the serviceability of the new technical approaches used in SESTLAC. These assumptions are made by using the method of "frozen coefficients" and method of movements separation. This is possible due to the fact that at the upper level the adaptation processes proceed slower than the control processes of the main loop. Therefore, all dynamic links of the lower level can be often considered stationary, and analytical methods of the linear and nonlinear theory can be used for the analysis and synthesis. For the linear theory they are: the Laplace transformation; method of transfer functions and block diagrams; frequency characteristics method; D - partitioning in studying stability and increasing the stability reserve; state space method. For the nonlinear theory: the method of many-sheeted phaseplane and method of point mapping tAndronov, Vitt and Khaikin, 1959) which provide for partitioning the systems parameters space into domains with qualitatively similar behavior of phase trajectories; the method of harmonic linearization displaying the main relation ships in SESTLAC by means of deviding the movements into the fast and slow ones; the method of expansion in terms of a small parameter. The above assumptions make it necessary to check the theoretical results, and new proposals by means of mathematical (numerical) simulation. Besides, it is expedient to study the SESTLAC control processes in the electronic models of objects and actuators in real time. This provides for adjusting the developed controller, estimating the SESTLAC quality indexes by real data, as well as makes the work with SESTLAC easier for the experimenter studying physical and technological processes. IDENTIFICATION As a rule, in SESTLAC dynamiC objects (including plasma) very complex processes occur which are described by partial equations. Studying of these equations by numerical methods on digital computer requires a lot of machine time. To reduce this time, in many cases, other models of objects with distributed parameters are considered being the sets of ordinary differential equations. It turns out that due to their complexity these models are not suitable as well for developing control methods for the above objects. In ' this situation the structural identification problem must be solved. And the unitial system is tested by input actions of a convenient type, and the initial set of equations of large dimension is approximated by the low order dynamic link according to the results of observing the input and output variables. Such a problem is solved in (Mitrishkin and Savkina, 1984) for the model of plasma column equilibrium in a tokamak. By using the obtained structural identification result the problem is stated on estimating parameters, states and additive disturbance affecting an object. This problem can be solved by means of adaptive
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state estimators with continuous time, recurrent algorithms with discrete time, as well as by the accumulated data. The estimation results can be used directly to control the object, if the estimation is performed in the abservations mode, or to correct the program actions in the open control loop, if the estimation is performed by the accumulated data. The adaptive state estimators are included into the closed adaptation loops of the SESTLAC upper level, which causes a number of additional difficulties due to the required maintenance of stability and reliability of operation of the system on the whole. The SESTLAC design causes the necessity for solving the identification problems for actuators as well, since they are technical objects in which complex processes occur (including commutation ones). Such actuators include, for example, thyristor invertors used in furnishing magnetic fields in tokamaks. It turns out that these devices can be represented by relay elements with a time insensitivity zone (Mitrishkin, 1985). CONCLUSION The considered class of systems - SESTLAC - can be used to solve the automatic control problems of dynamic objects of various physical nature. On the one hand, it is an efficient technical instrument for the solution of the above problems, and on the other hand, this class of systems permits the development of new control principles, contains original spproaches for the first time utilized to control complex technical objects. In particular, these approaches are: methods of constructin8 adaptive self-excited control systems (Gachinsky and colleagues, 1985) and principles of adaptive minimization of oscillations amplitude in them (Mitrishkin, 1989); principles of compensation for external disturbance in relay self-excited systems (Gribov and coworkers, 1985); principles of automatic optimization of feedback systems and methods of constructing automatic optimizers (Mitrishkin, 1977, 1985; Merzlikin and Mitrishkin, 1978); methods of constructing elementary models of dynamic objects by input-output correspondence (Mitrishkin and Savkina, 1984) and methods of constructing adaptive state estimators providing for simultaneous estimation of additive disturbance and object parameters (Mitrishkin, 1985; Gribov and coworkers, 1986). REFERENCES Andronov, A.A., A.A. Vitt, and S.E. Khaikin (1959). Theory of oscillations. Phizmatgiz, Moscow. 916 pp. Feldbaum, A.A. (1965). Problems of selfadjusting (adaptive) systems. In V. A. Trapeznikov (Ed.), Self-Adjusting Systems. Nauka, Moscow. pp.5-22. Feldbaum, A.A. (1972). On problems of dual control theory. In Ya.Z. Tsypkin (Ed.), Automatic s stems 0 timization methods. Energia, Moscow. pp. - 09. Gachinsky, E.Ye., A.I. Drozdov, Yu.V. Mitrishkin, and M. Yu. Cherkashin (1985).
Adaptation in technical objects control systems. In V.A. Trapeznikov, and I.V. Prangishvily (Ed.), Control systems and their application. Institute of Control Sciences, Moscow. pp.10-22. Gvozdkov, Yu.V., Yu.V. Mitrishkin, M.Yu. Cherkashin, and V.A. Chuyanov (1985). Plasma control in experimental thermonuclear units. Preprint of Institute of Control Sciences, Moscow. 50 pp. t Gribov, Yu.V., Yu.A. Kostsov, Yu.V. Mitrishkin, V.A. Chuyanov, and K.G.Shakhovets (1985). Dynamics of plasma column and stabilization of its position in tokamak by analog models. Preprint of Institute of Atomic Energy - 4113/7, Moscow, 32 pp. Gribov, Yu.V., E.A. Kuznetsov, Yu.V. Mitrishkin, and V.A. Chuyanov (1986). Relay system for controlling plasma column position in tokamak. Voprosy atomnoi nauki i tekhniki, Series: Termo,jaterny sintez, .1" 51 - 57. Merzlikin, V.M., and Yu.V. Mitrishkin (1978). Automatic search system with self-adjustment of external feedback coefficient. In Ya.Z. Tsypkin (Ed.), Modelling and control in developing systems. Nauka, Moscow. pp.41-48. Mitrishkin, Yu.V. (1977). Automatic feedback optimization in a nonstationary unstable plant. Automatics and Telemechanics, 2, 66-76. Mitrishkin, Yu.V. (1978). Optimizing the solar energy conversion mode. Automatics and Telemechanics,l, 53=597 Mitrishkin, Yu.V. (1985). Dynamic objects control by automatic tuning. Nauka, Moscow. 158 pp. Mitrishkin, Yu.V. (1989). Minimizing the autooscillation amplitude in a switching control system with a stable linear dynamic part. Automatics and Telemechanics, 2, 91-102. Mitrishkin, Yu.V., and I.S. Savkina (1984). On a model of plasma equilibrium in a tokamak. Automatics and Telemechanics, 1, 66-76. Tsypkin, Ye.Z. (1968). Adattation and learning in automatic Sys ems. Nauka, Moscow. 400 pp.
Self- Excited Systems
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Controller
----------, --
Adaptation Idevice
I
I
Upper level
Lower level
L-i f
z
x*
Control unit of main loop
Fig. 1.
y
u
Ac tuator
Ob j ect
Generalized block diagram of a SESTLAC system
Searchless adaptive control
Adaptive control with situation identification
Adapti ve control with search
Combined adaptive control
Upper level
R
R
R
S
S
S
Si
Si
Si
RS
Lower level
R
S
RS
R
S
RS
R
S
RS
R
CONTROL Fig. 2.
-c
RS RS
S
RS
REGULATION (R) SEARCH (S)
Control system structures of SESTLAC
x
SELF-EXCITED SYSTEMS OF TWO-LEVEL ADAPTIVE CONTROL Vu. V. Mitrishkin institute of CU llt ml SCifll CfS, M usculi', USSR
Abstract. The class of piecewise linear self-excited systems of twolevel adaptive control (SESTLAC) is presented. The paper considers the SESTLAC features; their field of application; issues of classification, synthesis and analysis. The paper deals with the dynamic objects control problems solved by SESTLAC, as well as the problems of constructing elementary models of SESTLAC objects and methods of estimating their parameters and external disturbance. A number of new technical designs used in SESTLAC are listed. Keywords. Adaptive control; adaptive systems; self-exicited oscillations; dual control; identification; feedback control.
In some cases SESTLAC are the systems of dual control (Feldbaum, 1972).
FEATURES AND FIELD OF APPLICATION The SESTLAC systems were developed to solve important and interesting scientific and technical problems, e.g., in the field of modern power - engineering. The SESTLAC class is technical systems whose characteristic features (attributes) were established during the development of c ontrol devices for feedbsck systems in controlling the dynamic objects with concentrated and distributed parameters. These features are: SESTLAC systems are hierarchical: they c onform to two control le vels. At the lower level the main control aim is achieved: maintaining within certain limits or varying some of the object coordinates by a certain Isw. At the upper level the problem of adaptation for disturbance, is solved (Feldbsum, 1965; Tsypkin, 1968). SESTLAC operate in the mode of self-exc i ted oscillations set up at the lower or upper (or both) control levels (Androno v , Vit t and Khaikin, 1959). Self-excited oscillations in SESTLAC are caused by the presence of nonlinesr links with discontinuous chara c teristics in the executive and (or) control devices. The SESTLAC discontinuous characteristics can be of a relay type, both with a static c onnection between the output and input, and with the presence of the insensitivity time zone, or they can change the systems structure, in particular, the feedback sign (Mitrishkin, 1985).
First, at the SESTLAC upper adaptive level the devices for determing the states and parameters of an object in the observation mode can be used. In particular, these devices can be adaptive state estimators where the object model parameters are automatically adjusted for the object psrameters (Gribov and co-workers, 1986). In SESTLAC with identification the input action at the self-excited oacillations frequencies tests the object and simultaneously controls it. Second, in case of optimization in SESTLAC by means of the automatic search through self-excited oscillations, a direction is determined and movement is simultaneously performed towards the extremum of the performance index. And the oscillations are combined test ' and operation movements (Mitrishkin, 1985) • Among the problems solved by means of SESTLAC in the field of power-engineering, above all are the problems of controlling a high-temperature plasma in open and closed (tokamaks) magnetic traps (Gvozdkov and others, 1985), a s well aa the problems of automatic optimization of energy transformation in independent power supply systems (Mitrishkin, 1978). During stabilization of the plasma groove instability in the open magnetic trap by means of the feedback linear system, relaxation oscillations of the plasma density are set up. In (Mitrishkin, 1977) the author considers the statement and aolution of the problem of automatic optimization of the plssma threshold density by means of various types of automatic optimizers adjusting the feedback system parameters.
An object in SESTLAC is linear, and it is affected both by coordinate and parametric disturbances. The nonlinear links with discontinuous characteristics impart the features of piecewise linear systems to SESTLAC, i.e., the systems with a state space consisting of domains where dynamic equations of movements are linear (Andronov, Vitt and Khaikin, 1959).
In order to contain plasma in tokamaks when the discharge periods are long, it is necessary to stabilize the position of 15 1
Yu. V. l\litrishkin
plasma column. The stabilization is performed by means of the external magnetic fields. Their generation in modern tokamaks requires large power provided by special actuators with the increased speed and efficiency - thyristor switches. The utilization of thyristor switches for the plasma equilibriwn of clcsed systems with discontinuous control, which leads to the excitement of oscillations at the SESTLAC lower level. And since the parameters of plasma vary over a considerablY wide range during its containment ill a toksmak, then adaptation methods are expedient to provide a high-quality stabilization (Gvozdkov and others, 1985; Mitrishkin, 1985). In independent power supply systems the automatic pulse optimizers with switch elements in the power section are used to coordinate the mode of operation of the primary energy converter and storage battery and load (Mitrishkin, 1978, 1985). Since the volt-ampere characteristic of primary converters is nonstationary, and the load is of a varying character, then in order to improve the accuracy of tracking the maximum power of a primary converter, it is possible to purposely vary the parameters of the pulse optimizer control section. Such adaptation can decrease, for example, the hunting losses. GENERALIZED BLOCK DIAGRAM Fig. 1 shows a generalized block diagram of a SESTLAC system. It is the general diagram for adaptive systems (Feldbaum, 1965; Gachinsky snd colleagues, 1985). The system lower level contains the main (closed) control loop including an object, actuator and section of the controller. The lower level can include additional loops as well, embracing, for example, the controller and actuator. In SESTLAC the controller is devided into two sections: the main section belongs to the lower control level, and adaptive section - to the upper level. Signals in SESTLAC can be both scalar and vector values depending on the object character and control problem statement. In Fig. 1 these signals are: x, u - output signal and input action of the object respectively; f - additive disturbance in the right side of the object model differential equations as a time function (or vector-function);x*- reference action; y - input signal of the actuator; z - output signal of the controller adaptive section affecting the controller main section. COMBINATION OF CONTROL PRINCIPLES Various control laws can be used at the lower and upper level of SESTLAC: regulation and search. Therefore, it is expedient to classify the SESTLAC systems by the mutual combination of these laws at different levels, similarly to the general classification of adaptive systems proposed in (Gachinsky and colleagues, 1985). Depending on the kind of control used at the upper level, SESTLAC can be devided into the searchless, search, combined, and those with the situation iden-
tification (Fig. 2). DYNAMIC OBJECTS CONTROL PROBLEMS The SESTLAC systems are designed to control non-stationary dynamic objects. SESTLAC with self-excited oscillations at the lower level are considered most interesting due to the peculiarities of solving regUlation and extremal control prpblems by the main loop. In the control ' problems the object output value x (Fig.1) must approach reference action x* as close as possible, oscillating with respect to it, no matter if x* is constant (stabilization), x* variation law is known (program) and value x* is arbitrary in time (tracking). In the search problems output value must oscillate with respect to the value corresponding to the performance index extremum. As a rule, the control problems in SESTLAC have two interrelated statements. First, it is required to transfer the dynamic object control from the initial conditions to the assigned state space domain (or the assigned state). Second, it is required to Keep the object in this domain. And the object transfer from one state to another can be performed by means of SESTLAC or in the forced operation mode (without oscillations in the main loop), or in the oscillations mode. In the required state space domain the object can be kept in the oscillations mode only, since the oscillations break-down causes the object moving beyond the required domain and can make the control process unacceptable. It is expedient to point out the problems of suppressing various kinds of instabilities of dynamic objects by SESTLAC. Such problems are more and more frequent as the theory and technology of automatic c ontrol de velop, particularly, in the controlled thermonuclear synthesis. The instabilities considerably complicate the object transfer and its keeping in the required state, since, in particular, the unstable object parameters fluctuation can cause the oscillations break-down in SESTLAC. Another important problem solved by SESTLAC is to counteract the disturbance effects (in some cases uncontrollable). The disturbance can cause the oscillations break-down, increase in statical and dynamic control errors. SYNTHESIS The SESTLAC synthesis is based upon the development of the lower level control loop. It is at the lower level that the control main goal must be achieved, and the upper level must favor the efficient achievement of this goal. To solve the lower level control problems the classical control theory methods can be used. They provide, for example, the development of regulators for the main section of a control unit with . P-, PD-, PID-laws. To increase the accuracy and speed in controlling nonstationary objects, it is necessary to vary the controllers parameters in accordance with the measured parameters of an object.
Self-Excited Systems
To solve the extreme problems of the lower level in SESTLAC the automatic optimization methods are used which lead to the development of one-channel and multichannel optimizers (Gachinsky and colleagues, 1985; Mitrishkin, 1985). The control upper level can vary the search step, when moving towards the extremum, and the frequency of test actions thus increasing the accuracy and speed of search. In some cases the upper level control laws can extend the stability domain in the lower level automatic search system parameters space, and even transfer the system from the unstable search domain to the stability domain (Merzlikin and Mitrishkin, 1978). In synthesizing the lower level control laws the invariant theory methods can be used. They provide for the development of the main control loop whose operation is invariant to the disturbance effect. As a result, a combined system is synthesized composed of a deviation control principle and disturbance control principle. Thus, for example, in (Gribov and co-workers, 1985) a SESTLAC system is developed where the invariance is achieved to the additive disturbance effect in the average component of the oscillation process. To synthesize the SESTLAC main loop the optimal control theory methods can be used. They provide for finding the best modes of operation of the main loop and creating the control laws for these modes. The control laws in SESTLAC (including optimal ones) under the conditions of the object nonstationary state are realized by means of adaptation methods (Tsypkin, 1968). By varying the lower level operation algorithm by means of the upper level, one can optimize the performance index (or indexes) of the main loop operation. In particular, an optimal system has been developed in (Mitrishkin, 1989), where by means of adaptive variation of the control unit main section parameters the oscillations amplitude is minimized. The SESTLAC synthesis necessarily provides for the utilization of the identification methods for constructing the models of objects under control and actuators, as well as for estimating their states and parameters. ANALYSIS Adequate models of SESTLAC objects with concentrated and distributed parameters are described by the systems of ordinary differential equations. Besides, in actuator and controller the analog elements are mainly used in which the processes are also rather accurately represented by ordinary differential equations. Therefore the theory of ordinary differential equations is a theoretical basis in the development of SESTLAC. This theory makes the basis for the known analytical methods of studying linear and nonlinear control systems. All these methods are to this or that extent applicable to SESTLAC (Mitrishkin, 1985). And the difficulties of the SESTLAC theoretical studying sometimes are unresolvable, since SESTLAC are generally described by complex nonlinear sets of differential equations with the coefficients variable in time. Therefore,
153
various assumptions have to be used which make it possible to justify the serviceability of the new technical approaches used in SESTLAC. These assumptions are made by using the method of "frozen coefficients" and method of movements separation. This is possible due to the fact that at the upper level the adaptation processes proceed slower than the control processes of the main loop. Therefore, all dynamic links of the lower level can be often considered stationary, and analytical methods of the linear and nonlinear theory can be used for the analysis and synthesis. For the linear theory they are: the Laplace transformation; method of transfer functions and block diagrams; frequency characteristics method; D - partitioning in studying stability and increasing the stability reserve; state space method. For the nonlinear theory: the method of many-sheeted phaseplane and method of point mapping tAndronov, Vitt and Khaikin, 1959) which provide for partitioning the systems parameters space into domains with qualitatively similar behavior of phase trajectories; the method of harmonic linearization displaying the main relation ships in SESTLAC by means of deviding the movements into the fast and slow ones; the method of expansion in terms of a small parameter. The above assumptions make it necessary to check the theoretical results, and new proposals by means of mathematical (numerical) simulation. Besides, it is expedient to study the SESTLAC control processes in the electronic models of objects and actuators in real time. This provides for adjusting the developed controller, estimating the SESTLAC quality indexes by real data, as well as makes the work with SESTLAC easier for the experimenter studying physical and technological processes. IDENTIFICATION As a rule, in SESTLAC dynamiC objects (including plasma) very complex processes occur which are described by partial equations. Studying of these equations by numerical methods on digital computer requires a lot of machine time. To reduce this time, in many cases, other models of objects with distributed parameters are considered being the sets of ordinary differential equations. It turns out that due to their complexity these models are not suitable as well for developing control methods for the above objects. In ' this situation the structural identification problem must be solved. And the unitial system is tested by input actions of a convenient type, and the initial set of equations of large dimension is approximated by the low order dynamic link according to the results of observing the input and output variables. Such a problem is solved in (Mitrishkin and Savkina, 1984) for the model of plasma column equilibrium in a tokamak. By using the obtained structural identification result the problem is stated on estimating parameters, states and additive disturbance affecting an object. This problem can be solved by means of adaptive
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state estimators with continuous time, recurrent algorithms with discrete time, as well as by the accumulated data. The estimation results can be used directly to control the object, if the estimation is performed in the abservations mode, or to correct the program actions in the open control loop, if the estimation is performed by the accumulated data. The adaptive state estimators are included into the closed adaptation loops of the SESTLAC upper level, which causes a number of additional difficulties due to the required maintenance of stability and reliability of operation of the system on the whole. The SESTLAC design causes the necessity for solving the identification problems for actuators as well, since they are technical objects in which complex processes occur (including commutation ones). Such actuators include, for example, thyristor invertors used in furnishing magnetic fields in tokamaks. It turns out that these devices can be represented by relay elements with a time insensitivity zone (Mitrishkin, 1985). CONCLUSION The considered class of systems - SESTLAC - can be used to solve the automatic control problems of dynamic objects of various physical nature. On the one hand, it is an efficient technical instrument for the solution of the above problems, and on the other hand, this class of systems permits the development of new control principles, contains original spproaches for the first time utilized to control complex technical objects. In particular, these approaches are: methods of constructin8 adaptive self-excited control systems (Gachinsky and colleagues, 1985) and principles of adaptive minimization of oscillations amplitude in them (Mitrishkin, 1989); principles of compensation for external disturbance in relay self-excited systems (Gribov and coworkers, 1985); principles of automatic optimization of feedback systems and methods of constructing automatic optimizers (Mitrishkin, 1977, 1985; Merzlikin and Mitrishkin, 1978); methods of constructing elementary models of dynamic objects by input-output correspondence (Mitrishkin and Savkina, 1984) and methods of constructing adaptive state estimators providing for simultaneous estimation of additive disturbance and object parameters (Mitrishkin, 1985; Gribov and coworkers, 1986). REFERENCES Andronov, A.A., A.A. Vitt, and S.E. Khaikin (1959). Theory of oscillations. Phizmatgiz, Moscow. 916 pp. Feldbaum, A.A. (1965). Problems of selfadjusting (adaptive) systems. In V. A. Trapeznikov (Ed.), Self-Adjusting Systems. Nauka, Moscow. pp.5-22. Feldbaum, A.A. (1972). On problems of dual control theory. In Ya.Z. Tsypkin (Ed.), Automatic s stems 0 timization methods. Energia, Moscow. pp. - 09. Gachinsky, E.Ye., A.I. Drozdov, Yu.V. Mitrishkin, and M. Yu. Cherkashin (1985).
Adaptation in technical objects control systems. In V.A. Trapeznikov, and I.V. Prangishvily (Ed.), Control systems and their application. Institute of Control Sciences, Moscow. pp.10-22. Gvozdkov, Yu.V., Yu.V. Mitrishkin, M.Yu. Cherkashin, and V.A. Chuyanov (1985). Plasma control in experimental thermonuclear units. Preprint of Institute of Control Sciences, Moscow. 50 pp. t Gribov, Yu.V., Yu.A. Kostsov, Yu.V. Mitrishkin, V.A. Chuyanov, and K.G.Shakhovets (1985). Dynamics of plasma column and stabilization of its position in tokamak by analog models. Preprint of Institute of Atomic Energy - 4113/7, Moscow, 32 pp. Gribov, Yu.V., E.A. Kuznetsov, Yu.V. Mitrishkin, and V.A. Chuyanov (1986). Relay system for controlling plasma column position in tokamak. Voprosy atomnoi nauki i tekhniki, Series: Termo,jaterny sintez, .1" 51 - 57. Merzlikin, V.M., and Yu.V. Mitrishkin (1978). Automatic search system with self-adjustment of external feedback coefficient. In Ya.Z. Tsypkin (Ed.), Modelling and control in developing systems. Nauka, Moscow. pp.41-48. Mitrishkin, Yu.V. (1977). Automatic feedback optimization in a nonstationary unstable plant. Automatics and Telemechanics, 2, 66-76. Mitrishkin, Yu.V. (1978). Optimizing the solar energy conversion mode. Automatics and Telemechanics,l, 53=597 Mitrishkin, Yu.V. (1985). Dynamic objects control by automatic tuning. Nauka, Moscow. 158 pp. Mitrishkin, Yu.V. (1989). Minimizing the autooscillation amplitude in a switching control system with a stable linear dynamic part. Automatics and Telemechanics, 2, 91-102. Mitrishkin, Yu.V., and I.S. Savkina (1984). On a model of plasma equilibrium in a tokamak. Automatics and Telemechanics, 1, 66-76. Tsypkin, Ye.Z. (1968). Adattation and learning in automatic Sys ems. Nauka, Moscow. 400 pp.
Self- Excited Systems
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Controller
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Adaptation Idevice
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Upper level
Lower level
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Control unit of main loop
Fig. 1.
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Ac tuator
Ob j ect
Generalized block diagram of a SESTLAC system
Searchless adaptive control
Adaptive control with situation identification
Adapti ve control with search
Combined adaptive control
Upper level
R
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S
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Lower level
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CONTROL Fig. 2.
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REGULATION (R) SEARCH (S)
Control system structures of SESTLAC
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