The order of the differential equation describing the system is equal to the minimum number required. The three simultaneous first-order differential equations are required along with three state variables, if the third order third-order differential equation describes the system. The order of the differential equation is the order of the denominator of the transfer function after canceling the common factors in the numerator and denominator, from the perspective of the transfer function.
In most of the cases, to determine the number of state variables through another way is to count the number of independent energy-storage elements in the system. the order of the differential equation and the number of state variables is equal to the number of these energy-storage elements.
In many cases it is also possible to complete the writing of the state equations, as the derivatives of the state variables cannot be expressed as the linear combinations of the reduced number of state variables. Even if we select the minimum number of state variables, it will not be linear independent, so we will be not able to solve all the other system variables. In addition, the worst part of it will be that we will not be able to complete the writing of the state equations. Often there are more than the than the minimum number of state variables are included in the state vector then the requirement.
Here, the two possible cases exist. Often the state variables are chosen to be the Physical variables of a system; such as position and velocity in the mechanical system. The possibility of these cases to be arising is where these variables although linearly independent are decoupled. In order to solve for any of the other linearly independent variables or any other dependent system variable, some linearly independent variables are not required. In the case of the mass and viscous damper whose differential equation is M dv/dt + Dv = f(t), and where v is the velocity of the mass. One state equation is required to define this system in the state space with velocity as the state variable, as it is a first order equation. In addition, as there is only one energy-storage element, mass, there is a requirement of only one state variable to represent this system in state space. However, the mass has an associated position, which is linearly independent of velocity.
In order to include the position in the state vector along with the velocity, we just have to add the position as a state variable that is linearly independent of the other state variable, velocity. In the Figure 3.4 it has been illustrated fully what is happening.
The first block is the transfer function which is equivalent to M dv{t) / dt + Dv(t) = f(t) whereas the second block shows that we integrate the output velocity to yield the output displacement. Therefore, for the displacement as an output, the denominator, or characteristic equation, has to be increased in order to 2, the product of the two transfer functions. Many times, by including additional state variables, the writing of the state equations is simplified.
In another case, when the added variable is not linearly independent of the other members of the state vector, the size of the state vector arises. This event is resulted when the variable is selected as a state variable but in addition, its dependence on the other state variable is not immediately apparent.
For an example, in order to select the state variables the energy-storage elements may be used and it may be hard to recognize the dependence of the variable associated with one energy-storage element on the variables of other energy storage elements. Therefore, the system matrix’s dimension is unnecessarily increased and the solution for the state vector is more difficult.
Now, we find the state-space representation for a mechanical system. It is more convenient for us to work with the mechanical systems in order to obtain the state equations directly from the equations of motion rather than from the energy storage elements. For an example, consider an energy-storage element such as a spring, where F = Kx. The derivative of a physical variable as in the case of electrical networks, where i = C dv/dt for capacitors, and v = L di/dt for inductors is not included in this relationship. Therefore, in the mechanical system, we change our selection of the state variables to be the position and velocity of each point of the linearly independent motion.
In this example, there are three energy storage elements and there will be four state variables too. An additional linearly independent state variable is also included for the convenience of writing the state equations. It is left on the students to show how the system yields a fourth-order transfer function, if we relate the displacement of either mass to the applied force, and a third-order transfer function if we relate the velocity of either mass to the applied force.
In most of the cases, to determine the number of state variables through another way is to count the number of independent energy-storage elements in the system. the order of the differential equation and the number of state variables is equal to the number of these energy-storage elements.
In many cases it is also possible to complete the writing of the state equations, as the derivatives of the state variables cannot be expressed as the linear combinations of the reduced number of state variables. Even if we select the minimum number of state variables, it will not be linear independent, so we will be not able to solve all the other system variables. In addition, the worst part of it will be that we will not be able to complete the writing of the state equations. Often there are more than the than the minimum number of state variables are included in the state vector then the requirement.
Here, the two possible cases exist. Often the state variables are chosen to be the Physical variables of a system; such as position and velocity in the mechanical system. The possibility of these cases to be arising is where these variables although linearly independent are decoupled. In order to solve for any of the other linearly independent variables or any other dependent system variable, some linearly independent variables are not required. In the case of the mass and viscous damper whose differential equation is M dv/dt + Dv = f(t), and where v is the velocity of the mass. One state equation is required to define this system in the state space with velocity as the state variable, as it is a first order equation. In addition, as there is only one energy-storage element, mass, there is a requirement of only one state variable to represent this system in state space. However, the mass has an associated position, which is linearly independent of velocity.
In order to include the position in the state vector along with the velocity, we just have to add the position as a state variable that is linearly independent of the other state variable, velocity. In the Figure 3.4 it has been illustrated fully what is happening.
The first block is the transfer function which is equivalent to M dv{t) / dt + Dv(t) = f(t) whereas the second block shows that we integrate the output velocity to yield the output displacement. Therefore, for the displacement as an output, the denominator, or characteristic equation, has to be increased in order to 2, the product of the two transfer functions. Many times, by including additional state variables, the writing of the state equations is simplified.
In another case, when the added variable is not linearly independent of the other members of the state vector, the size of the state vector arises. This event is resulted when the variable is selected as a state variable but in addition, its dependence on the other state variable is not immediately apparent.
For an example, in order to select the state variables the energy-storage elements may be used and it may be hard to recognize the dependence of the variable associated with one energy-storage element on the variables of other energy storage elements. Therefore, the system matrix’s dimension is unnecessarily increased and the solution for the state vector is more difficult.
Now, we find the state-space representation for a mechanical system. It is more convenient for us to work with the mechanical systems in order to obtain the state equations directly from the equations of motion rather than from the energy storage elements. For an example, consider an energy-storage element such as a spring, where F = Kx. The derivative of a physical variable as in the case of electrical networks, where i = C dv/dt for capacitors, and v = L di/dt for inductors is not included in this relationship. Therefore, in the mechanical system, we change our selection of the state variables to be the position and velocity of each point of the linearly independent motion.
In this example, there are three energy storage elements and there will be four state variables too. An additional linearly independent state variable is also included for the convenience of writing the state equations. It is left on the students to show how the system yields a fourth-order transfer function, if we relate the displacement of either mass to the applied force, and a third-order transfer function if we relate the velocity of either mass to the applied force.
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