Well, you are correct that the order of the system and the number of states should be the same thing. However, perhaps you reversed the order of the process of determining each of these. Often, it is preferable to determine the state equations. Then, based on the number of the states, you know the order of the system. In this case we have 4 state equations and so the order is 4.
Alternatively, there are often ways to determine the order of the system correctly, and then to find state equations, however, my experience is that often a mistake can be made. Sometimes the system has a constraint that reduces the order down from what you think it should be, and this constraint may not be obvious at the start. However, all constraints become obvious when you analyze the system and derive state equations.
One thing we learn with mechanical systems is that every point particle has 2 states (position and velocity, or position and momentum - your choice) for each degree of freedom (i.e. x, y and z directions are 3 separate degrees of freedom). Rigid bodies made from many point particles have additional degrees of freedom due to rotation, and non rigid bodies have even more degrees of freedom due to deformations. In this example we have two masses that we treat as point particles with one degree of freedom each (constraints of the track prevent motion in the other two directions), which implies 2 states each, so we usually expect 4 states here.
However, your intuition that there might be only 2 states is not a crazy notion. The position state is a simple definition equation of the form dx/dt=v. That is, position states are just the integral of velocity. Very often, if we don't need a position type of variable in our analysis, we can remove such a state, if other state equations don't need that state variable. In this example, I notice that x3 is a state that is not critical because the other state equations dont' need it. Hence, one could reduce the order by removing x3, provided that you don't care what x3 is in your analysis. However, the person who developed these equations obviously cares what x3 is because he defined one of the outputs to be y2=x3. Hence, we know why x3 was included as a state.
The general theme of these questions is that the "state space approach" is a formulation with great flexibility. There are some basic rules that must be obeyed, but then one is free to add more things, like non-critical states, additional inputs and any outputs we like.
Alternatively, there are often ways to determine the order of the system correctly, and then to find state equations, however, my experience is that often a mistake can be made. Sometimes the system has a constraint that reduces the order down from what you think it should be, and this constraint may not be obvious at the start. However, all constraints become obvious when you analyze the system and derive state equations.
One thing we learn with mechanical systems is that every point particle has 2 states (position and velocity, or position and momentum - your choice) for each degree of freedom (i.e. x, y and z directions are 3 separate degrees of freedom). Rigid bodies made from many point particles have additional degrees of freedom due to rotation, and non rigid bodies have even more degrees of freedom due to deformations. In this example we have two masses that we treat as point particles with one degree of freedom each (constraints of the track prevent motion in the other two directions), which implies 2 states each, so we usually expect 4 states here.
However, your intuition that there might be only 2 states is not a crazy notion. The position state is a simple definition equation of the form dx/dt=v. That is, position states are just the integral of velocity. Very often, if we don't need a position type of variable in our analysis, we can remove such a state, if other state equations don't need that state variable. In this example, I notice that x3 is a state that is not critical because the other state equations dont' need it. Hence, one could reduce the order by removing x3, provided that you don't care what x3 is in your analysis. However, the person who developed these equations obviously cares what x3 is because he defined one of the outputs to be y2=x3. Hence, we know why x3 was included as a state.
The general theme of these questions is that the "state space approach" is a formulation with great flexibility. There are some basic rules that must be obeyed, but then one is free to add more things, like non-critical states, additional inputs and any outputs we like.
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