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3.4 Equivalent transformations of VSV models and canonical representations

Lecture



As noted in § 3.1.1, various ways are possible to select state variables of a dynamic system. The ambiguity of this choice determines the non-uniqueness of the input-state-output models [M4], [M5] (or [M6], [M7]) corresponding to a particular input-state model [M1] ([M2] or [M3]), since the choice of other state variables leads to a different model of BCB. On the other hand, the original model of BCB can be specially transformed, which is usually associated with changes in the basis (system of coordinates) of the state space   3.4 Equivalent transformations of VSV models and canonical representations . This kind of transformation is called equivalent, or similarity transformation .



3.4.1. Equivalent conversion . Let the input-output model of a single-channel system (control object) be given by the operator equation



(3.93) a ( p ) y ( t ) = b ( p ) u ( t ),



Where



  3.4 Equivalent transformations of VSV models and canonical representations .



and   3.4 Equivalent transformations of VSV models and canonical representations - the roots of the characteristic equation (system poles), or the equation [M3], where



(3.94)   3.4 Equivalent transformations of VSV models and canonical representations ,



and the model ENE is obtained in the form [М6], [М7]. We introduce a new (transformed) state vector:



(3.95)   3.4 Equivalent transformations of VSV models and canonical representations ,



Where   3.4 Equivalent transformations of VSV models and canonical representations - transformation matrix (similarity), satisfying the condition   3.4 Equivalent transformations of VSV models and canonical representations . Then there is an inverse transformation



(3.96)   3.4 Equivalent transformations of VSV models and canonical representations .



Differentiating (3.95) and substituting (3.96), [М6] we find:



(3.97)   3.4 Equivalent transformations of VSV models and canonical representations



and



(3.98)   3.4 Equivalent transformations of VSV models and canonical representations .



The resulting expressions are rewritten as



[M6 * ]   3.4 Equivalent transformations of VSV models and canonical representations ,



[M7 * ]   3.4 Equivalent transformations of VSV models and canonical representations ,



Where



(3.99)   3.4 Equivalent transformations of VSV models and canonical representations



(matrix like A ),



(3.100)   3.4 Equivalent transformations of VSV models and canonical representations ,



(3.101)   3.4 Equivalent transformations of VSV models and canonical representations .



Such matrices have the following properties:



but)   3.4 Equivalent transformations of VSV models and canonical representations   3.4 Equivalent transformations of VSV models and canonical representations   3.4 Equivalent transformations of VSV models and canonical representations



b)   3.4 Equivalent transformations of VSV models and canonical representations   3.4 Equivalent transformations of VSV models and canonical representations   3.4 Equivalent transformations of VSV models and canonical representations



The model [М6 * ], [М7 * ] is called the equivalent (similar) model [М6], [М7]. Fairly obvious property:



(3.102)   3.4 Equivalent transformations of VSV models and canonical representations   3.4 Equivalent transformations of VSV models and canonical representations ,



those. for such systems, the connections of the output and input variables are preserved, and, therefore, the BB [M1], [M2], [M3] models.



3.4.2. Canonical representations of models VSV . The simplest input-state-output models corresponding to the initial equations of the system [М6], [М7] are called canonical representations ( forms ) .



Diagonal shape is a model represented by the equations of state



(3.103)   3.4 Equivalent transformations of VSV models and canonical representations



and the output equation



(3.104)   3.4 Equivalent transformations of VSV models and canonical representations .



(Fig. 3.24). The model can be written in a compact form [M6 *] [M7 *], where



  3.4 Equivalent transformations of VSV models and canonical representations ,   3.4 Equivalent transformations of VSV models and canonical representations ,   3.4 Equivalent transformations of VSV models and canonical representations .



Systems with different poles are diagonalized.   3.4 Equivalent transformations of VSV models and canonical representations   3.4 Equivalent transformations of VSV models and canonical representations . At the same time (see clause 3.1.3) of the matrix of the main and transformed system are related by



(3.105)   3.4 Equivalent transformations of VSV models and canonical representations ,



those. transformation matrix is ​​like



(3.106)   3.4 Equivalent transformations of VSV models and canonical representations .



  3.4 Equivalent transformations of VSV models and canonical representations



Fig. 3.24. The block diagram of the canonical (diagonal) form



The model of a fully controlled system [10] can be reduced to a controlled (Frobenius)) canonical form



(3.107)   3.4 Equivalent transformations of VSV models and canonical representations



and



(3.108)   3.4 Equivalent transformations of VSV models and canonical representations ,



(Fig. 3.25). This form corresponds to the vector-matrix equations [M6 *] [M7 *], in which



  3.4 Equivalent transformations of VSV models and canonical representations   3.4 Equivalent transformations of VSV models and canonical representations ,   3.4 Equivalent transformations of VSV models and canonical representations ,



C * = [ b n b n -1 ... b 2   b 1 ] ,



  3.4 Equivalent transformations of VSV models and canonical representations and   3.4 Equivalent transformations of VSV models and canonical representations - the coefficients of equation (3.107),



a T = [ -a n -   a n -1 ... -a 2   -a 1 ] , 0 = [0 0 ... 0 0] T ,



I is the unit matrix of size ( n- 1)   3.4 Equivalent transformations of VSV models and canonical representations ( n- 1). State matrix   3.4 Equivalent transformations of VSV models and canonical representations is called the accompanying matrix of a polynomial   3.4 Equivalent transformations of VSV models and canonical representations or Frobenius matrix.



  3.4 Equivalent transformations of VSV models and canonical representations



Fig. 3.25. Canonical controlled form





The transformation matrix to the canonical controllable form is as



(3.109) P = U * U -1 ,



where U and U * are the controllability matrices of the initial [10] and canonical [10] models, respectively. For the case of n = 3, the following holds:



  3.4 Equivalent transformations of VSV models and canonical representations .



The model of a fully observable system (see [10]) can be reduced to the observable (Frobenius)) canonical form of the form :



(3.110)   3.4 Equivalent transformations of VSV models and canonical representations



and



(3.111)   3.4 Equivalent transformations of VSV models and canonical representations ;



(Fig. 3.26). This canonical form corresponds to the vector-matrix equations [M6 *] [M7 *], where   3.4 Equivalent transformations of VSV models and canonical representations - accompanying (Frobenius) matrix of the species



  3.4 Equivalent transformations of VSV models and canonical representations   3.4 Equivalent transformations of VSV models and canonical representations ,   3.4 Equivalent transformations of VSV models and canonical representations ,   3.4 Equivalent transformations of VSV models and canonical representations ,



  3.4 Equivalent transformations of VSV models and canonical representations



Fig. 3.26. Canonical observable form



The transformation matrix is ​​as



(3.112) P = ( Q * ) -1 Q,



where Q and Q * are the observability matrices of the original and canonical models [10]. For the case of n = 3, the following holds:



  3.4 Equivalent transformations of VSV models and canonical representations .


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Mathematical foundations of the theory of automatic control

Terms: Mathematical foundations of the theory of automatic control