**The Infinite Time Regulator Problem**

**1. In infinite time regulator for the final time is:**

a) t1

b) t0

c) Infinite

d) Zero

**Answer:** c

**Explanation:** In infinite time regulator is the extended version of the state regulator and the final time for the infinite time regulator is infinite in this case.

**2. When time and the state variables tend to infinity then the system becomes:**

a) Unstable

b) Stable

c) Marginally stable

d) Conditionally stable

**Answer:** b

**Explanation:** The terminal penalty term has no significance and it does not appear in the performance index and terminal time if tends to infinity then the system becomes stable.

**3. In finite time regulator:**

a) There is no restriction on the controllability of the plant

b) Performance index is infinite

c) Instability imposes problem in finite-interval control

d) No restriction on controllability and performance index

**Answer:** a

**Explanation:** In finite time regulator there is no restriction on the controllability of the plant. This is because is always finite and instability does not impose any problems in finite-time control.

**4. Performance index can become infinite if:**

a) One or more state are uncontrollable

b) The uncontrollable states are uncontrollable

c) The unstable states are reflected in system performance index

d) All of the mentioned

**Answer:** d

**Explanation:** Performance index can become infinite if one or more states are uncontrollable and unstable states are reflected in system performance index.

**5. In Infinite regulator the optimal control cannot be differentiated from other controls.**

a) True

b) False

**Answer:** a

**Explanation:** In infinite regulator the optimal control may have same output for all the controls as the same solution for the infinite regulators will exist.

**6. The differential term in the riccati equation in infinite regulator is:**

a) Finite

b) Positive

c) Infinite

d) Zero

**Answer:** d

**Explanation:** As the final time is not defined hence the differential term of the riccati equation becomes zero and the equation we get is the modified riccati equation.

**7. The solution of the modified riccati equation may be:**

a) Independent

b) Dependent

c) Unique

d) Non-unique

**Answer:** d

**Explanation:** The solution of the modified Riccati equation may not be unique as the solution depends upon the requirement that P must be positive and definite.

**8. If Q is positive semi- definite:**

a) The optimal closed loop system is asymptotically stable

b) The asymptotic stability is not guaranteed

c) The system is stable

d) The system is unstable

**Answer:** b

**Explanation:** If the plant is controllable and bounded and in infinite state regulators the asymptotic stability is not guaranteed even if the Q is positive and semi-definite.

**9. Tracking problem can be in:**

a) Infinite state regulators

b) Finite state regulator

c) Sub-optimal state regulators

d) Non-linear regulators

**Answer:** c

**Explanation:** The application of the optimal regulators is that the minimum time, energy, fuel and tracking problems are designed.

**10. The regulators that convert the non-linear stability into linear stability:**

a) Linear

b) Sub-optimal

c) Finite

d) Infinite

**Answer:** d

**Explanation:** In infinite state regulators stability is not guaranteed and hence the nonlinear stability is converted into to the linear stability.

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