Concept of state-state equation for dynamic systems
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State-space Representation of Continuous Dynamical System
Transfer function expresses the relationship between the output and the input of the system. The main advantage of this disclosure is to obtain the equation of which can easily determine the behavior of the system.
State space describes a system using conditions and their changes over time. Výstupné veličiny (Figure 1) sústavy y(t) sú funkciou stavových x(t) a vstupných veličín u(t). The output variables (Figure 1) of the system are the function of the state values and input values. Vector of output of the model are the state matrix, which indicate the relationship between the variables describing the behavior of the system.
In this description, it is possible to observe the state of the system at the time, which sometimes allows estimating the behavior the next time step.
Under the effect of input variables u1, u2,...., um is the state vector u(t), the output signal is determined through the response of the system thru output vector y(t) and indicates the status of the state vector x(t).
If we have a system according the m inputs and r outputs, we may describe a system of n first order differential equations.
It is described by state equation:
(1)
where initial conditions x(0)=xS, output values of system are the function of state variables and input values. The output equation of the system is in form:
(2)
where:
xT = (x1,x2,...., xn) - vector of state variables,
uT = (u1, u2,...., um) – vector of input variables,
yT = (y1, y2, ..., yr) - vector of output variables,
fT = (f1, f2,...., fn)– nonlinear vector functions,
n – number of state variables,
m – number of input,
r – number of output.
The equation (1) is called the equation of dynamics, the state equation of the system and the equation (2) is called the equation of output of the system.
These two equations describe the continuous nonlinear time variant system.
If the functions f and g are not dependend explicitly on the time t, the system description is obtained in the form:
(3)
(4)
These are the time invariant systems, their characteristics do not change in time.
Transfer function expresses the relationship between the output and the input of the system. The main advantage of this disclosure is to obtain the equation of which can easily determine the behavior of the system.
State space describes a system using conditions and their changes over time. Výstupné veličiny (Figure 1) sústavy y(t) sú funkciou stavových x(t) a vstupných veličín u(t). The output variables (Figure 1) of the system are the function of the state values and input values. Vector of output of the model are the state matrix, which indicate the relationship between the variables describing the behavior of the system.
In this description, it is possible to observe the state of the system at the time, which sometimes allows estimating the behavior the next time step.
Under the effect of input variables u1, u2,...., um is the state vector u(t), the output signal is determined through the response of the system thru output vector y(t) and indicates the status of the state vector x(t).
If we have a system according the m inputs and r outputs, we may describe a system of n first order differential equations.
It is described by state equation:
(1)
where initial conditions x(0)=xS, output values of system are the function of state variables and input values. The output equation of the system is in form:
(2)
where:
xT = (x1,x2,...., xn) - vector of state variables,
uT = (u1, u2,...., um) – vector of input variables,
yT = (y1, y2, ..., yr) - vector of output variables,
fT = (f1, f2,...., fn)– nonlinear vector functions,
n – number of state variables,
m – number of input,
r – number of output.
The equation (1) is called the equation of dynamics, the state equation of the system and the equation (2) is called the equation of output of the system.
These two equations describe the continuous nonlinear time variant system.
If the functions f and g are not dependend explicitly on the time t, the system description is obtained in the form:
(3)
(4)
These are the time invariant systems, their characteristics do not change in time.
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