**The State Regulator Problem**

**1. The optimization method based on dynamic programming views :**

a) Control problem as the multistage decision problem

b) Control input as a time sequence of decisions

c) A sampled data system gives rise to sequence of transformations of the original state vector

d) All of the mentioned

**Answer:** d

**Explanation:** The optimization method is based on dynamic programming views control problem as the multistage decision problem and control input as time sequence of decisions.

**2. The choice of control vector is considered decision of interest in :**

a) Multistage process of N stages

b) Minimizing the performance index

c) Optimal policy or sequence

d) All of the mentioned

**Answer:** d

**Explanation:** The choice of control vector is considered decision of interest as it the part of multistage process of N stages and this is optimal policy or sequence and this minimizes the performance criteria.

**3. The principle of invariant imbedding is :**

a) The optimal control sequence is function of initial state

b) The optimal control sequence is function of number of stages N

c) Control problem is imbedded with the family of problems with fixed initial value and final state

d) The optimal control sequence is function of final state

**Answer:** c

**Explanation:** In principle of invariant imbedding we do not regard the control problem as an isolated problem with fixed initial value and N but rather imbed it with the family of problems.

**4. The principle of optimality :**

a) The optimal control sequence is function of initial state

b) The optimal control sequence is function of number of stages N

c) The principle maintains the N-stage decision process

d) Find one control value at a time until optimal policy is determined

**Answer:** c

**Explanation:** The principle of optimality reduces the N-stage decision process into N-stage single state process, which state that an optimal control policy has the property that whatever the initial state and conditions are final state must follow the optimal policy.

**5. The calculation in multistage process must always start from the first stage :**

a) True

b) False

**Answer:** b

**Explanation:** It is permissible to find the last decision as the initial calculation step in multi stage decision process.

**6. The optimal control at each state is ________ combination of states.**

a) Linear

b) Non-linear

c) Time variant

d) Time invariant

**Answer:** a

**Explanation:** The optimal control at each state is linear combination of states and thus giving the linear state variable feedback control policy.

**7. Feedback is :**

a) Linear

b) Non-linear

c) Time variant

d) Time invariant

**Answer:** c

**Explanation:** Feedback is time varying all the optimal policy converts a linear time invariant plant with time invariant quadratic performance index into a linear time varying feedback system.

**8. Realization of the optimal control policy :**

a) Feedback of the state variables

b) Feedback of the control variables

c) Control constraint

d) Control function

**Answer:** a

**Explanation:** Once the optimal control policy has been determined, its realization is the second phase of the optimal control problem its realization seeks feedback of the state variables.

**9. If the plant states are not variable for measurement, then it is possible to construct physical device state observer if :**

a) Produces output at its plant states

b) Plants equations must satisfy the conditions of observability

c) Both a and b

d) Input must be present

**Answer:** c

**Explanation:** It produces at it output the plant states, when driven by the both the plant input and output.

**10. Value of the performance index must be finite if :**

a) System is controllable

b) System is observable

c) System is stable

d) System is unstable

**Answer:** d

**Explanation:** The contribution of uncontrollable and unstable states is always finite provided the control interval is always finite.