• Equations: pictures only - do not use google docs equation bar as it is invisible for mobile: here is a test: .
• Discussion in comments
• If an example is needed: let’s provide at least two solutions here or a scheme only (it looks fishy when everyone compute the same matrix as an example or provide the same methods for computing expm(At)).
• This will probably be the document which will be extensively used by us during the exam, so let’s look after it.
• Arenta po****ło. (Means: Aren’t is not good)
• Edit locked.

Pls pay attention - proszę zapłacić za uwagę

Dont boat yourself - nie łódź sie

Keep clear the doors - trzymaj przeźroczyste drzwi

Thanks from mountains - z góry dzięki

Sorry non-polish speakers, these are jokes only Poles will understand.

BUT Here we need to place Poles correctly so...

Ode to Arent

Exam will be uncontrollable.

May your cheating be unobservable!

I hope 3.0 is reachable.

And deficit - stabilizable.

[ ][ ][ ][ ][ ][ ][ ][ ][ ][ ][ ][ ]

👍👌

1. Write the definition of exp(A*t). Present two selected methods for computing exp(A*T).

2. Write an Input/State/Output representation of a linear dynamical system. Present a general solution of this differential equation and justify your answer.

Input/State/Output representation:

(we need a step-by-step solution originating from state-space form or from solution to state space form as in lecture notes)

3. Write the definition of controllability. Present the Kalman's controllability rank condition and  the Hautus test. Prove one of the mentioned controllability tests.

Definition of controllability:
        VERSION 1: A system with internal state vector x is called controllable if and only if the system states can be changed by changing the system input.
        VERSION 2: A state x0 is controllable at time t0 if for some finite time t1 there exists an input u(t) that transfers the state x(t) from x0 to the origin at time t1. [source]
        VERSION 3: A dynamical system is controllable if and only if for any x0, xk ∈ Rn there exists a control signal u:[0, tk] → Rm, which moves this system from the state x(0) = x0 to the state x(tk) = xk in a finite time tk. [source: lecture notes]

Kalman controllability test:

Hautus controllability test:


Proof of Kalman test: (pls, shorter than in lecture notes)

Proof of Hautus test:

4. Write the definition of an A-invariant space. Prove that the image of the Kalman controllability matrix is an A-invariant space. Provide a geometric interpretation of controllability in terms of A-invariance and the image of B.

A-invariant space        

Below there is just an attempt to understand the proof from above.

5. Write the definition of a reachable system. Prove that each reachable state of the state space is a linear combination of columns of the Kalman controllability matrix.

6. Write the definition of input - output equivalent representations. Then write the definitions of: controllability canonical form, controllable canonical form, observable canonical form and discuss their fundamental properties.

Every system in controllable canonical form is controllable?

Yes, if a system can be written in controllable canonical form, it is controllable.

7. Write theorems on decomposition of the pair (A,B) and on decomposition of the pair (C, A). Prove one of these theorems.

8. Explain the pole placement control problem. Closed-loop interpretation of controllability: what is the relationship between controllability and the freedom of assigning the closed-loop poles to a state feedback control system? Justify it.

Controllability is a necessary condition for free shaping of the characteristic polynomial. Proof:

9. Write and explain the Ackermann's formula. How to apply the Ackermann's formula to implement a pole-placement control for multi-input systems?

μ*(A) = Δnew(A) - characteristic polynomial for expected (proposed) eigenvalues with λ=A (matrix) eg:

we want a system to have the following poles: λ1 = -1, λ2 = -2, then the expected polynomial is: (λ+1)(λ+2) = λ2 + 3λ + 2. So Δnew(A) = A2 + 3A + 2I

M and N can be chosen randomly (there is very high probability that such pair is controllable)

10. Explain the concept of system stabilizability. Write the definition of the stabilizable eigenvalues. What is the relationship between stabilizability and stabilizable eigenvalues? What is the relationship between stabilizability and controllability?

Concept of system stabilizability

Can we find an u(t) so that the system approaches x = 0?

Controllable and stabilizable eigenvalues:

Relation between stabilizability and controllability:

11. Write the definition of observability. Present the Kalman and the Hautus observability tests. Prove one of them.

A system is observable if the initial state can be obtained (”observed”) from the knowledge of the input and the output.

Proof of Kalman test:

When the above is true, the theorem (1) below is also true.

12. Provide the geometric interpretation of observability in terms of A-invariance and the kernel of C. Justify it.

13. Explain the concept of an observer. Discuss the construction of the Luenberger observer. Propose a sufficient condition for existence of a Luenberger observer and justify it.

An algorithm mapping u, y to xˆ is called an observer. A graphical illustration of the observer concept is shown in Figure 1. The task can be summarized as follows. Formulate the necessary and sufficient conditions for the existence of an observer and construct such an observer. By saying construct we mean the proposition of a set of differential equations involving u, y and xˆ where u and y are being set and xˆ is the solution (such that the difference x(t) − xˆ(t) is small).

Luenberger observer construction:

14. Explain the concept of a compensator, including the separation principle and the certainty equivalence principle. Write down the theorem on fundamental properties of a control system with a compensator and prove it.

Compensator is a dynamical system consisting of serially connected two subsystems: a static state feedback controller and an observer. Compensator design is based on the certainty equivalence principle^2 and the separation principle^3. The general concept of a compensator is presented in Figure 2, in the part circled by a dashed line.

15. Write and discuss the Kalman decomposition theorem

VERSION 1

VERSION 2

16. Explain the problem of Linear Quadratic Control. What is the necessary and sufficient condition for existence of the solution? Write the solution.
Problem formulation:

Generally speaking: F minimize this integral and still ensures system stability.

17. Write the Algebraic Riccati Equation and theorems on fundamental properties of the solutions of this equation.

IN ADDITION

(proof by contradiction is in notes but is not essential here)

18. Pontryagin's maximum (or minimum) principle. Application to minimum energy control and to time optimal control (simple examples).

19. Explain the idea of dynamic programming (including Bellman's principle of optimality).

* 3...N) continue this until you obtain Vo(xo)

.