Robust stabilization of the multilinked object

We consider the problem of robust control for a multilink object in the presence of the signal and parametrical uncertainty when the measurement of the derivatives of input and output signals of the local subsystems is not available. Proposed and justified the use of auxiliary circuits for each of the subsystems that eliminates the use of the vector of regression during the generate of the control signal, where by the reduced order closed-loop control system. The advantage of the proposed robust control law is that it remain unchanged when exposed to unknown disturbances, and in their absence. In the first case, the system is dissipative, while in the second case, it one is asymptotically stable.


Introduction
Problem of control of uncertain objects is one of the classical problems of control theory. Complex multiply linked objects take a special place among object of control, the control of which is complicated by the linkages between local subsystems. The large number of the interconnected subsystems influencing the each other complicates the traditional control problems, and the requirement of the decentralization comes to the fore [1,2]. The utilization of the decentralized algorithms corresponds to the nature of large interconnected systems, because it expects the distribution of the system components in space. And in addition to this, the decentralized management structure allows getting more qualitative and trusted control system, because it brings the governing body to the object and simplifies the system structure considerably [3,4,5].
In this paper, we consider that class of multilinked objects, which mathematical models can be brought to a canonical form [6,7]. To assess the perturbations introduced auxiliary circuit and the observer derivative. Compensation of the perturbations produced by the control law with opposite sign to the assessment of disturbances. The control law is constructed in such way as to offset the impact of local interconnections of subsystems [8,9,10,11]. There are developed a lot of methods of construct of adaptive and robust systems under parametric uncertainty of the object in the modern control theory. There are developed method extended errors, the method bypass, the algorithms of high order. At the core of these methods is the use of the filter or shunt device. A control signal is generated as the scalar product of vector of adjustable parameters and vector of regression. However, the task of creating a simple to implement and low-dimensional algorithms is still relevant. In this paper, we consider the stabilization problem for a dynamic object.

Statement of the problem
Let us examine the interconnected system with its dynamic processes which are described by the equations in the local subsystems so: -are linear differentiation operators whose elements depend on the vector of unknown parameters Ξ ∈ ξ ; Ξ -is a known aggregate of possible values of the vector ξ ; -is a scalar controlled variable of i-th subsystem which is accessible for measurement; -is a row matrix of the appropriate orders. In this case, the equation of the control object is presented in canonical form.
The equations (1) describe the dynamic processes in the local subsystems, and (2) in the cross coupling. Their transfer functions look like this as in

Robust stabilization of the multilinked dynamic object
We first solve the problem of the asymptotic robast stabilization when there are no external disturbances in the object (1), i.e. 0 ) , We have to do is to design the control system for which one condition will meet -is a linear differentiation operator with a constant rates. In the generation of control law will take estimate of the output signal with opposite sign. So it form is where a number 0 > which is realized by the observer [12] . ), ( ; the vector i H is chosen by special method so that the matrix -is an identity matrix of order ( ) ( ) . It is obvious, that the control law can be technically realizable, as it contains the known or measurable values. At the time, the solution of the problem gives the following statement. The Statement 1 If the hypotheses A.1 -A.5 are implemented, then the control law (5) with the observer (7) implements of the target condition (5).
Proof of Proposition 1. Inputting the control (5) to the equation of object (1) with (6), we'll get the last input-output model in the matrix vectorial form Following the [13], we introduce into consideration nullity vector i η as and write the equation (8), (9) in the composite matrix vectorial form and at the same time matrixes pi A will be Hurwitz. Let us take the Lyapunov function in the form of in which sij i P P, -are the positive definite symmetric matrixes satisfy the matrix relation: Because the matrixes pi A and sij A are Hurwitz, there are matrixes i P and sij P , that satisfy the matrix relation (12). Let us calculate the total derivative of the function (12) on the trajectories of the system (10), (2), with (12). Then the derivative of the Lyapunov function will look like , 2 the synthesized system is stable, because the target condition is satisfied (4).

Robust stabilization of the multilinked dynamic object under bounded disturbances
However, in practice it is difficult to create the situation for control of technical objects in ideal conditions when it is not exposed to external disturbances. The case where external disturbances are bounded is the simplest. Next, let us solve the problem of a decentralised stabilization when the external disturbances are present in the object (1), i.e. 0 ) , We had to design a control system for which next condition will be satisfied , In accordance with the approach presented in [13,14] let us ask one local control law in the form of (5) Implementing the control law (5) requires getting the estimate ) (t y i and its 1 − − i i m n derivatives, for which we will use the observer (7). It is obvious, that the control law can be technically realizable, as it contains the known or measurable values.
The Statement 2. If the hypotheses A.1 -A.5 are implemented, then the control law (5) with the observer (7) implements the limited nature of system trajectory (14).
Observation. But it should be noted, that choosing the number i θ of greater value, and the value χ of smaller, we can achieve the target condition (14).
Proof of Proposition 2. Let us convert the equation (14) to the vectorial matrix form where the matrix ,..., . Following the [13], we introduce the vector (9). Then from (7) and (16) we have   . To take account the fact that the matrixes are block diagonal, we will produce that positive definite symmetric matrixes i P must satisfy the equations for every subsystem Since the matrixes i F and sij A are Hurwitz we see that there are the matrixes i P and sij M , with the ratio (19). Let us take the Lyapunov function where H -are the positive definite symmetric matrixes, at the same time To take account the fact that the matrix H is a block diagonal, we will produce that positive definite symmetric matrixes i H must satisfy the equations for every subsystem , Since the matrixes i A 0 are Hurwitz we see that there are the matrixes i H which satisfy these matrix relation. To take account the estimated derivative of the functions (19), used (20) and the last inequality, we will calculate the total derivative of the Lyapunov function V=V1+V2, and we'll get the inequality , )) ( ), ( (  (5) greater. Unfortunately, we cannot get a more accurate estimate.

Example
Consider the multilinked control plant, dynamic processes which are described by the equations The control law is in the form of General block diagram of the control object shown in fig. 1. Here we have the multilinked control object consisting of K local subsystems. Control signal and the output signals from all the other K-1 local subsystems are inputs of each local subsystem. The local control unit generates a control signal, using an estimate of the state vector of the object that is obtained with the observer. Thus is implemented a decentralized control. This object is synthesized with a decentralized robust controller for stabilization of each of the local subsystems, the structure of which is shown in Fig.2. The control signal is generated with a opposite sign to the assessment of variables of the internal state. Compensation of external and structural disturbances is possible because introducing the auxiliary circuit. The disadvantage of the proposed control law is that the absolute magnitude of the coefficients K and α is impossible to find analytically. Their value is chosen at the modeling stage.
There are follows external disturbances in the object Consider the control object, dynamic processes in which case are described by the equation  Fig. 4 shows a model diagram of the first subsystem of the object (1) dissipative stabilization.

Discusion
The advantage of the proposed robust control law (5) is that it remains the same in the presence of unknown perturbation actions and in their absence. Only in the first case the system will be dissipative, while the second case it will be asymptotically robust.