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3.1 The concept of state space and the state-exit model

Lecture



  3.1 The concept of state space and the state-exit model



3.1.1. State variables Consider an autonomous dynamic system [M1a] with the output   3.1 The concept of state space and the state-exit model where   3.1 The concept of state space and the state-exit model . Note that for autonomous system solution   3.1 The concept of state space and the state-exit model contains only free component:   3.1 The concept of state space and the state-exit model . We introduce variables



(3.1)   3.1 The concept of state space and the state-exit model , i = 1,2, ... n



with initial values   3.1 The concept of state space and the state-exit model determined by   3.1 The concept of state space and the state-exit model and give the following definition [7] .



  3.1 The concept of state space and the state-exit model



State variables are called linearly independent variables x i ( t ) such that their values   3.1 The concept of state space and the state-exit model at time t 0 uniquely determine the state of the system at any time   3.1 The concept of state space and the state-exit model i.e. allow you to find the values ​​of the output variable y ( t ) at arbitrary times t according to the formula



(3.2)   3.1 The concept of state space and the state-exit model .



The procedure for finding the values ​​of a function   3.1 The concept of state space and the state-exit model ( t ) for time points   3.1 The concept of state space and the state-exit model called prediction (see clause 1.1.1). The possibility of prediction is a natural requirement of quality control, which determines the importance of the introduced concept for the nonautonomous (controlled) systems considered below. At the same time, a distinctive feature of state variables is that to predict the behavior of the system at any time t > t 0 (and control a non-autonomous system) there is enough information about state variables at time t 0 and no knowledge of the process history is required, i.e. functions x i ( t ) with t < t 0. The latter serves as the basis for constructing procedures (algorithms) for forecasting and controlling dynamic systems based on the current values ​​of state variables (see § 4.3).



The state variables of an autonomous system can be chosen, in particular, the phase variables of the system, i.e. output variable y ( t ) and n-1 of its derivatives   3.1 The concept of state space and the state-exit model ( t )   3.1 The concept of state space and the state-exit model . We introduce variables



(3.3)   3.1 The concept of state space and the state-exit model ,   3.1 The concept of state space and the state-exit model



with initial values



(3.4)   3.1 The concept of state space and the state-exit model .



The output of the stationary autonomous system [M1a], i.e. free part of the process   3.1 The concept of state space and the state-exit model = 0, for the case of unequal roots of the characteristic equation, it is determined by the formula (see Section 2.2)



(3.5)   3.1 The concept of state space and the state-exit model ,



where the coefficients C i depend on the initial values ​​of the output variable and its derivatives, or taking into account the introduced notation:



(3.6)   3.1 The concept of state space and the state-exit model .



Thus, the behavior of the system under consideration   3.1 The concept of state space and the state-exit model is uniquely determined by the initial values ​​of the variables x i and, therefore, by definition, these variables are state variables. The total number of state variables is   3.1 The concept of state space and the state-exit model i.e. the order of the differential equation [M1a]. Linear combinations of variables x i , supplemented to an already selected set, are not state variables, since they lead to a linear dependence of the variables.



Considering the introduced notation, we transform the equation [M1a] to the normal Cauchy form. Differentiating in time the equation (3.5) and substituting the obtained expressions (3.3) and [M1a], we find the so-called equations of state of an autonomous system



(3.7)   3.1 The concept of state space and the state-exit model



The output variable y ( t ) is related to state variables by a trivial expression ( output equation )



(3.8)   3.1 The concept of state space and the state-exit model .



The equations of state (3.7) and output (3.8) are the simplest example of a state-output (CB) model.



Remark 3.1. The choice of dynamic state variables is ambiguous. Not only phase variables can be taken as such variables.   3.1 The concept of state space and the state-exit model ,   3.1 The concept of state space and the state-exit model , but also physical variables of the system such as displacement, speed, current, voltage, etc. (see 4.1), as well as any other n linearly independent variables, obtained, for example, as linear combinations of phase and / or physical coordinates.



Naturally, the choice of state variables determines the structure and parameters of the state-output model. In addition to the above method of constructing such a model in the standard form (3.7), (3.8), the CB model can be obtained as a set of models of real physical processes, often corresponding to the elementary links of the first order (see Section 2.3).



3.1.2. State-output and transient models. In the most general case, the equations of state of an autonomous system are presented in the Cauchy normal form, i.e. as a system   3.1 The concept of state space and the state-exit model homogeneous differential equations



[M4a]   3.1 The concept of state space and the state-exit model



Where   3.1 The concept of state space and the state-exit model ,   3.1 The concept of state space and the state-exit model ,   3.1 The concept of state space and the state-exit model - constant or time-dependent coefficients (parameters), and the output equation relating the output variable of the system y ( t ) with the variables x i ( t ) has the form



[M5]   3.1 The concept of state space and the state-exit model ,



Where   3.1 The concept of state space and the state-exit model - coefficients (parameters). It is easy to show (see below) that the variables x i ( t ) are indeed state variables and, therefore, the equations [M4a] and [M5] represent the most general state-output model of a linear autonomous dynamic system.



The vector x = x ( t ) of dimension n , whose elements are state variables x i = x i ( t ), i.e.



  3.1 The concept of state space and the state-exit model



(3.9)   3.1 The concept of state space and the state-exit model =   3.1 The concept of state space and the state-exit model ,



called a state vector . The vector x is an element of n - dimensional linear (vector) space.   3.1 The concept of state space and the state-exit model which is called the state space :   3.1 The concept of state space and the state-exit model



The equations [M4a], [M5] can be written in vector-matrix form:



[M6a]   3.1 The concept of state space and the state-exit model ,



[M7]   3.1 The concept of state space and the state-exit model ,



Where   3.1 The concept of state space and the state-exit model ,   3.1 The concept of state space and the state-exit model - vector of initial states (initial conditions),



  3.1 The concept of state space and the state-exit model - matrix size system   3.1 The concept of state space and the state-exit model ,   3.1 The concept of state space and the state-exit model - size exit matrix   3.1 The concept of state space and the state-exit model .



In the particular case when the equations of the BC model are presented in the form (3.7) (3.8), we get



  3.1 The concept of state space and the state-exit model ,   3.1 The concept of state space and the state-exit model .



By solving a system of differential equations [M4a] with initial conditions   3.1 The concept of state space and the state-exit model called a feature set



(3.10)   3.1 The concept of state space and the state-exit model ,



which for t = t 0 satisfy the initial conditions, and for any   3.1 The concept of state space and the state-exit model - equations [M4a]. Accordingly, the solution of the equation [М6а] will be a vector function



(3.11)   3.1 The concept of state space and the state-exit model .



The solution can be represented as



(3.12)   3.1 The concept of state space and the state-exit model ,



Where   3.1 The concept of state space and the state-exit model - the fundamental (transitional) matrix of the system [М6а]. Substituting (3.12) into the output equation [M7], we obtain the expression for calculating the output variable



(3.13)   3.1 The concept of state space and the state-exit model .



For stationary systems, the transition matrix is



(3.14)   3.1 The concept of state space and the state-exit model .



Putting   3.1 The concept of state space and the state-exit model , we will find:



(3.15)   3.1 The concept of state space and the state-exit model



and



(3.16)   3.1 The concept of state space and the state-exit model .



Remark 3.2. Analysis of equations (3.14) - (3.16) shows the following.



1. Output variable   3.1 The concept of state space and the state-exit model ( t ) at any time   3.1 The concept of state space and the state-exit model uniquely determined by n initial values   3.1 The concept of state space and the state-exit model and therefore by definition the variables   3.1 The concept of state space and the state-exit model really are state variables.



2. Prehistory of the system (its movement at   3.1 The concept of state space and the state-exit model ) does not affect the behavior of the system when   3.1 The concept of state space and the state-exit model .



If for some initial conditions and   3.1 The concept of state space and the state-exit model there is an identity



(3.17)   3.1 The concept of state space and the state-exit model ,



where x * = const, then the value x = x * is called the equilibrium state , or the equilibrium position , of the autonomous system [M6a]. Obviously, in the equilibrium state,



(3.18)   3.1 The concept of state space and the state-exit model



and therefore



(3.19)   3.1 The concept of state space and the state-exit model .



Provided that det A   3.1 The concept of state space and the state-exit model 0, we find that the only equilibrium position of the [M6a] system is the origin of the state space R n , i.e.



x * = 0,



and with det A = 0, there are nontrivial sets of equilibrium states (straight lines, planes, that is, subspaces that satisfy equation (3.19)).



After substituting x * = 0 into the output equation [M7], we find the equilibrium value of the output variable (see § 2.2.2)



y * = 0.



Formulas (3.14) - (3.16) determine the transient processes of the system -   3.1 The concept of state space and the state-exit model time functions   3.1 The concept of state space and the state-exit model . Graphically, they can be represented as:



  • time diagrams (see clause 2.2);
  • integral curves.



The integral curve (phase trajectory) is a line described by a state vector.   3.1 The concept of state space and the state-exit model in the state space   3.1 The concept of state space and the state-exit model when the variable changes   3.1 The concept of state space and the state-exit model ,   3.1 The concept of state space and the state-exit model i.e. hodograph vector functions x ( x 0 , t   ) by parameter   3.1 The concept of state space and the state-exit model . Phase portrait - a set of phase trajectories corresponding to different values ​​of the initial conditions   3.1 The concept of state space and the state-exit model .





  3.1 The concept of state space and the state-exit model   3.1 The concept of state space and the state-exit model



Fig. 3.1. Integral curve in R n and phase portrait



The concepts introduced above are generalized to the class of multichannel (multiply connected, see § 2.1.3) systems, which are characterized by several output variables y j ,   3.1 The concept of state space and the state-exit model . The general model of a multichannel system includes the equations of state [M4a] and   3.1 The concept of state space and the state-exit model output equations



[M5m]   3.1 The concept of state space and the state-exit model



Where   3.1 The concept of state space and the state-exit model - coefficients (parameters).



Define   3.1 The concept of state space and the state-exit model -dimensional vector of outputs



(3.20)   3.1 The concept of state space and the state-exit model



as a vector space of output variables R   3.1 The concept of state space and the state-exit model and write the equation [M5m] in a compact vector-matrix form [M7], i.e.



[M7m]   3.1 The concept of state space and the state-exit model ,



Where   3.1 The concept of state space and the state-exit model - matrix output size   3.1 The concept of state space and the state-exit model . Thus, the state-output model of the multichannel system is represented by the equations [M4], [M5m] or vector-matrix equations [M6a] and [M7m].



3.1.3. Properties of state-exit models . Let us analyze the solutions of the equation of state [M6a], [M7] and the associated transients of the autonomous dynamic system (3.12) - (3.16).



First we define



  • eigenvalues ​​(eigenvalues) of the state matrix A as n numbers   3.1 The concept of state space and the state-exit model ;
  • characteristic equation



(3.21)   3.1 The concept of state space and the state-exit model   3.1 The concept of state space and the state-exit model ;



  • matrix eigenvectors as   3.1 The concept of state space and the state-exit model ,



  3.1 The concept of state space and the state-exit model



and also (for the case of real eigenvalues   3.1 The concept of state space and the state-exit model ) eigenspaces of the system as sets (straight lines, planes, etc.)



(3.22)   3.1 The concept of state space and the state-exit model ,



Where   3.1 The concept of state space and the state-exit model - real numbers. Recall that in this case, the eigenvectors satisfy the equations



(3.23)   3.1 The concept of state space and the state-exit model .



Matrix function



(3.24)   3.1 The concept of state space and the state-exit model =   3.1 The concept of state space and the state-exit model



called the matrix exponent. Matrix exponent of the diagonal matrix



  3.1 The concept of state space and the state-exit model ;



calculated by a simple formula



(3.25)   3.1 The concept of state space and the state-exit model .



In a more general case (provided   3.1 The concept of state space and the state-exit model ) from the expression (3.23) we obtain:



  3.1 The concept of state space and the state-exit model   3.1 The concept of state space and the state-exit model   3.1 The concept of state space and the state-exit model



and therefore matrix A is connected to a diagonal matrix   3.1 The concept of state space and the state-exit model by formula



(3.26)   3.1 The concept of state space and the state-exit model ,



Where



  3.1 The concept of state space and the state-exit model .



Then the matrix exponent is like



(3.27)   3.1 The concept of state space and the state-exit model .



Now, taking into account equation (3.27), we rewrite the formula for calculating the state vector (3.15) as



(3.28)   3.1 The concept of state space and the state-exit model   3.1 The concept of state space and the state-exit model   3.1 The concept of state space and the state-exit model   3.1 The concept of state space and the state-exit model   3.1 The concept of state space and the state-exit model .



Considering that   3.1 The concept of state space and the state-exit model , запишем



  3.1 The concept of state space and the state-exit model



and therefore



  3.1 The concept of state space and the state-exit model .



Тогда уравнение (3.28) принимает вид



(3.29)   3.1 The concept of state space and the state-exit model .



We introduce the notation



(3.30)   3.1 The concept of state space and the state-exit model ( x 0 )



и запишем формулу (3.29) в виде разложения по собственным векторам



(3.31)   3.1 The concept of state space and the state-exit model   3.1 The concept of state space and the state-exit model ,



где векторы   3.1 The concept of state space and the state-exit model принадлежат собственным подпространствам системы   3.1 The concept of state space and the state-exit model и называются собственными составляющими решения x ( t ), или модами вектора состояния системы.



Замечание 3.3. Если начальное значение вектора состояния принадлежит собственному подпространству   3.1 The concept of state space and the state-exit model i.e.   3.1 The concept of state space and the state-exit model then



(3.32)   3.1 The concept of state space and the state-exit model   3.1 The concept of state space and the state-exit model



and therefore



(3.33)   3.1 The concept of state space and the state-exit model   3.1 The concept of state space and the state-exit model ,



those. траектория системы целиком лежит в собственном подпространстве R i .



Такого рода подпространства пространства состояний R n относятся к классу инвариантных множеств динамической системы.



Проанализируем поведение выходной переменной y ( t ) . Подставляя уравнение ( 3.31 ) в [M7] находим:



(3.34)   3.1 The concept of state space and the state-exit model   3.1 The concept of state space and the state-exit model ,



где y i ( t ) - моды выходной переменной.



Сравнивая последнее уравнение с выражением (2.20), получим, что неопределенные коэффициенты C i могут быть рассчитаны как



(3.35)   3.1 The concept of state space and the state-exit model .



Более того,



(3.36)   3.1 The concept of state space and the state-exit model ,



those. полюсы системы p i совпадают с собственными числами матрицы   3.1 The concept of state space and the state-exit model . Отсюда следует, что совпадают и характеристические уравнения (2.3) и (3.21), или



(3.37)   3.1 The concept of state space and the state-exit model .



Thus, the following properties of state-output models are obtained.



Property 3.1.



  3.1 The concept of state space and the state-exit model .



Property 3.2.



  3.1 The concept of state space and the state-exit model .



Property 3.3.



  3.1 The concept of state space and the state-exit model .



Property 3.4.



  3.1 The concept of state space and the state-exit model .



Property 3.5.



  3.1 The concept of state space and the state-exit model .


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Mathematical foundations of the theory of automatic control

Terms: Mathematical foundations of the theory of automatic control