1. (COEN507) LECTURE III SLIDES By M. Abdullahi
Department of Electrical and Computer Engineering
Ahmadu Bello University, Zaria
CONTROL SYSTEM ENGINEERING III
(COEN507)
LECTURE III
SLIDES
By M. Abdullahi

2. MODERN CONTROL SYSTEMS STATE-SPACE CONCEPTS
CAYLEY-HAMILTON THEOREM
The theorem is more computationally convenient for large systems compared to the other 2 methods
“Every Square Matrix A satisfies its own Characteristic Equation” If
is the characteristic equation of A, then
The theorem provides a simple procedure for evaluating the function of a matrix

3. MODERN CONTROL SYSTEMS STATE-SPACE CONCEPTS
Given a matrix polynomial in terms of the λ as f(λ) and the characteristic polynomial as q(λ)
If f(λ) is divided by q(λ) then:
Where R(λ) is the remainder polynomial of the form:
If we evaluate f(λ) at the eigenvalues (λi); then q(λ)=0 and we have

4. MODERN CONTROL SYSTEMS STATE-SPACE CONCEPTS
The coefficients can be obtained by successively substituting into equation for q(λ)
Substituting A for λ:
Since q(A) is identically zero, it follows that:
which is the desired result

5. MODERN CONTROL SYSTEMS STATE-SPACE CONCEPTS
The formal procedure of evaluation of the matrix polynomial f(A) is given as:
Find the eigenvalues of matrix A
If all the eigenvalues are distinct, solve n simultaneous equations given by f(λi) for the coefficients
If A possesses an eigenvalue λk of order m, then only one independent equation can be obtained by substituting λk into equation for f(λi). The remaining (m-1) linear equations, which must be obtained in order to solve for αi can be found by differentiating both sides of:

6. MODERN CONTROL SYSTEMS STATE-SPACE CONCEPTS
Since
It follows that:
The coefficients α1 obtained and equation (f(A)=R(A)) yield the required result

7. MODERN CONTROL SYSTEMS STATE-SPACE CONCEPTS
The Cayley-Hamilton theorem allows us to attack the problems of computation of eAt , where A is a constant n x n matrix.
The power series for the scalar eλt :
converges for all finite λ and t.
It follows from this that the matrix power series
converges for all A and for all finite t.

8. MODERN CONTROL SYSTEMS STATE-SPACE CONCEPTS
The matrix polynomial f(A)=eAt can be expressed as a polynomial in A of degree (n-1) using the technique presented
Ex: Find

9. MODERN CONTROL SYSTEMS STATE-SPACE CONCEPTS
STATE OBSERVER
In the design of Control Systems, it is assumed that all state variables are accessible for feedback
However, this is not the case in practice and as such there must be mechanisms to allow us to estimate for the unavailable state variables e.g introducing a state observer
Its essence is to ensure that the state variables are accessible for measurement and control purposes

10. MODERN CONTROL SYSTEMS STATE-SPACE CONCEPTS
There are 3 steps to ensuring accessibility of the state variables:
Assume that all state variables are measureable and utilize them in a full-state feedback control law
Construct an observer to estimate the states that are not directly sensed and available as outputs
Appropriately connect the observer to the full-state feedback control law

11. MODERN CONTROL SYSTEMS STATE-SPACE CONCEPTS
An OBSERVER is “a device that can be driven by the available inputs and outputs to approximate the state vector and whose output can then be used to implement the feedback control law”
A control system with an observer is as shown

12. MODERN CONTROL SYSTEMS STATE-SPACE CONCEPTS
If the observer ‘observes’ all the state variables even if some are not available for direct measurement, it is called a FULL-ORDER OBSERVER.
If the observer assumes that certain state variables are already available as system outputs and do not need to be estimated, it is called a REDUCED-ORDER OBSERVER.
The design criterion should be to minimize the difference between y=Cx (system output) and the output as constructed by the observed vector
This is equivalent to minimizing

13. MODERN CONTROL SYSTEMS STATE-SPACE CONCEPTS
However, since x may not be accessible always, we attempt instead to minimize
The design criterion then is: “Design Ke which is an n x 1 vector so that is minimized”

14. MODERN CONTROL SYSTEMS STATE-SPACE CONCEPTS
Key to the determination of the state observer is the determination of an appropriate observer gain matrix Ke
Feedback signal through Ke serves as a correction signal to the plant.
If the unknowns to be accounted for are significant, the feedback signal must be large
If the output contains noise or is unreliable, the feedback must be small
Several Kes must be tested based on several different characteristic equations and the best one chosen

15. MODERN CONTROL SYSTEMS STATE-SPACE CONCEPTS
The vector can be made to decay if the eigenvalues of (A-KeC) are properly selected and if the system is observable, then they can be chosen arbitrarily.
We will consider two approaches to determining Ke:
We solve the characteristic equation:
and equate it to the characteristic equation formed by the given eigenvalues in order to determine the coefficients of Ke.

16. MODERN CONTROL SYSTEMS STATE-SPACE CONCEPTS
Consider a linear system
Assume that the states x1, x2 and x3 are not accessible for feedback. An observer system is to be designed to reconstruct x so that the eigenvalues are at (-3, -4, -2).

17. MODERN CONTROL SYSTEMS STATE-SPACE CONCEPTS
To solve the problem in Matlab, we make use of the following commands in the CLI:
syms
eye
poly

18. MODERN CONTROL SYSTEMS STATE-SPACE CONCEPTS
2. We can also use the Ackermann’s formula

19. MODERN CONTROL SYSTEMS STATE-SPACE CONCEPTS
The full-order state observer is given by:
Ex:
Consider the system:
where:
Assume the desired eigenvalues of the observer matrix µ1=µ2=-10, determine the full-order state observer

20. MODERN CONTROL SYSTEMS STATE-SPACE CONCEPTS
To solve the problem in Matlab, we make use of the following commands in the CLI:
syms
eye
transpose (‘)
acker

21. MODERN CONTROL SYSTEMS STATE-SPACE CONCEPTS
LINEAR QUADRATIC OPTIMAL REGULATOR
The LQR method provides a systematic way of computing the state feedback gain matrix,Ke.
Once we can obtain an optimal solution, the system can be STABLE irrespective of the stability of the open-loop system.
NOTE: We seem more interested in stability in the classical domain and ignore optimality of performance
Given:
We need to determine matrix Ke of the optimal system

22. MODERN CONTROL SYSTEMS STATE-SPACE CONCEPTS
This is in order to minimize the performance index:
Where Q and R are positive definite matrices (pdm) and account for the relative importance of the errors
(u*Ru) accounts for expenditure of the energy of the system.
NOTE: A given matrix
is a pdm iff:

23. MODERN CONTROL SYSTEMS STATE-SPACE CONCEPTS
It can be shown that:
The Reduced-Riccati Matrix (RRM) equation is given by:
The LQR methodology is as follows:
Solve the RRM to obtain P, which must be a pdm. This implies system stability
Substitute P into the equation for Ke. The resulting matrix is the optimal gain matrix

24. MODERN CONTROL SYSTEMS STATE-SPACE CONCEPTS
Consider the system shown. Assuming the control signal to be: , determine the optimal feedback gain matrix k such that the following performance index is minimized
where:
u x x1 -k ∫

25. MODERN CONTROL SYSTEMS STATE-SPACE CONCEPTS
To solve the problem in Matlab, we make use of the following commands in the CLI:
lqr

26. MODERN CONTROL SYSTEMS STATE-SPACE CONCEPTS
LYAPUNOV STABILITY
Consider a system:
Three concepts of
stability are consi-
dered with respect
to non-linear systems
The system is STABLE at the origin if, for every initial state x(t0) which is close to the origin, x(t) remains near the origin for all t

27. MODERN CONTROL SYSTEMS STATE-SPACE CONCEPTS
The system is ASYMPTOTICALLY STABLE if x(t) approaches the origin as t → ∞
(Every motion starting in S(r) converges to the origin as t → ∞)
The system is ASYMPTOTICALLY STSBLE IN THE LARGE (Global Stability) if it is asymptotically stable for every initial state regardless of how near or far it is from the origin (Every motion will approach the origin)
The primary method of testing stability of non-linear systems is the Lyapunov’s Direct Method, which is concerned with determining the Lyapunov Function

28. MODERN CONTROL SYSTEMS STATE-SPACE CONCEPTS
Given
If there exists a scalar function V(x) that satisfies the following properties
Then the system is STABLE at the origin
NOTE: Inability to find the required V(x) does not imply instability but that the attempt to establish stability has failed

29. MODERN CONTROL SYSTEMS STATE-SPACE CONCEPTS
A scalar function f(x) is Positive Definite if:
Then g(x) = -f(x) is said to be a Negative Definite Function
A matrix Q is semi-definite if any of the det[Q]=0
A matrix Q is negative definite (or semi-definite) if –Q is positive definite (or semi-definite)
If Q is positive definite so also will be Q2 and Q-1

30. MODERN CONTROL SYSTEMS STATE-SPACE CONCEPTS
A scalar function V(x) that satisfies the following conditions is said to be a Lyapunov Function

31. MODERN CONTROL SYSTEMS STATE-SPACE CONCEPTS
The Lyapunov direct method stability criterion:
For a linear system
Suppose that u=0 and there exists two pdms P>0 and Q>0, it is stated thus:
“A linear system is asymptotically stable at the origin iff given any symmetric, pdm Q, there exists a symmetric pdm P which is the unique solution of: “
The choice of Q can be made arbitrarily (as far as it is a pdm) but it is most common to set Q=I, the identity matrix.

32. MODERN CONTROL SYSTEMS STATE-SPACE CONCEPTS
The test of P’s positive definite status is carried out using Sylvester’s theorem:
“The necessary and sufficient conditions for a matrix P to be positive are that all successive principal minors of P be positive”

33. MODERN CONTROL SYSTEMS STATE-SPACE CONCEPTS
Using the Liapunov Direct Method, determine the stability-in-the-large of a system described by the following: with
NOTE: The lyap command is used to solve this problem in the Matlab CLI