Table of Contents

## Is the system controllable?

Definition: An LTI system is controllable if, for every x�(t) and every finite T > 0, there exists an input function u(t), 0 < t ≤ T , such that the system state goes from x(0) = 0 to x(T ) = x� . 1This controllability from the origin is often called reachability.

## What does it mean if a system is controllable?

Complete state controllability (or simply controllability if no other context is given) describes the ability of an external input (the vector of control variables) to move the internal state of a system from any initial state to any final state in a finite time interval.

**How do you show a system is controllable?**

Complete Output Controllability: The system given in equation (1) is said to be completely output controllable or simply output controllable if any final output y(N) can be reached from any initial state x(0) by applying an unconstrained input sequence u(k), k = 0,1,2,··· ,N, for some finite N.

**Is this system controllable and observable?**

The concepts of controllability and observability are very similar. In fact, there is a concrete relationship between the two. We can say that a system (A, B) is controllable if and only if the system (A’, C, B’, D) is observable.

### How do you know if a system is Stabilizable?

(Stabilizable system): The pair (A,B) is stabilizable if it is algebraically equivalent to a system in the standard form for uncontrollable systems with n = m (i.e, Au does not exist) or with Au a stability matrix. 4 4 ] = 1 =⇒ (A, B) is not controllable!

### Can an unstable system be controllable?

If the system is only uncontrollable and unstable but the uncontrollable part is stable (this is strictly the definition of stabilizability), the unstable part can be stabilized by using a feedback over the controllable states. A standard car with 4 wheels is stable and controllable.

**How do you know if a system is observable?**

Consider a physical system modeled in state-space representation. A system is said to be observable if, for any possible evolution of state and control vectors, the current state can be estimated using only the information from outputs (physically, this generally corresponds to information obtained by sensors).

**What is Kalman test?**

Kalman’s test is a method to find controllability and observability of a system with matrix formation in a particular manner for both the test. For this lesson, basic knowledge of matrices like determinant and rank will be very useful.

## What is controllable canonical form?

1.1 Controllable Canonical Form. The controllable canonical form arranges the coefficients of the transfer func- tion denominator across one row of the A matrix:

## Why are poles placed?

Placing poles is desirable because the location of the poles corresponds directly to the eigenvalues of the system, which control the characteristics of the response of the system. The system must be considered controllable in order to implement this method.

**Is the system observable?**

A system is said to be observable if, for any possible evolution of state and control vectors, the current state can be estimated using only the information from outputs (physically, this generally corresponds to information obtained by sensors).

**Is a controllable system Stabilizable?**

A standard car with 4 wheels is stable and controllable. A standard car with 3 wheels is unstable but controllable, i.e. it is stabilizable because all unstable modis are controllable (you only need to change the balance).