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4.2 Models of driver units and external influences

Lecture



For the description of the external environment, the objects of tracking and generation of driving forces in program control systems, there is a need to design additional dynamic models ( external action generators ), the output of which is perturbing   4.2 Models of driver units and external influences or specifying the y * ( t ) effect (see § 1.2.1). For smooth functions   4.2 Models of driver units and external influences and y * ( t ) the corresponding disturbing and driving effects generators (driving blocks) can be obtained in the class of autonomous linear models similar to those previously considered [M1a], [M2a] or [M6a], [M7]



  4.2 Models of driver units and external influences



Thus, the master block of a BZ (generator of master actions) can be described by a homogeneous differential equation of the form



(4.24)   4.2 Models of driver units and external influences



with constant coefficients   4.2 Models of driver units and external influences i and initial values



(4.25)   4.2 Models of driver units and external influences ,   4.2 Models of driver units and external influences ,



or corresponding vector-matrix equations of state



(4.26)   4.2 Models of driver units and external influences



and exit



(4.27)   4.2 Models of driver units and external influences ,



Where   4.2 Models of driver units and external influences - s - dimensional vector of tasks (generator state) with initial values ​​of coordinates   4.2 Models of driver units and external influences .



  4.2 Models of driver units and external influences



The generator of smooth disturbing influences (model of the external environment of the Armed Forces) is described by a homogeneous equation



(4.28)   4.2 Models of driver units and external influences



with constant coefficients   4.2 Models of driver units and external influences i   and initial values



(4.29)   4.2 Models of driver units and external influences ,   4.2 Models of driver units and external influences



or in a compact form



(4.30)   4.2 Models of driver units and external influences



(4.31)   4.2 Models of driver units and external influences



Where   4.2 Models of driver units and external influences - r - dimensional vector of the state of the environment with the initial values ​​of the coordinates   4.2 Models of driver units and external influences .



The impacts generated by the considered models correspond to the solutions of differential equations (4.26), (4.27) and (4.30), (4.31), i.e. functions



(4.32)   4.2 Models of driver units and external influences



and



(4.33)   4.2 Models of driver units and external influences ,



respectively. In particular cases, using such models can be obtained:



  • polynomial effects


(4.34)   4.2 Models of driver units and external influences ,



Where   4.2 Models of driver units and external influences - constants, defined as



(4.35)   4.2 Models of driver units and external influences ;



  • harmonic effects


(4.36)   4.2 Models of driver units and external influences ,



where a i ,   4.2 Models of driver units and external influences i and   4.2 Models of driver units and external influences i - constants, corresponding to amplitudes, phases and frequencies of harmonics, etc.



To build an impact model for a given function y * ( t ) (or   4.2 Models of driver units and external influences ) you can use the method of sequential differentiation of the corresponding analytical expressions



  4.2 Models of driver units and external influences



(4.37) y = y * ( t )



(or f = f (t)), represented, for example, in the form (4.34) or (4.36).



Example 4.1 . To obtain a model of a constant signal



(4.38) y * (t) = C 0 ,



  4.2 Models of driver units and external influences



let's differentiate the last expression by time. Get the first order equation



(4.39)   4.2 Models of driver units and external influences



with initial value y * (0) = С 0 .



  4.2 Models of driver units and external influences



Model of linearly increasing signal (uniform motion)



(4.40)   4.2 Models of driver units and external influences ,



is obtained by a double differentiation. In the first step we get



(4.41)   4.2 Models of driver units and external influences ,



and on the second - the desired differential equation



(4.42)   4.2 Models of driver units and external influences .



To obtain the initial conditions from equation (4.40), we find y * ( 0 ) = С 0 , and from equation (4.41) -   4.2 Models of driver units and external influences * (0) = C 1 .



  4.2 Models of driver units and external influences



Equation (4.42) is reduced to the form of a state model (a system of equations in the form of Cauchy). To do this, in addition to the main state variable that coincides with the output variable y * ( t ), the second variable V * ( t ) (travel speed) is introduced:



(4.43)   4.2 Models of driver units and external influences



with initial value   4.2 Models of driver units and external influences . Equation (4.42) can be rewritten as



(4.44)   4.2 Models of driver units and external influences .



The obtained equations (4.43) and (4.44) describe the desired model of the state of the master block.



Note that the solution of equation (4.42) or system (4.43), (4.44) (i.e., expression (4.40)) can be written as



(4.45)   4.2 Models of driver units and external influences .



  4.2 Models of driver units and external influences



The model is quadratically increasing signal



(4.46)   4.2 Models of driver units and external influences



(uniformly accelerated motion) is obtained as a result of the procedure of three-fold differentiation of expression (4.46) and has the form



(4.47)   4.2 Models of driver units and external influences



with initial conditions



  4.2 Models of driver units and external influences



y * (0) = C 0 ,   4.2 Models of driver units and external influences * (0) = C 1 and   4.2 Models of driver units and external influences . To build a state model of the considered master block, state variables y * ( t ) , V * ( t ) , a * (t) are introduced, where   4.2 Models of driver units and external influences - acceleration, and after their differentiation in time and corresponding substitutions, a system of equations is obtained



(4.48)   4.2 Models of driver units and external influences ,



(4.49)   4.2 Models of driver units and external influences ,



(4.50)   4.2 Models of driver units and external influences



with initial values ​​of y * (0) = C 0 , V * (0) = C 1 , a * (0) = C 2 .



Note that the solution of equation (4.47) or system (4.48), (4.49) (i.e., expression (4.46)) can be written as



(5.51)   4.2 Models of driver units and external influences





  4.2 Models of driver units and external influences



Example 4.2 . To generate the simplest harmonic effect



with frequency   4.2 Models of driver units and external influences i.e. signal



  4.2 Models of driver units and external influences ,



the model of single-frequency linear oscillator is used (conservative, see p. 2.3):



(4.52)   4.2 Models of driver units and external influences .



Amplitude amplitude A and phase shift   4.2 Models of driver units and external influences determined by the initial values ​​of the model y (0) and   4.2 Models of driver units and external influences .

  4.2 Models of driver units and external influences

Model (4.52) is easily reduced to Cauchy form.



(4.53)   4.2 Models of driver units and external influences



with exit



(4.54)   4.2 Models of driver units and external influences .



  4.2 Models of driver units and external influences



Here the following notation is entered:



(4.55)   4.2 Models of driver units and external influences



In the particular case to build a signal generator



(4.56)   4.2 Models of driver units and external influences



state variables are entered



(4.57)   4.2 Models of driver units and external influences



(4.58)   4.2 Models of driver units and external influences



with initial values



(4.59)   4.2 Models of driver units and external influences ,   4.2 Models of driver units and external influences .



After time differentiation of expressions (4.57), (4.58) and corresponding substitutions, we obtain the system of equations (4.53).



Remark 4.1. If state variables are selected as



(4.60)   4.2 Models of driver units and external influences ,



  4.2 Models of driver units and external influences



(4.61)   4.2 Models of driver units and external influences



with initial values   4.2 Models of driver units and external influences ,   4.2 Models of driver units and external influences , then the state model of the master oscillator takes the form



(4.62)   4.2 Models of driver units and external influences



For generation of nonsmooth functions y * (t) and f (t) are used.



  • nonlinear dynamic models [10];
  • variable structure models [10];
  • non-autonomous linear models of the form:


(4.63)   4.2 Models of driver units and external influences



Where   4.2 Models of driver units and external influences - non-smooth (may be discontinuous) input effect of the model.



Example 4.3. To generate a signal y * ( t ) with a trapezoidal velocity graph   4.2 Models of driver units and external influences (Fig. 4.6) you can use the model:



(4.64)   4.2 Models of driver units and external influences



  4.2 Models of driver units and external influences





Fig. 4.6. Non-Smooth Signal Processes



or - in the form of Cauchy



(4.65)   4.2 Models of driver units and external influences ,



(4.66)   4.2 Models of driver units and external influences ,



where a * = a * ( t ) is the type acceleration signal



(4.67)   4.2 Models of driver units and external influences



where a 1 > 0 and a 2 > 0 are constant.


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Mathematical foundations of the theory of automatic control

Terms: Mathematical foundations of the theory of automatic control