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The Control Under Uncertainty Conditions: History and Perspective
7th IFAC Conference on Manufacturing Modelling, Management, and Control International Federation of Automatic Control June 19-21, 2013. Saint Petersburg, Russia
The Control Under Uncertainty Conditions: History and Perspective Stanislav V. Emelyanov ∗ V. V. Fomichev ∗∗ A. S. Fursov ∗∗∗ ∗
Institute for Systems Analysis of RAS, Moscow, 9, Prospekt 60-let Oktyabrya, 117312 Russia (e-mail: [email protected]; [email protected]). ∗∗ Lomonosov Moscow State University, Moscow, Leninskie gory, MSU, Dep. CMC, 119991 Russia (e-mail: [email protected]). ∗∗∗ Lomonosov Moscow State University, Moscow, Leninskie gory, MSU, Dep. CMC, 119991 Russia (e-mail: [email protected]).
Abstract: Within a problem of control of dynamic plants under uncertainty are considered both classical methods of the theory of systems with variable structure, and their modern development in the form of the theory of new types of feedback. The exact structure of feedback in control variable structure systems is clarified. Existence of two types of feedback is revealed: one classical feedback by the general regulation error, and another, new type, the coordinateto-operational feedback. Thus feedback is responsible for a set of positive dynamical properties for control variable structure systems. Also the perspective directions of further development of the theory of new types of feedback are considered. In particular, methods of simultaneous control by families of dynamic plants are considered are considered. Keywords: Feedback, stabilization, control system, variable structure system, sliding mode, new feedback types, coordinate-to-operator feedback, simultaneous stabilization. 1. INTRODUCTION
fluence of a head wind aboard the plane can be an example of coordinate disturbance and the plane with changing geometry of a wing represents the system being under the influence of parametrical disturbance.
Problem of the control under uncertainty is one of main problems of the automatic control theory. It has an old story. As known, a controllable plant is characterized by input-output relationship W , that is the relation between sets of input and output signals
In modern control theory the control plant is described by systems of differential equations. For example, a linear scalar stationary object under the action of external perturbations can be represented by the following equations ½ x˙ = (A + ∆A)x + (b + ∆b)u + f (1) y = (c + ∆c)x, x ∈ Rn , u, y ∈ R1 ,
y = W u. The plant can be graphically represented as a block with the input u and the output y (fig. 1).
where a differential equation is an equation of state (vector x is a state vector of the object), and static ratio is an output equation. Matrices ∆A, ∆b, ∆c characterize parametric perturbations, f — coordinate perturbation. Let’s assume that only ranges of changes of external uncertainty (probably enough large) are known. That means that for (1) are known majorants a0 , b0 , c0 , f0 , which realize following estimations
Fig. 1
The object input u is called a control, and its input y — a variable coordinate. In fact, plant input is a some allocated coordinate, setting which you can change the behavior of the object. The goal of control is usually defined by desired behavior of the output coordinate of the plant.
|∆A| ≤ a0 ,
|∆c| ≤ c0 ,
|f | ≤ f0 .
With all the possible actions on the plant, the primal control problem can be formulated as follows: to provide a given operating mode of the object under perturbations operating in a wide range.
As a rule, the plant of control during its functioning is affected by perturbations of two types: coordinate and operational. Coordinate perturbations make an additive contribution to input or internal signals of plant, and operational perturbations are caused either change of parameters of plant (parametrical perturbations), or change of its dynamic order (structural perturbations). That conducts to change an input-output characteristies of an plant. In978-3-902823-35-9/2013 © IFAC
|∆b| ≤ b0 ,
Operating mode of plant is understood as some prescribed behavior of its controlled coordinate and desirable dynamics of the plant. Properties of dynamics characterize quality of the control system. Changing of the output variable is given by coordinate setter (fig. 2), 7
10.3182/20130619-3-RU-3018.00363
2013 IFAC MIM June 19-21, 2013. Saint Petersburg, Russia
control and in the coordinates; discrete variable structure systems; solution to the problem of tracking with high degree of astatism in case unknown influence; achievement of complete independence motion equations in a closed loop control system from some uncertainty factors; construction information and filtering devices, in particular, differentiators of multiple differentiation in case of a information deficit; systems of identification and optimization under uncertainty. Theory of VSS also has significant potential for application problems. Its methods are used in the autopilot, in the elements of industrial automation, robotics, in instrumentation and amplification equipment.
Fig. 2
where g(t) is a function (signal) that defines the desired behavior of the regulated output; goal of control is to provide a convergence to zero of the contol error: |e(t)| = |g(t) − y(t)| → 0 (fig. 3).
We now pose the following question: what is the fundamental feature of the device with variable structure controller, which allows to confer a closed system very positive dynamic and static properties under the essential uncertainty and to provide a theory of VSS as an independent direction in general feedback theory?
Fig. 3
Under the influence of external perturbations information about which is often insufficient, the communication between an input and an output of the plant becomes ambiguous and uncertain. That very complicate the solution of a problem of control.
As a rule, the most important advantages of the VSS carried out in sliding mode (although in the VSS can be implemented other types of movements), a very common response to the above question is the following: it’s all in the use of sliding mode.
Problem of the control under uncertainty is well developed section of the control theory. There is a large number of methods in this direction. Detailed analysis of features and conditions of applicability each of them, and, especially, their comparison, represents a difficult problem. Therefore we will consider one of the most difficult cases, namely, a problem of the control for the plant under the essential uncertainty conditions, being characterized by strong variability of parameters of plant.
Some looking ahead, let’s say that the sliding mode is not the prime cause of all the advantages of variable structure systems, but only one of many possible means of implementation of some more general principle. 2. PRINCIPLE OF FEEDBACK
About 60 years ago the idea of variable structure feedback controllers was put forward. It appeared very effective at the solution of a wide range problems of control and optimization under the essential uncertainty and also led to creation of the whole direction in the feedback theory — so-called variable structure systems (VSS).
Let’s now turn to a question of construction of control algorithm providing a given operating mode of the plant. If the plant is stable, its mathematical model is known exactly and it is not affected by any external perturbations, then it is possible to create control as a time function u(t). The similar principle of construction of the control algorithm carries the name of the program control principle.
The idea of a principle of structure variability consists in spasmodic change of communications between functional elements of a controller depending on a phase condition of the closed control system which in that case is named a control variable structure system (VSS). If the linear plant and functional elements of a controller are considered, it is possible to interpret corresponding VSS as set of linear subsystems and rules of transition from one element of this set to another. This rules depends from crossing by a phase point of dividing hyperplanes in phase space of system which name discontinuity surfaces. If the plant or functional elements are nonlinear, then the combination of nonlinear subsystems and, accordingly, nonlinear surfaces or discontinuity manifolds is difficult problem. Both in the first and in the second case the VSS is nonlinear dynamic system, it is described by differential equations with discontinuous second member of equation. The synthesis of VSS is reduced to the choice of discontinuity surfaces and initial set of subsystems that guarantee the solution of the control problem.
If the plant is affected by unknown coordinate perturbations which have essential impact on the operated coordinate, then the program control principle is inapplicable. In this case the next methods of compensation of coordinate perturbations possible — either to change an object design for each type of perturbations (but for this purpose it is necessary to know them in advance), or to form an input variable u for compensating this perturbation. The last method is possible when the perturbation f can be measured. Control method, in which the control u is generated based on the measured coordinate perturbation is called the principle of compensation. Essential lack of the principle of compensation is that it can’t be applied in case when there is no opportunity to measure coordinate perturbation and also if the plant is unstable. Both discussed principle do not use information about the result of control, i. e. about the output variable of the control object. In this case, comes to the aid a fundamental principle of control — the principle of feedback.
Examples of the use of the principle of variable structure for solution a large range of control problems are well known, so there is simply list of them: control on a state and an output in the linear SISO and MIMO plants which parametres and exterior forces are unknown and can change largely; control of systems with delay in the
The principle of feedback is based on the construction of the control algorithm, using information about the results 8
2013 IFAC MIM June 19-21, 2013. Saint Petersburg, Russia
of control. Thus operating signal u is formed according to the following scheme (fig. 4).
necessary to modify the control algorithm (the regulator R). There is a new problem: to find a method affecting on the regulator, allowing to change the control algorithm to compensate parametric perturbations, i. e. to ensure the invariance of the control system in reference to these perturbations. There are two possible approaches in the case of coordinate perturbations. One consists in estimation (identification) of changes in dynamics of the object, caused by parametrical perturbations, and further tuning of the controller. Various effective methods of the adaptive control theory can be used for this purpose. Another way is to use the principle of feedback that does not require exact knowledge of parametric perturbations.
Fig. 4
On this scheme there is an operator R which sets control algorithm. Let the operators W and R be linear and stationary, then the output of the covered by the feedback object W is related to a setting influence g and an additive noise f by the following ratio
Within the classical feedback theory one of possible ways to compensate parametric perturbations is the usage of the strong feedback. However, the invariance of the control system for the unknown perturbations is achieved by increasing the gain, which is undesirable because of existence of natural amplitude limits for signals in the system. But then the problem is to achieve the effect of large gain in the presence of amplitude limits without increasing the gain in the real system. It would be desirable to remain within the feedback theory, that is to solve the problem of the construction of control algorithm through new feedback. To solve this problem we need to extend the concept of feedback.
′
y=
W WR g+ f, 1 + WR 1 + WR
(W ′ describes additive occurrence of noise f ). The relation shows that the feedback changes operators of transmission from inputs g and f to the output y without any interference into the design of object — only by skillful use of information about the output. Thus, the instability of the plant is no longer a barrier to the construction of the control algorithm, as its stability can be achieved by using external feedback. It is possible to enumerate such known algorithms of compensation of nonmeasurable coordinate perturbations by means of feedback as, for example, strong feedback method, relay regulator, methods of providing the set order of astatism of the closed object.
3. PRINCIPLE OF DUALITY Let’s address to analogies to understand what has to be the nature of changing the control algorithm of the system.Let’s remember the so-called principle of a duality which assumes that the same object can have the dual nature (in other words it can behave in different ways, depending on the nature of the interaction with another objects). There is one well-known example.
Principle of the feedback is one of the main directions of the modern feedback theory. In comparison with a case of coordinate perturbations, there is qualitatively other situation with a plant under parametric perturbations (fig. 5).
Neumann’s principle or the principle of joint storage of programs and data in the computer’s memory reveals the dual nature of a number. It consists in the following: both programs (commands), and data are stored in the same memory, thus it is possible to carry out the same actions over commands, as over data. It should be noted that Neumann’s principle was really the revolutionary idea which importance it is difficult to overestimate. Originally the program was set by installation of crossing points on the special switching panel. This was a very time-consuming task and sometimes it takes a few days to change the program (while the actual calculation lasted no more than a few minutes). Presence of the given set of executable commands and programs was a specifics of the first computer systems. Today, such an architecture is used to simplify the design of the computing device. So, desktop calculators, in principle, are devices with a fixed set of executable programs. They can be used for mathematical calculations, but it is impossible to apply them for text processing, computer games, for viewing graphics or video. Changing the firmware for such devices require almost complete their transformation and is impossible in most cases. Everything was changed by the
Fig. 5
In this case for system with feedback it isn’t possible to carry out regulator R synthesis in a ”program” way. The main difficulty lies in the fact that information on the parametric uncertainty can’t be used in the control algorithm (due to the fact that it is not a priori known). Therefore feedback for stabilization indefinite object has to make system very robast which is hardly possible, or somehow change itself during the control process. Thus, due to unpredictable changes of the operator of the object in the process of its functioning (and as a consequence, changes in the dynamics of a closed-loop system), it is 9
2013 IFAC MIM June 19-21, 2013. Saint Petersburg, Russia
a shift in the ψ-cell are organized on a some surface which is in the phase space of the closed system σ(x) = 0, on which should be organized sliding mode. In fact, in ψcell is a switching of structures in the phase space of the closed system.
idea of the storage of computer programs in the general memory. Neumann was the first who realized that the program can be stored as a set of zeros and ones, and in the same memory space as numbers processed by it. Lack of a fundamental difference between the program and the data opens up a whole range of possibilities. For example, a program in its processing can also be reprocessed which allows to set in a program the rules for obtaining some parts of it (so an execution of the loops and subroutines is organized in a program). Moreover, the commands of one program can be obtained as the results of the execution of another program. In other words, the entered principle made it possible for the computer itself to form a program in accordance with the results of calculations. 4. THE DUAL NATURE OF THE SIGNALS IN THE CONTROL SYSTEM Using the example of the dual nature of a number it is feasible to suggest the idea: using the principle of a duality in the feedback theory. Primary signals (coordinates) that are processed in the control system are presented as functions of time and control algorithm implemented by the regulator, is an operator that converts these functions. But in turn, this operator can be represented as an element of a set of stabilizing feedbacks. Wherein it is possible to parameterize set this operators by one or more parameters and to set a one-to-one correspondence between the parameters and controllers. Now the choice of algorithm of management, in fact, will mean a choice of the corresponding parameter (or of the vector of parameters). Thus, changing parameter, we change the control algorithm. But the parameter changing in time is already admissible to consider as the same signal which can be processed and formed in a control system.
Fig. 6
Here explicitly dependence of control u is seen on two variables: from the control error e and some variable σ, characterizes the deviation of the motion in the system from the surface σ(x) = 0. In the canonical theory of VSS the target of feedback synthesis is them consistently zero: first variable σ = 0, and then the main error e = 0. This is due to the fact that the position of the sliding surface σ(x) = 0 in the phase space of the system affects the transients in the closed control system and, therefore, the choice of the sliding surface σ(x) = 0 defines the quality of the transition processes in the system, in other words, the dynamics of a closed system and that the equality e = 0 provides the required static in the closed-loop control system. In fact, the condition σ = 0 provides a sliding mode along the surface σ(x) = 0. This motion is invariant to parametric perturbations. While outside this surface the motion of a closed system, of course, will depend on these disturbances.
Thus we come to an important principle revealing the dual nature of the signals in the control system — binarity principle. • Signals in the control system have a dual nature and and can act either as variable (transfer information), over which transformations are carried out, or as operators defining these transformations • Absence of a fundamental difference between the signals-coordinates (ordinary variables) and signalsoperators allows to change the control algorithm basing on operating conditions 5. FEATURES OF FEEDBACK IN THE VSS The binarity principle allows to unerstand the nature of efficiency of VSS. Let’s consider the block diagram of the standard VSS, solving task of stabilizing under the action of parametric perturbations (fig.6).
Fig. 7
Thus, in case of using the VSS and generating sliding modes, wich invariant with respect to existing parametric perturbations, it is necessary to specify in advance the dynamics of a closed system.
Here W is an plant of control, Ψ — variable structure controller (ψ-cell), G — coordinate setpoint (in case of the stabilization task g ≡ 0), ∆W — parametric perturbations, f — coordinate perturbations. Discontinuous VSS feedback u = ψ(σ)e is arranged on an error tracking e = g − y,
Therefore, in accordance with the principle of regulation by the deviation, it is natural to introduce the operator D, which determines a setpoint of the dynamic properties of a closed system. In this case, the value of σ = De 10
2013 IFAC MIM June 19-21, 2013. Saint Petersburg, Russia
determines the deviation of the actual behavior of transients in the closed system of control from its desired behavior, where σ = 0.
multiplying. But if one of the signals is a C-signal (z), and the other is an O-signal (y, fig. 10), then the multiplier’s action consists of transforming the C-signal by an operator which is defined by the O-signal, so an operator y obtains to a signal z(t) and the result is a C-signal x = y(z(t)). In fact, we can assume that to the signals z and y is obtained an operator multiplication x = y ◦ z. Note that the action of the operator may also be reduced to a simple multiplication of the signals.
These considerations allow to open explicite structure of the ψ-cell and redraw Fig. 1 in the following form (fig. 7). Here, R1 and R2 are the operators setting the required feedback regulation between the error e and the control u. In the scheme of Fig. 7 is clearly seen having a second (additional to the main) loop with his setter D, controller ˜ , consisting of the original plant, R2 and generic plant W covered with the main feedback with the controller R1 , parameterized signal µ, i. e. R(µ).
Depending on the types of inputs and outputs, there are four main types of dynamic objects (fig. 11).
For solution of problem of operator R2 choice. Let’s consider in more detail the binarity principle. 6. BASIC CONCEPTS: TYPES OF ELEMENTS For the development of a new approach to the construction of control algorithms based on the introduced binarity principle it is necessary to enter some of new concepts and designations.
Fig. 11
1. A CC-object. The input and the output are signalscoordinates.
Let’s denote signals-operators by double arrows and call them O-signals for short. Ordinary signals-coordinates (Csignals) as before will be denoted by single arrows (fig. 8).
2. A CO-object. The input is a signal-coordinate and the output is a signal-operator. 3. An OC-object. The input is a signal-operator and the output is a signal-coordinate.
Fig. 8
4. An OO-object. The input and the output are signalsoperators.
An important role in the approach is played by an element having two independent entrances for two types of signals — O-signals and C-signals. Let’s call such an element a binary element (fig. 9).
The following ideas are offered: • Firstly, it is necessary to synthesize not the stabilizing operator of feedback, but the algorithm of its construction, i. e. to generate appropriate signalsoperators. In this case not signal but the operator will be regulator. • Secondly, as synthesis of a control system has to happen under the uncertainty conditions, it is necessary to use feedback in an additional contour of generation for the formation of signals-operators. • Thirdly, as parametrical perturbations cause change of dynamics of the closed object, then to use a setter of a desirable dynamics of an object — operator setter — for a contour of generation of signalsoperators (let’s notice that in coordinate contour of feedback for the output coordinate of the object is used the coordinate setter). • Fourthly, to try to use known methods of the classical theory of feedback in a contour of generation of signals-operators for formation of these signals.
Fig. 9
In fact, the binary element is a dynamic or static element, which parameters are changing in the process of functioning of the control system at the expense of O-signal µ. Thus an operator B (transforming C-signal x), is changing. An example of such a binary element is bilinear element (fig. 10), carrying out multiplication of input signals.
7. OPERATOR SETTER One of the offered above ideas consists in use of an operator setter in the additional contour of feedback. The basic purpose of a setter consists in maintenance of the set dynamics of the plant and invariancy of this dynamics towards active parametrical perturbations. Actually, thus it is a question of ensuring the specified quality of the
Fig. 10
If on an entrance of the multiplier two C-signals are given, then they just multiply and the output is the result of 11
2013 IFAC MIM June 19-21, 2013. Saint Petersburg, Russia
For a better visual separation of the functions performed by these signals in the diagram it is appropriate to denote the first variable, as is customary in the classical control theory, by a single arrow, and the second kind of variables by a double.
closed control system. The quality of the control system can be defined in various ways. One of possibly ways is to set the dynamics of the control error e(t) by a differential equation, to which this error has to obey. Setting dynamics of the error, we set desirable dynamics of the closed object of control. Certainly, thus it is necessary to specify a set of setting coordinate signals g(t), for which the error has to have the desirable dynamics. On this basis, let’s set quality of a control system by means of an operator setter (fig. 12) defining change of
Fig. 12
the dynamics of the object (here D(s) is an operator, setting dynamics). Let’s notice that the output signal of an operator setter is a signal-operator and its action we will understand according to the following equivalent schemes (fig. 13) which mean applying the operator D(s) to the
Fig. 14
Such a designation, without changing anything on the merits, allows to identify visually on the schemes the new type of coordinate-to-operator feedback. Fig. 7 in this notation takes the following form (Fig. 14), in which the coordinate-to-operator feedback contour is now clearly visible.
Fig. 13
variable e(t), i. e. σ = De; purpose of the control is to provide a condition σ(t) → 0. For example, if D(s) is given d + q or in symbolic in the form of a differential operator dt d form s + q (s = dt ), then De = e˙ + qe = σ. If σ → 0, then the control error e(t) tends to a solution of the equation e˙ + qe = 0, and then dynamics of object becomes close to dynamics of object of the first order. By analogy with a static error e(t) let’s call the σ(t) dynamic error.
Analysis of the block diagram in the new notation reveals that the fundamental basis of the independence movement in the sliding mode from the parameters of the plant is not the sliding mode itself but a new type of feedback. In the simplest case, the controller R2 of the additional circuit is a relay element responsible for the emergence of ideal sliding mode. Note that in practice such a control algorithm generates real sliding mode, which has a number of drawbacks. However, the use of other types of controllers R2 can afford to get other, including smooth, motion of a closed system. This is one of the factors that lead to promising perspectives of new types of feedback to control plants under uncertainty. Another such factor is possibility of sharing the common controller synthesis problem into several sub-tasks associated with different control objectives for each control loop.
Let’s notice that representation of an operator setter in the same form as coordinate setter allows to define more accurately its role in an additional contour of feedback. This setter defines desirable dynamics for comparison with the current behavior of object. Thus setting dynamics D(s) can be changed in the process of object functioning as well as setting influence g(t).
9. DEVELOPMENT OF THE VSS THEORY BY SIMULTANEOUS STABILIZATION METHODS
8. COORDINATE-TO-OPERATOR FEEDBACK IN VVS
Theory of the VSS provides effective methods of constructing control algorithms for the state vector (particularly stabilization) allowing to compensate parametric perturbations acting on the plant. But the essential assumption is immutability of dynamic degree (the dimension of the phase vector) of the plant during its functioning. This assumption is primarily due to the fact that in case of stabilization the feedback is clearly dependent on the phase variables of the plant. If the order of the stabilized plant changes abruptly, and therefore changes its set of variables, the next question arises: which form of feedback are stabilizing such an plant. However, during the functioning of the plant, its order can abruptly change, for example, because of the loss of efficiency of its individual subsystems. Actually, in this case we can talk about the
This new feedback µ = R2 σ is naturally to call the coordinate-to-operator feedback, its “output” is the feedback operator R1 of the main loop cotrol. Thus there was suggested a division of signals circulating in the control system to the signals of two different types: — signals-variables, in this case, for example, e(t), which is transformed; — signals-operators, in this case, µ, which determine the conversion of dynamic links. 12
2013 IFAC MIM June 19-21, 2013. Saint Petersburg, Russia
problem of stabilization by state vector of a finite family of dynamic plants of different degrees. Currently, one of the branches of the theory of automatic control studying also the problem of stabilization of plants of different orders, is the theory of simultaneous stabilization. Applying the method of extending the dynamic degree, consisting in the completion of each plant of the family by an independent stable subsystem, we can reduce the task of simultaneous stabilization of plants of different degrees to the simultaneous stabilization of plants of the same degree. Then, based on the methods of the theory of variable structure systems the variable structure controller stabilizing by state vector a given finite family of linear dynamic plants can be constructed.
−1 −1 1 3 A2 = 4 −1 −2 , b2 = 1 , 1 −2 −5 −2 Γ1 = {2, 3}, Γ2 = {1, 3, 4}, Γ = {1, 2, 3, 4}, n = 4. Let’s extend both systems to the same order n = 4. To the first system we must add the coordinates x1 , x4 and the corresponding differential equations, and to the second — x2 and the corresponding differential equation.
Let’s take ¶ 0 1 , A¯2 = −2, −2 −3 which will determine the differential equations for the additional coordinates. Then 0 0 0 1 0 0 2 −3 0 1 A˜1 = , ˜b = , 0 0 1 2 1 −2 -2 0 0 -3 0 −1 0 −1 1 3 0 -2 0 0 , ˜b2 = 0 A˜2 = 4 1 0 −1 −2 −2 1 0 −2 −5 and, thus, the systems x˙ 1 = x4 , x˙ = 2x − 3x + u, 2 2 3 , x ˙ = x + 2x − 2u, 3 2 3 x˙ 4 = −2x1 − 3x4 x˙ 1 = −x1 − x3 + x4 + 3u, x˙ = −2x , A¯1 =
Let us consider briefly the main idea of the expansion of the degree of dynamic systems. We consider k dynamic plants of different degrees with a scalar input x˙ j = Aj xj + bj u, j = 1, . . . , k, (2) where Aj ∈ Rnj ×nj , bj ∈ Rnj ×1 , u ∈ Rn , xj = (xi1 , . . . , xinj ) ∈ Rnj , i1 < . . . < inj , {i1 , . . . , inj } ⊆ {1, . . . , n}, where n = max{in1 , . . . , ink }. Denote an ordered set of indices {i1 , . . . , inj } through Γj , and the set {1, . . . , n} as Γ, i. e. Γj — is a subset nj of indices from Γ, which are ”used” in the j-th system. We denote by x ˜j a n vector from R , all components with indices from the set of Γ \ Γj are equal to zero. In fact, the vector xj — is the vector x ˜j with ”remoted” zero components. For systems (2) it is necessary to find a common controller in the form of state feedback u = u(x), u(0) = 0, x ∈ Rn , stabilizing all these plants. There has to be a global asymptotic stability of the closed systems x˙ j = Aj xj + bj u(˜ xj ), j = 1, . . . , k. (3) The main idea of this approach is as follows. Let A¯j (j = 1, . . . , k) be matrices of size n − |Γj |. Denoted by the A˜j = (Aj , A¯j , Γj ) matrix of order n, in which the intersection of rows and columns with the numbers i1 , . . . , inj ∈ Γj are the matrix Aj , in the intersection of rows and columns with indices in the Γ \ Γj is the matrix A¯j of the order n − |Γj |, and everywhere else are zeros. Similarly, a ˜bj = (bj , Γj ) denote the column vector from Rn , in whose elements with indices i1 , . . . , inj is a column bj and everywhere else are zeros. Using this notation, the system x˙ = A˜j x + ˜bj u can be considered as an extension of the system x˙ j = Aj xj + bj u by adding an independent subsystem of the order n − |Γj |. As an explanation of the procedure for extending dynamic order let’s give an example. Example. We consider ½ x˙ 2 = 2x2 − 3x3 + u, x˙ 3 = x2 + 2x3 − 2u, Then A1 =
µ
2
¶
,
b1 =
µ
1 −2
¶
2
x˙ = 4x1 − x3 − 2x4 + u, 3 x˙ 4 = x1 − 2x3 − 5x4 − 2u. are the expansions of systems (4).
Note that added coordinates do not depend on the old and do not influence them directly (but can influence through control u = u(x), x ∈ R4 ). Moreover, if the A¯1 A¯2 — Hurwitz (as suggested in the example), then added subsystems are asymptotically stable, and it is natural to ask the following question: if a control in the form of feedback on the expanded state vector u = u(x1 , x2 , x3 , x4 ) stabilizes expanded systems, whether it will stabilize the original system? The following theorem allows us to reduce simultaneous stabilization problem of a finite family of dynamic plants (2) of different degrees n1 , . . . , nq to the problem of simultaneous stabilization of plants of the same degree n by continuous universal stabilizer.
two systems x˙ 1 = −x1 − x3 + x4 + 3u, x˙ 3 = 4x1 − x3 − 2x4 + u, (4) x˙ 4 = x1 − 2x3 − 5x4 − 2u.
2 −3 1 2
µ
Theorem 1. Suppose that for some set of stable matrices A¯∗1 , . . . , A¯∗k controller u(x) (u ∈ C(Rn )) is a universal stabilizer for a family of plants of degree n x˙ = A˜j x + ˜bj u, A˜j = (Aj , A¯∗j , Γj ), j = 1, . . . , k. (5) Then it simultaneously stabilizes the family of linear plants (2).
, 13
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Theorem analogous to theorem 1 can be formulated for universal variable structure stabilizers of form ( + u (x), if σ(x) > 0, u(x) = (6) u− (x), if σ(x) < 0, where σ(x) = cx is a rupture surface, c ∈ R1×n , u+ (x), u− (x) are continuous homogeneous degrees α of function (u+ (mx) = mα u+ (x), u− (mx) = mα u− (x), α > 0). Theorem 2. Suppose that for some set of stable matrices A¯∗1 , . . . , A¯∗k variable structure controller (6) is the universal stabilizer for the family of plants (5) of the degree n. Then it simultaneously stabilizes the family of linear plants (2). For the universal stabilizer in the form of linear feedback u(x) = −Kx it is possible to prove stronger theorems. Theorem 3. The controller u(x) = −Kx iff is a universal regulator for a family of plants (2), when it simultaneously stabilizes the family (5) of extended linear plants of degree n for a set of stable matrices A¯∗1 , . . . , A¯∗k . Theorem 3 can be extended as follows. Theorem 4. Let for some set of stable matrices A¯∗1 , . . . , A¯∗k the controller u(x) = −Kx be the universal stabilizer for the family of extended plants (5), then it simultaneously stabilizes this family for any set of stable matrices A¯1 , . . . , A¯k . REFERENCES Emelyanov S.V. Automatische regelsysteme veranderlicher struktur. B.: Akad.-Verl., 1971. Emelyanov S.V., Korovin S.K. Control of Complex and Uncertain Systems. London: Springer-Verlag Ltd. 2000. Emelyanov S.V., Fomichev V.V., Fursov A.S. Simultaneous stabilization of linear dynamic plants by the variable-structure controller. Automation and Remote Control, 2012, v. 73, 7, p. 1126-1133. Bobyleva O N., Fomichev V V., Fursov A S. Sufficient conditions for the existence of a common stabilizer for a family of linear nonstationary plants. Differential Equations, 2012, v. 48, 7, p. 901-908. Korovin S.K., Il’In A.V., Fomichev V.V., Fursov A.S. A topological approach to the problem of existence of a common stabilizer for a family of dynamical systems. Doklady Mathematics, 2011, v. 84, 3, p. 911-916. Korovin S.K., Minyaev S.I., Fursov A.S. Approach to simultaneous stabilization of linear dynamic plants with delay. Differential Equations, 2011, v. 47, 11, p. 16121618. Korovin S.K., Fursov A.S. Simultaneous stabilization: Universal controller synthesis. Automation and Remote Control, 2011, v. 72, 9, p. 1852-1863.
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The Control Under Uncertainty Conditions: History and Perspective Stanislav V. Emelyanov ∗ V. V. Fomichev ∗∗ A. S. Fursov ∗∗∗ ∗
Institute for Systems Analysis of RAS, Moscow, 9, Prospekt 60-let Oktyabrya, 117312 Russia (e-mail: [email protected]; [email protected]). ∗∗ Lomonosov Moscow State University, Moscow, Leninskie gory, MSU, Dep. CMC, 119991 Russia (e-mail: [email protected]). ∗∗∗ Lomonosov Moscow State University, Moscow, Leninskie gory, MSU, Dep. CMC, 119991 Russia (e-mail: [email protected]).
Abstract: Within a problem of control of dynamic plants under uncertainty are considered both classical methods of the theory of systems with variable structure, and their modern development in the form of the theory of new types of feedback. The exact structure of feedback in control variable structure systems is clarified. Existence of two types of feedback is revealed: one classical feedback by the general regulation error, and another, new type, the coordinateto-operational feedback. Thus feedback is responsible for a set of positive dynamical properties for control variable structure systems. Also the perspective directions of further development of the theory of new types of feedback are considered. In particular, methods of simultaneous control by families of dynamic plants are considered are considered. Keywords: Feedback, stabilization, control system, variable structure system, sliding mode, new feedback types, coordinate-to-operator feedback, simultaneous stabilization. 1. INTRODUCTION
fluence of a head wind aboard the plane can be an example of coordinate disturbance and the plane with changing geometry of a wing represents the system being under the influence of parametrical disturbance.
Problem of the control under uncertainty is one of main problems of the automatic control theory. It has an old story. As known, a controllable plant is characterized by input-output relationship W , that is the relation between sets of input and output signals
In modern control theory the control plant is described by systems of differential equations. For example, a linear scalar stationary object under the action of external perturbations can be represented by the following equations ½ x˙ = (A + ∆A)x + (b + ∆b)u + f (1) y = (c + ∆c)x, x ∈ Rn , u, y ∈ R1 ,
y = W u. The plant can be graphically represented as a block with the input u and the output y (fig. 1).
where a differential equation is an equation of state (vector x is a state vector of the object), and static ratio is an output equation. Matrices ∆A, ∆b, ∆c characterize parametric perturbations, f — coordinate perturbation. Let’s assume that only ranges of changes of external uncertainty (probably enough large) are known. That means that for (1) are known majorants a0 , b0 , c0 , f0 , which realize following estimations
Fig. 1
The object input u is called a control, and its input y — a variable coordinate. In fact, plant input is a some allocated coordinate, setting which you can change the behavior of the object. The goal of control is usually defined by desired behavior of the output coordinate of the plant.
|∆A| ≤ a0 ,
|∆c| ≤ c0 ,
|f | ≤ f0 .
With all the possible actions on the plant, the primal control problem can be formulated as follows: to provide a given operating mode of the object under perturbations operating in a wide range.
As a rule, the plant of control during its functioning is affected by perturbations of two types: coordinate and operational. Coordinate perturbations make an additive contribution to input or internal signals of plant, and operational perturbations are caused either change of parameters of plant (parametrical perturbations), or change of its dynamic order (structural perturbations). That conducts to change an input-output characteristies of an plant. In978-3-902823-35-9/2013 © IFAC
|∆b| ≤ b0 ,
Operating mode of plant is understood as some prescribed behavior of its controlled coordinate and desirable dynamics of the plant. Properties of dynamics characterize quality of the control system. Changing of the output variable is given by coordinate setter (fig. 2), 7
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control and in the coordinates; discrete variable structure systems; solution to the problem of tracking with high degree of astatism in case unknown influence; achievement of complete independence motion equations in a closed loop control system from some uncertainty factors; construction information and filtering devices, in particular, differentiators of multiple differentiation in case of a information deficit; systems of identification and optimization under uncertainty. Theory of VSS also has significant potential for application problems. Its methods are used in the autopilot, in the elements of industrial automation, robotics, in instrumentation and amplification equipment.
Fig. 2
where g(t) is a function (signal) that defines the desired behavior of the regulated output; goal of control is to provide a convergence to zero of the contol error: |e(t)| = |g(t) − y(t)| → 0 (fig. 3).
We now pose the following question: what is the fundamental feature of the device with variable structure controller, which allows to confer a closed system very positive dynamic and static properties under the essential uncertainty and to provide a theory of VSS as an independent direction in general feedback theory?
Fig. 3
Under the influence of external perturbations information about which is often insufficient, the communication between an input and an output of the plant becomes ambiguous and uncertain. That very complicate the solution of a problem of control.
As a rule, the most important advantages of the VSS carried out in sliding mode (although in the VSS can be implemented other types of movements), a very common response to the above question is the following: it’s all in the use of sliding mode.
Problem of the control under uncertainty is well developed section of the control theory. There is a large number of methods in this direction. Detailed analysis of features and conditions of applicability each of them, and, especially, their comparison, represents a difficult problem. Therefore we will consider one of the most difficult cases, namely, a problem of the control for the plant under the essential uncertainty conditions, being characterized by strong variability of parameters of plant.
Some looking ahead, let’s say that the sliding mode is not the prime cause of all the advantages of variable structure systems, but only one of many possible means of implementation of some more general principle. 2. PRINCIPLE OF FEEDBACK
About 60 years ago the idea of variable structure feedback controllers was put forward. It appeared very effective at the solution of a wide range problems of control and optimization under the essential uncertainty and also led to creation of the whole direction in the feedback theory — so-called variable structure systems (VSS).
Let’s now turn to a question of construction of control algorithm providing a given operating mode of the plant. If the plant is stable, its mathematical model is known exactly and it is not affected by any external perturbations, then it is possible to create control as a time function u(t). The similar principle of construction of the control algorithm carries the name of the program control principle.
The idea of a principle of structure variability consists in spasmodic change of communications between functional elements of a controller depending on a phase condition of the closed control system which in that case is named a control variable structure system (VSS). If the linear plant and functional elements of a controller are considered, it is possible to interpret corresponding VSS as set of linear subsystems and rules of transition from one element of this set to another. This rules depends from crossing by a phase point of dividing hyperplanes in phase space of system which name discontinuity surfaces. If the plant or functional elements are nonlinear, then the combination of nonlinear subsystems and, accordingly, nonlinear surfaces or discontinuity manifolds is difficult problem. Both in the first and in the second case the VSS is nonlinear dynamic system, it is described by differential equations with discontinuous second member of equation. The synthesis of VSS is reduced to the choice of discontinuity surfaces and initial set of subsystems that guarantee the solution of the control problem.
If the plant is affected by unknown coordinate perturbations which have essential impact on the operated coordinate, then the program control principle is inapplicable. In this case the next methods of compensation of coordinate perturbations possible — either to change an object design for each type of perturbations (but for this purpose it is necessary to know them in advance), or to form an input variable u for compensating this perturbation. The last method is possible when the perturbation f can be measured. Control method, in which the control u is generated based on the measured coordinate perturbation is called the principle of compensation. Essential lack of the principle of compensation is that it can’t be applied in case when there is no opportunity to measure coordinate perturbation and also if the plant is unstable. Both discussed principle do not use information about the result of control, i. e. about the output variable of the control object. In this case, comes to the aid a fundamental principle of control — the principle of feedback.
Examples of the use of the principle of variable structure for solution a large range of control problems are well known, so there is simply list of them: control on a state and an output in the linear SISO and MIMO plants which parametres and exterior forces are unknown and can change largely; control of systems with delay in the
The principle of feedback is based on the construction of the control algorithm, using information about the results 8
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of control. Thus operating signal u is formed according to the following scheme (fig. 4).
necessary to modify the control algorithm (the regulator R). There is a new problem: to find a method affecting on the regulator, allowing to change the control algorithm to compensate parametric perturbations, i. e. to ensure the invariance of the control system in reference to these perturbations. There are two possible approaches in the case of coordinate perturbations. One consists in estimation (identification) of changes in dynamics of the object, caused by parametrical perturbations, and further tuning of the controller. Various effective methods of the adaptive control theory can be used for this purpose. Another way is to use the principle of feedback that does not require exact knowledge of parametric perturbations.
Fig. 4
On this scheme there is an operator R which sets control algorithm. Let the operators W and R be linear and stationary, then the output of the covered by the feedback object W is related to a setting influence g and an additive noise f by the following ratio
Within the classical feedback theory one of possible ways to compensate parametric perturbations is the usage of the strong feedback. However, the invariance of the control system for the unknown perturbations is achieved by increasing the gain, which is undesirable because of existence of natural amplitude limits for signals in the system. But then the problem is to achieve the effect of large gain in the presence of amplitude limits without increasing the gain in the real system. It would be desirable to remain within the feedback theory, that is to solve the problem of the construction of control algorithm through new feedback. To solve this problem we need to extend the concept of feedback.
′
y=
W WR g+ f, 1 + WR 1 + WR
(W ′ describes additive occurrence of noise f ). The relation shows that the feedback changes operators of transmission from inputs g and f to the output y without any interference into the design of object — only by skillful use of information about the output. Thus, the instability of the plant is no longer a barrier to the construction of the control algorithm, as its stability can be achieved by using external feedback. It is possible to enumerate such known algorithms of compensation of nonmeasurable coordinate perturbations by means of feedback as, for example, strong feedback method, relay regulator, methods of providing the set order of astatism of the closed object.
3. PRINCIPLE OF DUALITY Let’s address to analogies to understand what has to be the nature of changing the control algorithm of the system.Let’s remember the so-called principle of a duality which assumes that the same object can have the dual nature (in other words it can behave in different ways, depending on the nature of the interaction with another objects). There is one well-known example.
Principle of the feedback is one of the main directions of the modern feedback theory. In comparison with a case of coordinate perturbations, there is qualitatively other situation with a plant under parametric perturbations (fig. 5).
Neumann’s principle or the principle of joint storage of programs and data in the computer’s memory reveals the dual nature of a number. It consists in the following: both programs (commands), and data are stored in the same memory, thus it is possible to carry out the same actions over commands, as over data. It should be noted that Neumann’s principle was really the revolutionary idea which importance it is difficult to overestimate. Originally the program was set by installation of crossing points on the special switching panel. This was a very time-consuming task and sometimes it takes a few days to change the program (while the actual calculation lasted no more than a few minutes). Presence of the given set of executable commands and programs was a specifics of the first computer systems. Today, such an architecture is used to simplify the design of the computing device. So, desktop calculators, in principle, are devices with a fixed set of executable programs. They can be used for mathematical calculations, but it is impossible to apply them for text processing, computer games, for viewing graphics or video. Changing the firmware for such devices require almost complete their transformation and is impossible in most cases. Everything was changed by the
Fig. 5
In this case for system with feedback it isn’t possible to carry out regulator R synthesis in a ”program” way. The main difficulty lies in the fact that information on the parametric uncertainty can’t be used in the control algorithm (due to the fact that it is not a priori known). Therefore feedback for stabilization indefinite object has to make system very robast which is hardly possible, or somehow change itself during the control process. Thus, due to unpredictable changes of the operator of the object in the process of its functioning (and as a consequence, changes in the dynamics of a closed-loop system), it is 9
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a shift in the ψ-cell are organized on a some surface which is in the phase space of the closed system σ(x) = 0, on which should be organized sliding mode. In fact, in ψcell is a switching of structures in the phase space of the closed system.
idea of the storage of computer programs in the general memory. Neumann was the first who realized that the program can be stored as a set of zeros and ones, and in the same memory space as numbers processed by it. Lack of a fundamental difference between the program and the data opens up a whole range of possibilities. For example, a program in its processing can also be reprocessed which allows to set in a program the rules for obtaining some parts of it (so an execution of the loops and subroutines is organized in a program). Moreover, the commands of one program can be obtained as the results of the execution of another program. In other words, the entered principle made it possible for the computer itself to form a program in accordance with the results of calculations. 4. THE DUAL NATURE OF THE SIGNALS IN THE CONTROL SYSTEM Using the example of the dual nature of a number it is feasible to suggest the idea: using the principle of a duality in the feedback theory. Primary signals (coordinates) that are processed in the control system are presented as functions of time and control algorithm implemented by the regulator, is an operator that converts these functions. But in turn, this operator can be represented as an element of a set of stabilizing feedbacks. Wherein it is possible to parameterize set this operators by one or more parameters and to set a one-to-one correspondence between the parameters and controllers. Now the choice of algorithm of management, in fact, will mean a choice of the corresponding parameter (or of the vector of parameters). Thus, changing parameter, we change the control algorithm. But the parameter changing in time is already admissible to consider as the same signal which can be processed and formed in a control system.
Fig. 6
Here explicitly dependence of control u is seen on two variables: from the control error e and some variable σ, characterizes the deviation of the motion in the system from the surface σ(x) = 0. In the canonical theory of VSS the target of feedback synthesis is them consistently zero: first variable σ = 0, and then the main error e = 0. This is due to the fact that the position of the sliding surface σ(x) = 0 in the phase space of the system affects the transients in the closed control system and, therefore, the choice of the sliding surface σ(x) = 0 defines the quality of the transition processes in the system, in other words, the dynamics of a closed system and that the equality e = 0 provides the required static in the closed-loop control system. In fact, the condition σ = 0 provides a sliding mode along the surface σ(x) = 0. This motion is invariant to parametric perturbations. While outside this surface the motion of a closed system, of course, will depend on these disturbances.
Thus we come to an important principle revealing the dual nature of the signals in the control system — binarity principle. • Signals in the control system have a dual nature and and can act either as variable (transfer information), over which transformations are carried out, or as operators defining these transformations • Absence of a fundamental difference between the signals-coordinates (ordinary variables) and signalsoperators allows to change the control algorithm basing on operating conditions 5. FEATURES OF FEEDBACK IN THE VSS The binarity principle allows to unerstand the nature of efficiency of VSS. Let’s consider the block diagram of the standard VSS, solving task of stabilizing under the action of parametric perturbations (fig.6).
Fig. 7
Thus, in case of using the VSS and generating sliding modes, wich invariant with respect to existing parametric perturbations, it is necessary to specify in advance the dynamics of a closed system.
Here W is an plant of control, Ψ — variable structure controller (ψ-cell), G — coordinate setpoint (in case of the stabilization task g ≡ 0), ∆W — parametric perturbations, f — coordinate perturbations. Discontinuous VSS feedback u = ψ(σ)e is arranged on an error tracking e = g − y,
Therefore, in accordance with the principle of regulation by the deviation, it is natural to introduce the operator D, which determines a setpoint of the dynamic properties of a closed system. In this case, the value of σ = De 10
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determines the deviation of the actual behavior of transients in the closed system of control from its desired behavior, where σ = 0.
multiplying. But if one of the signals is a C-signal (z), and the other is an O-signal (y, fig. 10), then the multiplier’s action consists of transforming the C-signal by an operator which is defined by the O-signal, so an operator y obtains to a signal z(t) and the result is a C-signal x = y(z(t)). In fact, we can assume that to the signals z and y is obtained an operator multiplication x = y ◦ z. Note that the action of the operator may also be reduced to a simple multiplication of the signals.
These considerations allow to open explicite structure of the ψ-cell and redraw Fig. 1 in the following form (fig. 7). Here, R1 and R2 are the operators setting the required feedback regulation between the error e and the control u. In the scheme of Fig. 7 is clearly seen having a second (additional to the main) loop with his setter D, controller ˜ , consisting of the original plant, R2 and generic plant W covered with the main feedback with the controller R1 , parameterized signal µ, i. e. R(µ).
Depending on the types of inputs and outputs, there are four main types of dynamic objects (fig. 11).
For solution of problem of operator R2 choice. Let’s consider in more detail the binarity principle. 6. BASIC CONCEPTS: TYPES OF ELEMENTS For the development of a new approach to the construction of control algorithms based on the introduced binarity principle it is necessary to enter some of new concepts and designations.
Fig. 11
1. A CC-object. The input and the output are signalscoordinates.
Let’s denote signals-operators by double arrows and call them O-signals for short. Ordinary signals-coordinates (Csignals) as before will be denoted by single arrows (fig. 8).
2. A CO-object. The input is a signal-coordinate and the output is a signal-operator. 3. An OC-object. The input is a signal-operator and the output is a signal-coordinate.
Fig. 8
4. An OO-object. The input and the output are signalsoperators.
An important role in the approach is played by an element having two independent entrances for two types of signals — O-signals and C-signals. Let’s call such an element a binary element (fig. 9).
The following ideas are offered: • Firstly, it is necessary to synthesize not the stabilizing operator of feedback, but the algorithm of its construction, i. e. to generate appropriate signalsoperators. In this case not signal but the operator will be regulator. • Secondly, as synthesis of a control system has to happen under the uncertainty conditions, it is necessary to use feedback in an additional contour of generation for the formation of signals-operators. • Thirdly, as parametrical perturbations cause change of dynamics of the closed object, then to use a setter of a desirable dynamics of an object — operator setter — for a contour of generation of signalsoperators (let’s notice that in coordinate contour of feedback for the output coordinate of the object is used the coordinate setter). • Fourthly, to try to use known methods of the classical theory of feedback in a contour of generation of signals-operators for formation of these signals.
Fig. 9
In fact, the binary element is a dynamic or static element, which parameters are changing in the process of functioning of the control system at the expense of O-signal µ. Thus an operator B (transforming C-signal x), is changing. An example of such a binary element is bilinear element (fig. 10), carrying out multiplication of input signals.
7. OPERATOR SETTER One of the offered above ideas consists in use of an operator setter in the additional contour of feedback. The basic purpose of a setter consists in maintenance of the set dynamics of the plant and invariancy of this dynamics towards active parametrical perturbations. Actually, thus it is a question of ensuring the specified quality of the
Fig. 10
If on an entrance of the multiplier two C-signals are given, then they just multiply and the output is the result of 11
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For a better visual separation of the functions performed by these signals in the diagram it is appropriate to denote the first variable, as is customary in the classical control theory, by a single arrow, and the second kind of variables by a double.
closed control system. The quality of the control system can be defined in various ways. One of possibly ways is to set the dynamics of the control error e(t) by a differential equation, to which this error has to obey. Setting dynamics of the error, we set desirable dynamics of the closed object of control. Certainly, thus it is necessary to specify a set of setting coordinate signals g(t), for which the error has to have the desirable dynamics. On this basis, let’s set quality of a control system by means of an operator setter (fig. 12) defining change of
Fig. 12
the dynamics of the object (here D(s) is an operator, setting dynamics). Let’s notice that the output signal of an operator setter is a signal-operator and its action we will understand according to the following equivalent schemes (fig. 13) which mean applying the operator D(s) to the
Fig. 14
Such a designation, without changing anything on the merits, allows to identify visually on the schemes the new type of coordinate-to-operator feedback. Fig. 7 in this notation takes the following form (Fig. 14), in which the coordinate-to-operator feedback contour is now clearly visible.
Fig. 13
variable e(t), i. e. σ = De; purpose of the control is to provide a condition σ(t) → 0. For example, if D(s) is given d + q or in symbolic in the form of a differential operator dt d form s + q (s = dt ), then De = e˙ + qe = σ. If σ → 0, then the control error e(t) tends to a solution of the equation e˙ + qe = 0, and then dynamics of object becomes close to dynamics of object of the first order. By analogy with a static error e(t) let’s call the σ(t) dynamic error.
Analysis of the block diagram in the new notation reveals that the fundamental basis of the independence movement in the sliding mode from the parameters of the plant is not the sliding mode itself but a new type of feedback. In the simplest case, the controller R2 of the additional circuit is a relay element responsible for the emergence of ideal sliding mode. Note that in practice such a control algorithm generates real sliding mode, which has a number of drawbacks. However, the use of other types of controllers R2 can afford to get other, including smooth, motion of a closed system. This is one of the factors that lead to promising perspectives of new types of feedback to control plants under uncertainty. Another such factor is possibility of sharing the common controller synthesis problem into several sub-tasks associated with different control objectives for each control loop.
Let’s notice that representation of an operator setter in the same form as coordinate setter allows to define more accurately its role in an additional contour of feedback. This setter defines desirable dynamics for comparison with the current behavior of object. Thus setting dynamics D(s) can be changed in the process of object functioning as well as setting influence g(t).
9. DEVELOPMENT OF THE VSS THEORY BY SIMULTANEOUS STABILIZATION METHODS
8. COORDINATE-TO-OPERATOR FEEDBACK IN VVS
Theory of the VSS provides effective methods of constructing control algorithms for the state vector (particularly stabilization) allowing to compensate parametric perturbations acting on the plant. But the essential assumption is immutability of dynamic degree (the dimension of the phase vector) of the plant during its functioning. This assumption is primarily due to the fact that in case of stabilization the feedback is clearly dependent on the phase variables of the plant. If the order of the stabilized plant changes abruptly, and therefore changes its set of variables, the next question arises: which form of feedback are stabilizing such an plant. However, during the functioning of the plant, its order can abruptly change, for example, because of the loss of efficiency of its individual subsystems. Actually, in this case we can talk about the
This new feedback µ = R2 σ is naturally to call the coordinate-to-operator feedback, its “output” is the feedback operator R1 of the main loop cotrol. Thus there was suggested a division of signals circulating in the control system to the signals of two different types: — signals-variables, in this case, for example, e(t), which is transformed; — signals-operators, in this case, µ, which determine the conversion of dynamic links. 12
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problem of stabilization by state vector of a finite family of dynamic plants of different degrees. Currently, one of the branches of the theory of automatic control studying also the problem of stabilization of plants of different orders, is the theory of simultaneous stabilization. Applying the method of extending the dynamic degree, consisting in the completion of each plant of the family by an independent stable subsystem, we can reduce the task of simultaneous stabilization of plants of different degrees to the simultaneous stabilization of plants of the same degree. Then, based on the methods of the theory of variable structure systems the variable structure controller stabilizing by state vector a given finite family of linear dynamic plants can be constructed.
−1 −1 1 3 A2 = 4 −1 −2 , b2 = 1 , 1 −2 −5 −2 Γ1 = {2, 3}, Γ2 = {1, 3, 4}, Γ = {1, 2, 3, 4}, n = 4. Let’s extend both systems to the same order n = 4. To the first system we must add the coordinates x1 , x4 and the corresponding differential equations, and to the second — x2 and the corresponding differential equation.
Let’s take ¶ 0 1 , A¯2 = −2, −2 −3 which will determine the differential equations for the additional coordinates. Then 0 0 0 1 0 0 2 −3 0 1 A˜1 = , ˜b = , 0 0 1 2 1 −2 -2 0 0 -3 0 −1 0 −1 1 3 0 -2 0 0 , ˜b2 = 0 A˜2 = 4 1 0 −1 −2 −2 1 0 −2 −5 and, thus, the systems x˙ 1 = x4 , x˙ = 2x − 3x + u, 2 2 3 , x ˙ = x + 2x − 2u, 3 2 3 x˙ 4 = −2x1 − 3x4 x˙ 1 = −x1 − x3 + x4 + 3u, x˙ = −2x , A¯1 =
Let us consider briefly the main idea of the expansion of the degree of dynamic systems. We consider k dynamic plants of different degrees with a scalar input x˙ j = Aj xj + bj u, j = 1, . . . , k, (2) where Aj ∈ Rnj ×nj , bj ∈ Rnj ×1 , u ∈ Rn , xj = (xi1 , . . . , xinj ) ∈ Rnj , i1 < . . . < inj , {i1 , . . . , inj } ⊆ {1, . . . , n}, where n = max{in1 , . . . , ink }. Denote an ordered set of indices {i1 , . . . , inj } through Γj , and the set {1, . . . , n} as Γ, i. e. Γj — is a subset nj of indices from Γ, which are ”used” in the j-th system. We denote by x ˜j a n vector from R , all components with indices from the set of Γ \ Γj are equal to zero. In fact, the vector xj — is the vector x ˜j with ”remoted” zero components. For systems (2) it is necessary to find a common controller in the form of state feedback u = u(x), u(0) = 0, x ∈ Rn , stabilizing all these plants. There has to be a global asymptotic stability of the closed systems x˙ j = Aj xj + bj u(˜ xj ), j = 1, . . . , k. (3) The main idea of this approach is as follows. Let A¯j (j = 1, . . . , k) be matrices of size n − |Γj |. Denoted by the A˜j = (Aj , A¯j , Γj ) matrix of order n, in which the intersection of rows and columns with the numbers i1 , . . . , inj ∈ Γj are the matrix Aj , in the intersection of rows and columns with indices in the Γ \ Γj is the matrix A¯j of the order n − |Γj |, and everywhere else are zeros. Similarly, a ˜bj = (bj , Γj ) denote the column vector from Rn , in whose elements with indices i1 , . . . , inj is a column bj and everywhere else are zeros. Using this notation, the system x˙ = A˜j x + ˜bj u can be considered as an extension of the system x˙ j = Aj xj + bj u by adding an independent subsystem of the order n − |Γj |. As an explanation of the procedure for extending dynamic order let’s give an example. Example. We consider ½ x˙ 2 = 2x2 − 3x3 + u, x˙ 3 = x2 + 2x3 − 2u, Then A1 =
µ
2
¶
,
b1 =
µ
1 −2
¶
2
x˙ = 4x1 − x3 − 2x4 + u, 3 x˙ 4 = x1 − 2x3 − 5x4 − 2u. are the expansions of systems (4).
Note that added coordinates do not depend on the old and do not influence them directly (but can influence through control u = u(x), x ∈ R4 ). Moreover, if the A¯1 A¯2 — Hurwitz (as suggested in the example), then added subsystems are asymptotically stable, and it is natural to ask the following question: if a control in the form of feedback on the expanded state vector u = u(x1 , x2 , x3 , x4 ) stabilizes expanded systems, whether it will stabilize the original system? The following theorem allows us to reduce simultaneous stabilization problem of a finite family of dynamic plants (2) of different degrees n1 , . . . , nq to the problem of simultaneous stabilization of plants of the same degree n by continuous universal stabilizer.
two systems x˙ 1 = −x1 − x3 + x4 + 3u, x˙ 3 = 4x1 − x3 − 2x4 + u, (4) x˙ 4 = x1 − 2x3 − 5x4 − 2u.
2 −3 1 2
µ
Theorem 1. Suppose that for some set of stable matrices A¯∗1 , . . . , A¯∗k controller u(x) (u ∈ C(Rn )) is a universal stabilizer for a family of plants of degree n x˙ = A˜j x + ˜bj u, A˜j = (Aj , A¯∗j , Γj ), j = 1, . . . , k. (5) Then it simultaneously stabilizes the family of linear plants (2).
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2013 IFAC MIM June 19-21, 2013. Saint Petersburg, Russia
Theorem analogous to theorem 1 can be formulated for universal variable structure stabilizers of form ( + u (x), if σ(x) > 0, u(x) = (6) u− (x), if σ(x) < 0, where σ(x) = cx is a rupture surface, c ∈ R1×n , u+ (x), u− (x) are continuous homogeneous degrees α of function (u+ (mx) = mα u+ (x), u− (mx) = mα u− (x), α > 0). Theorem 2. Suppose that for some set of stable matrices A¯∗1 , . . . , A¯∗k variable structure controller (6) is the universal stabilizer for the family of plants (5) of the degree n. Then it simultaneously stabilizes the family of linear plants (2). For the universal stabilizer in the form of linear feedback u(x) = −Kx it is possible to prove stronger theorems. Theorem 3. The controller u(x) = −Kx iff is a universal regulator for a family of plants (2), when it simultaneously stabilizes the family (5) of extended linear plants of degree n for a set of stable matrices A¯∗1 , . . . , A¯∗k . Theorem 3 can be extended as follows. Theorem 4. Let for some set of stable matrices A¯∗1 , . . . , A¯∗k the controller u(x) = −Kx be the universal stabilizer for the family of extended plants (5), then it simultaneously stabilizes this family for any set of stable matrices A¯1 , . . . , A¯k . REFERENCES Emelyanov S.V. Automatische regelsysteme veranderlicher struktur. B.: Akad.-Verl., 1971. Emelyanov S.V., Korovin S.K. Control of Complex and Uncertain Systems. London: Springer-Verlag Ltd. 2000. Emelyanov S.V., Fomichev V.V., Fursov A.S. Simultaneous stabilization of linear dynamic plants by the variable-structure controller. Automation and Remote Control, 2012, v. 73, 7, p. 1126-1133. Bobyleva O N., Fomichev V V., Fursov A S. Sufficient conditions for the existence of a common stabilizer for a family of linear nonstationary plants. Differential Equations, 2012, v. 48, 7, p. 901-908. Korovin S.K., Il’In A.V., Fomichev V.V., Fursov A.S. A topological approach to the problem of existence of a common stabilizer for a family of dynamical systems. Doklady Mathematics, 2011, v. 84, 3, p. 911-916. Korovin S.K., Minyaev S.I., Fursov A.S. Approach to simultaneous stabilization of linear dynamic plants with delay. Differential Equations, 2011, v. 47, 11, p. 16121618. Korovin S.K., Fursov A.S. Simultaneous stabilization: Universal controller synthesis. Automation and Remote Control, 2011, v. 72, 9, p. 1852-1863.
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