The output of the system is considered near :
Explanation: Our main objective in the optimal control problem is to reduce the performance index of the system as minimize the output and make it near equal to zero.
A performance index written in terms of :
Generally the performance index be of any variables but in standard form we consider the performance index to be in 2 variable only.
Matrix Q is :
Explanation: Matrix Q defines positive definite or non-definite symmetric matrix which is used in the performance index so as to give equal weightage to each element.
Matrix R is :
Explanation: Matrix R defines positive definite or non-definite symmetric matrix which is used in the performance index so as to give equal weightage to each element.
The major requirement of making the output of the system small in output regulator problem is :
Explanation: We are concerned with making the output of the output regulator problem small this is achieved when the system is observable.
Minimum principle was given by:
Explanation: The minimum principle to minimize the performance index is given by Pontryagin.
Dynamic programming is developed by:
Explanation: Dynamic programming for minimizing the performance index is given by Bellman.
The performance index is reduced by:
Explanation: Once the performance index is calculated the next task is to find the control function which is used to minimize the performance index.
Minimum principle is based on :
Explanation: Minimum principle of Pontryagin is based on the concept of calculus of variations.
Dynamic programming is based on :
Explanation: Dynamic programming is based on principle of calculus, invariant imbedding and optimality and these are the basic laws of the nature and does not need complex mathematical development to explain its validity.
The main step for solving the optimal control problem:
Explanation: For solving the problem using optimal control problem various steps are required as first is to form the transfer function and then to compute the compensators and the major requirement is to minimize the quadratic function.
For minimizing the transfer function the condition is :
Explanation: In optimal control problems the main objective is to reduce the performance criterion which is used only when the second differentiation of the function must be negative.
For the stability in optimal control poles of the transfer function must be :
Explanation: For the stability point of view the basic definition continues that the poles must be located on the left half of s plane.
The method of choosing compensator is the configuration must be:
Explanation: The above mentioned are the various configurations of choosing a compensator.
When some of the states are inaccessible, then we may set the feedback coefficients equal to zero.
Explanation: This is done to adjust coefficients to realize the transfer function and if it is not possible then reconstruction of signals can be done.
Z-transform is used in:
Explanation: Z-transform by definition can is used in discrete case only both in optimal and normal control functions.
For the stable system in discrete optimal control systems:
Explanation: Poles in discrete system must be inside the unit circle and for causal system it must be outside the circle but no including the infinity.
The special case of the tracking problem with input equal to zero:
Explanation: For zero input output is zero if all the initial conditions are zero the response are due to non-initial conditions which are caused due to disturbances.
The primary objective of the output regulator problem is to damp out:
Explanation: The primary objective of the output regulator problem is to damp out the initial conditions quickly and also reduce the effect of excessive oscillations and overshoot.
The limitation of the transfer function approach are:
Explanation: The limitation of transfer function approach is that is it useful only for quadratic performance index and multi input and multi output systems are obvious and also it is ineffective for time varying and non-linear systems.