Lecture 34 - State Estimators or Observers - Electrical Engineering (EE) PDF Download

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Lecture 34 - State Estimators or Observers, Control Systems

1 State Estimators or Observers

• One should note that although state feedback control is very attractive because of precise computation of the gain matrix K , implementation of a state feedback controller is possible only when all state variables are directly measurable with help of some kind of sensors.
• Due to the excess number of required sensors or unavailability of states for measurement, in most of the practical situations this requirement is not met.
• Only a subset of state variables or their combinations may be available for measurements.

Sometimes only output y is available for measurement.

• Hence the need for an estimator or observer is obvious which estimates all state variables while observing input and output.

Full Order Observer: If the state observer estimates all the state variables, regardless of whether some are available for direct measurements or not, it is called a full order observer.

Reduced Order Observer: An observer that estimates fewer than “n” states of the system is called reduced order observer.

Minimum Order Observer: If the order of the observer is minimum possible then it is called minimum order observer.

2 Full Order Observers Consider the following system

x(k + 1) = Ax(k) + Bu(k) y(k) = C x(k)

where x ∈ R^n×1, u ∈ R^m×1 and y ∈ R^p×1.

Assumption: The pair (A, C ) is observable.

Goal: To construct a dynamic system that will estimate the state vector based on the information of the plant input u and output y.

2.1 Open Loop Estimator

The schematic of an open loop estimator is shown in Figure 1.

Figure 1: Open Loop Observer

The dynamics of this estimator are described by the following

where  is the estimate of x and  is the estimate of y.

Let  =  − x be the estimation error.

Then the error dynamics are defined by

with the initial estimation error as

If the eigenvalues of A are inside the unit circle then  will converge to 0. But we have no control over the convergence rate.

Moreover, A may have eigenvalues outside the unit circle. In that case  will diverge from 0. Thus the open loop estimator is impractical.

2.2 Luenberger State Observer 

Consider the system x(k + 1) = Ax(k) + B u(k). Luenberger observer is shown in Figure 2. The observer dynamics can be expressed as:

                   (1)

Figure 2: Luenberger observer

The closed loop error dynamics can be derived as:

It can be seen that x˜ → 0 if L can be designed such that (A − LC ) has eigenvalues inside the unit circle of z -plane.

The convergence rate can also be controlled by properly choosing the closed loop eigenvalues.

Computation of Observer gain matrix L

The task is to place the poles of |A − LC |. Necessary and sufficient condition for arbitrary pole placement is that the pair should be controllable.

Assumption: The pair (A, C ) is observable. Thus, from the theorem of duality, the pair (A^T , C ^T ) is controllable.

You should note that the eigenvalues of A^T − C^T L^T are same as that of A − LC . It is same as a hypothetical pole placement problem for the system, using a control law

Example:

The observability matrix

is non singular. Thus the pair (A, C ) is observable. The observer dynamics are

L should be designed such that the observer poles are at 0.2 and 0.3.

We design L^T such that A^T − C^T L^T has eigenvalues at 0.2 and 0.3.

Using Ackermann’s formula, L^T = [−0.5 20.06]. Thus

2.3 Controller with Observer 

The observer dynamics:

.

Combining with the system dynamics

Since the states are unavailable for measurements, the control input is

Putting the control law in the augmented equation

The error dynamics is

If we augment the above with the system dynamics, we get

where the dimension of the augmented system matrix is R^2n×2n. Looking at the matrix one can easily understand that 2n eigenvalues of the augmented matrix are equal to the individual eigenvalues of A − BK and A − LC .

Conclusion: We can reach to a conclusion from the above fact is the design of control law, i.e., A − B K is separated from the design of the observer, i.e., A − LC .

The above conclusion is commonly referred to as separation principle.

The block diagram of controller with observer is shown in Figure 3.

Figure 3: Controller with observer