2. Homework 9 Refer to the last example.
Chapter 8
State Feedback and State Estimators
Homework 9
Refer to the last example.
(a) Calculate the transfer function G(s) of the system.
(b) Calculate the steady-state value of the system to a step input, using the Final Value Theorem of Laplace Transform.
(c) Determine the gain K so that the steady-state response of KG(s) has zero error to a step input.
(d) Find out the relation between the transfer function gain K and the reference gain E.

3. Chapter 8
State Feedback and State Estimators
Solution of Homework 9
(a) Calculate the transfer function of the system in s-Domain.

4. Chapter 8
State Feedback and State Estimators
Solution of Homework 9
(b) Calculate the steady-state value of the step response of the system, using the Final Value Theorem of Laplace Transform.

5. Chapter 8
State Feedback and State Estimators
Solution of Homework 9
(c) Determine the gain K so that the steady-state response of KG(s) has zero error to a step input.
(d) Find out the relation between the transfer function gain K and the reference gain E.

6. Optimal Control: The Motivation
Chapter 10
Optimal Control
Optimal Control: The Motivation
Consider a car traveling on a straight line through a hilly road.
Case 1: How should the driver press the gas pedal in order to minimize the total traveling time while keeping the fuel consumption economical?
Case 2: How to travel with as cheap fuel consumption as possible, without regarding time?
Case 3: How if time is very important, no matter how much the fuel cost is.
Although the system is the same (i.e. the car), the solution for various cases can be very different. The way the driver behaves depends on the priority of time or cost, or the mix of both.

7. Optimal Control: The Motivation
Chapter 10
Optimal Control
Optimal Control: The Motivation
Previously, we already can design state-feedback control or output-feedback control, by placing the poles of the system in a certain location.
We do not consider any constraints until now about the location of the poles.
In order to get faster response and converge, the poles can be simply placed far to the left of imaginary axis, with bigger negative real parts.
But, having such poles means that the elements of k have to be large. This means requirement for large control signal, large capacity, and more energy.
Thus, there is a conflict between “fast response” and “low price.” The question is: how can we balance between good transient response and small control effort?

8. Chapter 10
Optimal Control
Optimal Control
Optimal control deals with the problem of finding a control law (in this case, in the form of feedback gain k) for a given system, such that a certain optimality on a criterion is achieved.
The resulting control law is optimal solution of a cost function or performance index.
A proper cost function (or performance index) will give a mathematical expression that includes all variables which need to be optimized. The variables can be state variables x, control input u, control error e, time t, etc.

9. Chapter 10
Optimal Control
Performance Indexes
A performance index is a mean to measure the control performance in the completion of a control objective.
Suppose the control objective is to bring a system modeled by:
in such a way and in a fixed time interval [to,tf], so that the components of the state vector are “small”.
A suitable performance index to be minimized would be:

10. Chapter 10
Optimal Control
Performance Indexes
If the control objective is to manipulate the system so that the components of output y(t) are to be small, then:
where Q = CTC is a symmetric positive semidefinite matrix.
A matrix is a positive semidefinite matrix if all the real parts of its eigenvalues are more or equal to zero, OR, if for all nonzero vector z with real entries, then zTQ z ≥ 0.

11. Chapter 10
Optimal Control
Performance Indexes
In order to control the inputs u(t) so that they are not too large, we use the following performance index:
where R is a symmetric positive semidefinite weight matrix.
Not all performance indexes can be minimized at the same time. A compromise can be taken by minimizing a convex combination of the indexes, e.g.:

12. Chapter 10
Optimal Control
Performance Indexes
The weighting constants can also be accommodated in the weight matrices Q and R as follows:
with

13. Chapter 10
Optimal Control
Weight Matrices Q and R
The weight matrices Q and R can be used to assign different weighting to each state and each input, in the calculation of the performance index.
Some examples are:

14. Some Performance Indexes for Optimal Control Problems
Chapter 10
Optimal Control
Some Performance Indexes for Optimal Control Problems
1. Minimum-time problems
2. Terminal control problems
: final state
: reference final state
3. Minimum control effort problems
4. Tracking problems

15. Chapter 10
Optimal Control
Optimal Control
In modern optimal control theory, the following cost function is mostly implemented:
This choice is based on a certain logics. It requires minimization of the square of input u (which in general means energy required to control a system), and minimization of the square of the state variables x.
Why should we minimize x, not e? The history of optimal control development starts from the implementation where state variables x should be minimized. In this implementation, the state is equal to the deviation from a set point zero.
Besides, the same thing applies also for linearized system. Here, the state Δx denotes the deviation from x0, the operating point where the linearization was taken.
In electric systems, i.e., the input can be electric current (i) or electric potential (V), which are quadratically proportional to electric power (P)

16. Chapter 10
Optimal Control
Optimal Control
An example for such implementation is the benchmark problem inverted pendulum. The states of the system: the deviation angle θ and the position x, should be guided to zero as fast as possible.
Or recall again, the water-tank system. The system is nonlinear due to h1/2. v1 qi qo V h

17. Chapter 10
Optimal Control
Optimal Control
The regulation of the state variables to zero can be modified by shifting the point of origin. Then, the state variables can be regulated to any constant values.
We will first discuss the optimal solution to minimize state variables x along with control effort u.
Afterwards, we will discuss the solution to minimize control error e along with control effort u.

18. Optimal Control: Minimizing State Variables
Chapter 10
Optimal Control
Optimal Control: Minimizing State Variables
Consider the n-dimensional single-variable state space equations:
The desired state vector is represented xd(t) = 0.
We will select a feedback controller, so that u(t) is some function of the measured state variables x(t), and therefore
Substituting into the first equation, we obtain

19. Optimal Control: Minimizing State Variables
Chapter 10
Optimal Control
Optimal Control: Minimizing State Variables
The performance index to be minimized is the one expressed in terms of the state vector
In minimizing J1, we let the final time of interest be tf = ∞.
To obtain the minimum value of J1, we postulate a decreasing magnitude of the state vector, or equivalently, postulate its negative first derivation.
We assume the existence of an exact differential so that
where P is yet to be determined.
P is chosen to be symmetric to simplify the algebra without any loss of generality.

20. Optimal Control: Minimizing State Variables
Chapter 10
Optimal Control
Optimal Control: Minimizing State Variables
Then, for a symmetric P, pij = pji, the differentiation on the left-hand side of the last equation can be completed as
But
thus If
then
as wished.

21. Optimal Control: Minimizing State Variables
Chapter 10
Optimal Control
Optimal Control: Minimizing State Variables
Further
Therefore, to minimize the performance index J1, the following two equations must be considered

22. Example 1: Optimal State Feedback
Chapter 10
Optimal Control
Example 1: Optimal State Feedback
Consider the open-loop control system shown in the figure below. The state variables are x1 and x2. The step response of the system is quite unsatisfactory due to undamped poles at the origin.
We will choose a feedback control in the form of
Therefore, inserting u(t) into the state equation,

23. Example 1: Optimal State Feedback
Chapter 10
Optimal Control
Example 1: Optimal State Feedback
To minimize the performance index J1, we write again the equation
Letting k1 equals an arbitrary stable value, say k1 = 2, then

24. Example 1: Optimal State Feedback
Chapter 10
Optimal Control
Example 1: Optimal State Feedback
The integral of the performance index is rewritten as
We will consider the case where the initial states are given as xT(0) = [1 2], so that
Substituting the values of P we already have,

25. Example 1: Optimal State Feedback
Chapter 10
Optimal Control
Example 1: Optimal State Feedback
To minimize J1 as a function of k2, we take the derivative w.r.t k2 and set it equal to zero
Thus, the system matrix for the compensated system is
Checking the characteristic equation of the compensated system,
Both poles are now with negative real value
The system is optimal for the given values of k1 and x(0)

26. Example 1: Validation of Answer
Chapter 10
Optimal Control
Example 1: Validation of Answer
Now, suppose we want to stabilize the given system and put the poles to certain locations by using a non-optimal feedback gain k.
Any positive value for k1 and k2 will guarantee the stability of the system
With no real input r(t) –the system only have pure state feedback connection– the output will decay to zero
The integral of squared states of the system using kopt will now be compared with any arbitrary chosen k
Note: k1 = 2 must be kept

27. Example 1: Validation of Answer
Chapter 10
Optimal Control
Example 1: Validation of Answer

28. Example 1: Validation of Answer
Chapter 10
Optimal Control
Example 1: Validation of Answer
: J1, k = [2 10]
: J1, kopt = [ ]
: J1, k = [2 2]
: J1, no k
Without any state feedback, the output of the double integrator will sum up to infinity
kopt results the lowest value of cost function J1

29. Chapter 10
Optimal Control
Homework 10
Consider again the control system as given before, described by
Assuming the linear control law
Determine the constants k1 and k2 so that the following performance index is minimized
Consider only the case where the initial condition is x(0)=[c 0]T and the undamped natural frequency (ωn) is chosen to be 2 rad/s.
Calculate the transfer function of the system if compensated with k
Determine the value of corresponding k (k1 or k2?) to obtain ωn as requested

30. Homework 10A Consider the system described by the equations
Chapter 10
Optimal Control
Homework 10A
Consider the system described by the equations
Determine the optimal control which minimizes the following performance index. (Hint: You may assume the value of the feedback gain, where available.)
Deadline: Wednesday, 26 November 2014.
Quiz 3: Lecture 8, 9, and 10.