Automatisk reglertenta Flashcards
Explain when the Jordan canonical form will be used
Jordan form will be used when we can’t write a system in its modal canonical form. This happens when at least two poles (eigenvalues) are the same. When at least two eigenvalues are the same A is not diagonalizable and we have to construct Jordan mini blocks for the eigenvalues that are the same.
Explain the concept of observability with the aid of discrete system and how the Observability matrix will decide the observability of the system.
Observability is about which states can be observed. If some states are unobservable it mean that we can’t see them in the output. Which means that even if the system is controllable and stable it is not possible to observe this system and we would need to construct an observer which makes the system more complex.
What does Non-minimum phase system mean?
A minimum-phase system is a system which is stable. It has all zeros and poles in the stable negative region. The system also has to be casual which means that poles has to have a higher order than the zeros. If zero-pole cancelation can be done the system is not written in minimum phase. A non minimum-phase system is therefore a system where zero-pole cancelation can be made.
Reduced order observers.
What is the advantage of using them?
What are the steps required to reduce the order?
When some states are known the need for a full order observer is not needed and then a reduced order will mean less work.
Idea behind Dead-beat controllers
where all eigenvalues are equal to zero. A fully controllable system includes a dead beat controller.
Dead beat corresponds in continuous time to have infinitely fast poles. A DBC uses alot of actuators
What is the disadvantage of Smith-compensator?
A smith compensator is a compensator which controls the system with pure time-delay. You have to discretize the system and look if T is multiple of the sampling period (delta)
Lyapunov function
A function esigned from model knowledge which represents the systems “energy”
This function and its derivative is used to prove stability
One part will show U always declining(udot < 0)
Duality
Duality of problem formulation
observability <-> controllability
constructing observability matrix on AT , BT gives controllability matrix. By calculating LQN on transposed matrixes we can also obtain kalman gain.
The relation between pole zero cancellation and non observable/controllable states.
When there is pole zero cancellations and if constructing it on control/observer canonical form the other can’t be achieved
control canonical => unobservable
Observer canonical => uncontrollable
A system with unstable non-controllable dynamics is not good?
An unstable system with uncontrollable dynamics means a system that from control perspective is unworkable since the system output will go to infinity. i.e the system will increase exponentially.
Asymptotic Stability
asymptotic stability exist for an equilibria where a (δ) exists and a (ε) , so that the state starts in δ and confined by ε. And when t –> inf x(t) –> xe.
Explain concept of smith compensator
A smith compensator is used to easily model or present a system that consist of a delay. an example is:
G_tot(s)=e^-tau(s)*G(s) where the exponential is the smith compensator
Explain difference between Modal canonical and Jordan canonical form.
Modal canonical is a way to represent a system with its unique eigenvalues are on the diagonal of the A-matrix of the SS-system. However some systems cant be written in modal due to having two or more eigenvalues with same value/multiplicity
Jordan canonical allows to have jordan mini blocks in the matrix with eigenvalues with same value in some block. This allows to work further with a system.
