Discuss the concept of controllability and explain the conditions for full state controllability of multivariable control system.
Answers
Answer:
Explanation:
Complete state controllability (or simply controllability if no other context is given) describes the ability of an external input (the vector of control variables) to move the internal state of a system from any initial state to any other final state in a finite time interval.
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Answer:
System Interaction
In the world of control engineering, there are a slew of systems available that need to be controlled. The task of a control engineer is to design controller and compensator units to interact with these pre-existing systems. However, some systems simply cannot be controlled (or, more often, cannot be controlled in specific ways). The concept of controllability refers to the ability of a controller to arbitrarily alter the functionality of the system plant.
The state-variable of a system, x, represents the internal workings of the system that can be separate from the regular input-output relationship of the system. This also needs to be measured, or observed. The term observability describes whether the internal state variables of the system can be externally measured.
Controllability
Complete state controllability (or simply controllability if no other context is given) describes the ability of an external input to move the internal state of a system from any initial state to any other final state in a finite time interval
We will start off with the definitions of the term controllability, and the related terms reachability and stabilizability.
Controllability
A system with internal state vector x is called controllable if and only if the system states can be changed by changing the system input.
Reachability
A particular state x1 is called reachable if there exists an input that transfers the state of the system from the initial state x0 to x1 in some finite time interval [t0, t).
Stabilizability
A system is Stabilizable if all states that cannot be reached decay to zero asymptotically.
We can also write out the definition of reachability more precisely:
A state x1 is called reachable at time t1 if for some finite initial time t0 there exists an input u(t) that transfers the state x(t) from the origin at t0 to x1.
A system is reachable at time t1 if every state x1 in the state-space is reachable at time t1.
Similarly, we can more precisely define the concept of controllability:
A state x0 is controllable at time t0 if for some finite time t1 there exists an input u(t) that transfers the state x(t) from x0 to the origin at time t1.
A system is called controllable at time t0 if every state x0 in the state-space is controllable.
Controllability Matrix
For LTI (linear time-invariant) systems, a system is reachable if and only if its controllability matrix, ζ, has a full row rank of p, where p is the dimension of the matrix A, and p × q is the dimension of matrix B.
[Controllability Matrix]
{\displaystyle \zeta ={\begin{bmatrix}B&AB&A^{2}B&\cdots &A^{p-1}B\end{bmatrix}}\in R^{p\times pq}} \zeta ={\begin{bmatrix}B&AB&A^{2}B&\cdots &A^{{p-1}}B\end{bmatrix}}\in R^{{p\times pq}}
A system is controllable or "Controllable to the origin" when any state x1 can be driven to the zero state x = 0 in a finite number of steps.
A system is controllable when the rank of the system matrix A is p, and the rank of the controllability matrix is equal to:
{\displaystyle Rank(\zeta )=Rank(A^{-1}\zeta )=p} Rank(\zeta )=Rank(A^{{-1}}\zeta )=p
If the second equation is not satisfied, the system is not .
MATLAB allows one to easily create the controllability matrix with the ctrb command. To create the controllability matrix {\displaystyle \zeta } \zeta simply type
{\displaystyle \zeta =ctrb(A,B)} \zeta =ctrb(A,B)
where A and B are mentioned above. Then in order to determine if the system is controllable or not one can use the rank command to determine if it has full rank.
If
{\displaystyle Rank(A)<p} Rank(A)<p
Then controllability does not imply reachability.
Reachability always implies controllability.
Controllability only implies reachability when the state transition matrix is nonsingular.
Determining Reachability
There are four methods that can be used to determine if a system is reachable or not:
If the p rows of {\displaystyle \phi (t,\tau )B(t)} \phi (t,\tau )B(t) are linearly independent over the field of complex numbers. That is, if the rank of the product of those two matrices is equal to p for all values of t and τ
If the rank of the controllability matrix is the same as the rank of the system matrix A.
If the rank of {\displaystyle \operatorname {rank} [\lambda I-A,B]=p} \operatorname {rank}[\lambda I-A,B]=p for all eigenvalues λ of the matrix A.