Suppose we are given a system

which is unstable (i.e. at least one of the roots of det(sI-A) = 0 is in the right half of the complex plane). We would like to know if there exists a control input of the form u = -Kx+v , where K is a constant vector and v is any external input, which will stabilize the system. In other words, we would like to feed the the state of the system back through the system input in such a way that the system from the new input v is stable.

In order to implement such a controller for a physical system, we must be able to measure our system state with sensors (e.g. potentiometers, tachometers, etc.). Therefore, assuming our full state is available for measurement, let us consider the type of systems for which a stabilizing state feedback controller may be obtained. Let us substitute the control law, u = -Kx+v , into our original system:

Our closed loop system is described by the following state-space equations:

With the feedback u = -Kx+v , the controlled system is stable if det(sI - (A-BK)) = 0 has all its roots in the left of the complex plane.

We will know demonstrate the ideas of state feedback and stabilization with the aid of the following example.

Example:

Consider the three spring-mass systems shown in the figure below:
(a) Spring-mass system with an external force u (input) .
(b) Spring-mass damper system with an external force v (input).
(c) Spring-mass system with state feedback. The external force is u . Input is v .

We will show in this example how state feedback is used to stabilize the spring-mass system. In addition, for a certain choice of the vector K the state feedback system is equivalent to the spring-mass-damper system which is stable.

Click here to see the spring-mass system stability animation

The state variables are position and velocity.

System (a):

System (b):

System (c):

The constants K₁ and K₂ are chosen so that the eigenvalues of the new A matrix have negative real parts (stable system). If we choose K = [0 b], we get

In this case system (c) becomes equivalent to the stable system (b).

Let us summarize the above results. System (a) is unstable since the two eigenvalues of the system have zero real part. One way to stabilize the system is to add a damper. Another way to stabilize system (a) is to add a state feedback controller. If the gain vector for the state feedback controller is K = [0 b ] system (c) becomes equivalent to system (b). Note that in system (c) we need to measure the state variables with sensors, whereas in system (b) there is no need to measure any variable. In addition, adding a damper to the system is a mechanical step towards stabilizing the system. On the other hand, in state feedback electrical components may be involved.