- state equations can be converted to transfer functions. The derivation follows.
- state equation coefficient matrices can be transformed to another equivalent for, if the state vector is rearranged.
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· The transfer function form can be put into a matrix form. In this case the denominator is the characteristic equation.
· The free (homogeneous) response of a system can be used to find the state transition matrix.
· The forced response (particular) response of the system can be found using convolution,
· As an example the homogeneous/free response of the system is shown below.
· The forced/particular solution is shown below,
· If a matrix is diagonizable, the diagonal matrix can be found with the following technique. This can be used for more advanced analysis techniques to create diagonal (and separable) system matrices.
· example,
· If there are repeated Eigenvalues in the system the Jordan Form can be used.
· The Eigenvectors can be used to calculate the system response.
· zeros of state space functions can be found using the state matrices.
- a system is observable iff the system state x(t) can be found by observing the input u and output y over a period of time from x(t) to x(t+h).
- If an input is to be observable it must be detectable in the output. For example consider the following state equations.
· This often happens when a system has elements that are decoupled, or when a pole and zero cancel each other.
· Observability can be verified formally for an LTI system with the following relationship.
· Another theorem for testing observability is given below. If any of the states satisfies the equation it is unobservable.
· Yet another test for observability is,
· If a system in unobservable, it is possible to make it observable by changing the model.
· A pole-zero cancellation is often the cause of the loss of observability.
· If all unstable modes are observable, the system is detectable.
- a system is controllable iff there is an input u(t) that will cause the system to go from any initial state to any final state in a finite time.
- stabilizable if it is controllable or if the uncontrollable nodes are stable.
- If an input is to be observable it must be detectable in the output. For example consider the following state equations.
· This can be verified with the
· Another test for controllability is,
· Yet another test for controllability is,
· For a system to be controllable, all of the states must be controllable.
· If a system in uncontrollable, it is possible to make it observable by changing the model.
· A pole-zero cancellation is often the cause of the loss of observability.
· if all unstable modes are controllable, the systems is said to be stabilizable.
· The principle of Duality requires that a system be completely observable to be controllable.
· Observers estimate system state variables when not all of the variables are directly observable.
· Observers use a limited set of system states that are available to identify other system states that are not observable.
· The separation principle ensures that the observer cannot effect the stability of the system it is observing.
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1.
How, J, "16.31 Feedback Control Course Notes", MIT Opencourseware website.