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3.2 Models of Managed Systems

Lecture




  3.2 Models of Managed Systems



3 .2.1. Models input-state-output. First, we consider a particular case of a controlled dynamic system with one input u ( t ) and one output y ( t ), described by the equation [M1a], where a 0 = 1. We introduce state variables (3.1). Differentiating (3.3) in time and substituting [M1a], we find the equation of state:



(3.38)   3.2 Models of Managed Systems



In this case, the output equation still has the form (3.8). Equations (3.38) and (3.8) are the simplest case of the input-state-output (VSB) model.



In the more general case, the model BC of the controlled dynamic system [M1] contains the equations of state of the form:



[M4]   3.2 Models of Managed Systems



and the output equation [M5]. To convert them to a compact vector-matrix form, it is necessary to determine the state vector   3.2 Models of Managed Systems ,   3.2 Models of Managed Systems matrices   3.2 Models of Managed Systems ,   3.2 Models of Managed Systems as well as size entry matrix   3.2 Models of Managed Systems



  3.2 Models of Managed Systems   3.2 Models of Managed Systems .



Then the equations [M4], [M5], describing the model input-state-output, take the form:



[M6]   3.2 Models of Managed Systems ,



[M7]   3.2 Models of Managed Systems ,



Where   3.2 Models of Managed Systems .



Model [M6] and [M7] connects the input   3.2 Models of Managed Systems and exit   3.2 Models of Managed Systems through intermediate variables   3.2 Models of Managed Systems .



In the particular case, when the model ENE is presented in the form (3.38), (3.8), we obtain the matrix



  3.2 Models of Managed Systems ,   3.2 Models of Managed Systems ,   3.2 Models of Managed Systems .



Similarly, we obtain a model of a VSV multichannel (multiply connected) system (see model [M2m]). In general, it contains the equations of state of the form



[M4m]   3.2 Models of Managed Systems



and output equations [М5m]. Define   3.2 Models of Managed Systems -dimensional control vector:   3.2 Models of Managed Systems   3.2 Models of Managed Systems and m- dimensional vector of outputs   3.2 Models of Managed Systems   3.2 Models of Managed Systems as well as matrices



  3.2 Models of Managed Systems ,   3.2 Models of Managed Systems



dimensions   3.2 Models of Managed Systems and   3.2 Models of Managed Systems , respectively. Then the equations [M4m] and [M5m] can be rewritten as [M6] and [M7].



  3.2 Models of Managed Systems



Consider a perturbed dynamic system, (see [M1f]), i.e. controlled system to the input of which the input signal additionally acts (disturbance)   3.2 Models of Managed Systems ( t ). The equation of state of such a system is written in the form:



[M4f]   3.2 Models of Managed Systems ,



where d i ,   3.2 Models of Managed Systems , are the coefficients, and the output equation preserves the form [M5]. The vector matrix form of the model [M4f] is:



[M6f]   3.2 Models of Managed Systems ,



[M7] y = Cx ,



Where   3.2 Models of Managed Systems   3.2 Models of Managed Systems .



If several disturbing influences f k act on the input of the system, then in the equation [M6f]   3.2 Models of Managed Systems - vector of perturbations, and   3.2 Models of Managed Systems .



In the particular case (see (3.35)), the equations of state of the perturbed system take the form



(3.39)   3.2 Models of Managed Systems



and in the equation [М6f]



  3.2 Models of Managed Systems .



Consider solutions of equations [M6], [M7], assuming   3.2 Models of Managed Systems . The solution of the equation of state [M6] can be represented as:



(3.40)   3.2 Models of Managed Systems ,



Where   3.2 Models of Managed Systems ( t ) is the free component ( transition process of an autonomous system), corresponding to the solutions of the homogeneous differential equation [M6a] and depending on the initial conditions   3.2 Models of Managed Systems ,   3.2 Models of Managed Systems ( t ) - forced component corresponding to the transition process of the system [M6] with zero initial conditions   3.2 Models of Managed Systems (system response to the input action u ( t )).



Substituting (3.40) into the output equation [M7], we get



(3.41)   3.2 Models of Managed Systems ,



Where



(3.42)   3.2 Models of Managed Systems .



Note that the matrix



(3.43)   3.2 Models of Managed Systems



is a weight (pulse transient) matrix (with   3.2 Models of Managed Systems - the weight function) and, therefore, the equation (3.42) coincides with the previously given expression (2.35).



For perturbed models, the NER solutions can be obtained in a similar form.



3.2.2. The transfer function (matrix) of the model BCB and block diagrams. The above equations describing the input-state-output models can be written in operator form (see 2.1). Consider the equations [M6a], [M7]. Using the differentiation operator p = d / dt , we write   3.2 Models of Managed Systems . Then from the equation of state [M6a] after the simplest algebraic transformations we find



(3.44)   3.2 Models of Managed Systems .



Substituting the last expression into the output equation [M7] we get



(3.45)   3.2 Models of Managed Systems .



We introduce the notation



(3.46)   3.2 Models of Managed Systems



and write the previous equation in the form



(3.47)   3.2 Models of Managed Systems .



Comparison with the equation [M3m] shows that the matrix integro-differential operator W ( p ) is nothing but the transfer matrix of the controlled dynamic system (see § 2.1.3).



Consider the properties of the operator (3.4 6). Matrix   3.2 Models of Managed Systems is called resolvent and can be represented as



(3.48)   3.2 Models of Managed Systems ,



Where   3.2 Models of Managed Systems - numeric matrices   3.2 Models of Managed Systems . Then



(3.49)   3.2 Models of Managed Systems ,



Where   3.2 Models of Managed Systems ;   3.2 Models of Managed Systems - matrix operator.



For the case of a single-channel system ( m =   3.2 Models of Managed Systems = 1) W ( p ) - transfer function. Taking into account equation (3.48), we find that



(3.50)   3.2 Models of Managed Systems -



characteristic polynomial of the system; but



(3.51)   3.2 Models of Managed Systems -



characteristic polynomial of the right side of the differential equation (see § 2.1.1). Consequently, the eigenvalues ​​of the matrix A coincide exactly with the roots of the characteristic equation (poles) of the system



(3.52)   3.2 Models of Managed Systems .



To build a structural scheme corresponding to the model of BCB, we rewrite the equation of state [М6] in the operator form



(3.53)   3.2 Models of Managed Systems



and also use the equation of output [M7]   3.2 Models of Managed Systems . The block diagram of the system takes the form shown in Fig. 3.2.



  3.2 Models of Managed Systems



Fig. 3. 2. The block diagram of the model ENE



In the particular case when the equations of state are written in the form (3.38), (3.8) we find



(3.54)   3.2 Models of Managed Systems



(3.55)   3.2 Models of Managed Systems .



  3.2 Models of Managed Systems



Fig. 3.3. Block diagram of the model (3.38), (3.8)



The structural diagram is shown in Fig. 3.3 and corresponds to the cononically controlled form of the linear system representation (see § 3.3.2)



3.2.3. Static mode Consider the behavior of the model ENE with a constant input (control) effects, i.e.   3.2 Models of Managed Systems . In this case, the solution of the differential equation [М6а] corresponds to the established component of the transition process and is sought in the form   3.2 Models of Managed Systems . Noticing that   3.2 Models of Managed Systems we find:



(3.56)   3.2 Models of Managed Systems .



Provided that   3.2 Models of Managed Systems (i.e.   3.2 Models of Managed Systems ), the algebraic equation (3.56) is uniquely solvable for x y :



(3.57)   3.2 Models of Managed Systems .



Substituting the found solution into the output equation [M7], we find the static characteristic of the system [M6a], [M7]



(3.58)   3.2 Models of Managed Systems .



Taking into account the expression (2.47), we can clearly write



(3.59)   3.2 Models of Managed Systems



and get the expression (2.46).



If the system is such that   3.2 Models of Managed Systems , then matrix A is irreversible and the system does not have a static mode (see § 2.2.5).



Example 3.1. Consider a second-order system ( n = 2), the explosive model of which is represented by the equation



(3.60)   3.2 Models of Managed Systems .



State variables are defined by expressions.



(3.61)   3.2 Models of Managed Systems ;   3.2 Models of Managed Systems



and model BCB are both



(3.62)   3.2 Models of Managed Systems   3.2 Models of Managed Systems ,



(3.63)   3.2 Models of Managed Systems .



The vector matrix form of the model is



(3.62a)   3.2 Models of Managed Systems   3.2 Models of Managed Systems ,



(3.63a)   3.2 Models of Managed Systems .



Example 3.2.   3.2 Models of Managed Systems - the circuit considered in clause 1.1.2 is described by a differential equation of the first order:



(3.64)   3.2 Models of Managed Systems .



We introduce the notation



  3.2 Models of Managed Systems



(3.65)   3.2 Models of Managed Systems



and



(3.66)   3.2 Models of Managed Systems .



Equation (3.64) takes the form



(3.67)   3.2 Models of Managed Systems ,



(3.68)   3.2 Models of Managed Systems ,



Where   3.2 Models of Managed Systems and   3.2 Models of Managed Systems .



Example 3.3. Consider a second order differential equation



  3.2 Models of Managed Systems ,



describing the motion of a material point (see example 2.3).



The equation is reduced to



(3.69)   3.2 Models of Managed Systems ,



Where   3.2 Models of Managed Systems . We introduce state variables



(3.70)   3.2 Models of Managed Systems ,



(3.71)   3.2 Models of Managed Systems



and find a model of VSV as



(3.72)   3.2 Models of Managed Systems   3.2 Models of Managed Systems ,



(3.73)   3.2 Models of Managed Systems .



The vector matrix form of the model is



  3.2 Models of Managed Systems ,



  3.2 Models of Managed Systems .



This is a special case of the previously considered model (3.62a), (3.63a).


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Mathematical foundations of the theory of automatic control

Terms: Mathematical foundations of the theory of automatic control