Control Systems - Chibum Lee | personal blog · Chibum Lee -Seoultech Control Systems...

Control Systems

State Estimation

[email protected]

Control SystemsChibum Lee -Seoultech

Outline

Dominant pole design

Symmetric root locus


State estimation

We are able to place the CLPs arbitrarily by feeding back all the

states: u = −Kx. But these may not all be available for measurement.

Hence we seek to estimate them by observing the plan input and

output, and use the estimated states in the control law:

Using our model of the plant, we could make open-loop estimates:


State estimation

However,

• our model may not be precise

• we don't know the plant's initial conditions

• the plant may be subject to unmodelled disturbances

• hence the estimated state is likely to diverge from the actual state

We therefore introduce feedback to try to get the estimated output

ŷ to track the measured output y


State estimation

is a vector of feedback gains

Thus, we form our estimates from

equations to beimplemented inestimator


Estimator dynamics

Given the plant input u and measured output y, the state estimate

evolves according to:

Our plant model is

Subtracting, and using the output equation, gives:

Denote the error in the state estimate by

Then the error dynamics are:

That is, given the assumptions of a perfect plant model, and no

disturbances, this suggests that the estimation error can be made

to go to zero in a stable and rapid manner by a suitable choice of L


Estimator dynamics

The estimated states should then continue to track the plant states

with zero error

In reality there will be estimation errors:

• plant disturbances

• model imperfections

These errors can be kept small if the eigenvalues of the error

system are well damped and relatively fast

compared with the closed-loop plant dynamics

Characteristic polynomial for error dynamics:

Desired characteristic polynomial:

Provided system is observable, we can solve for L so that aL(s) =

αe(s), using the same techniques as for K


Benefit of observer canonical form

The solution for the estimator gain matrix L is trivial if the

equations are in observer canonical form

The system matrix for the error equation is then:

This is still in observer c.f., and has the char. poly

Compare with desired char. poly:

Hence, by inspection: L1 = β1 – a1, L2 = β2 – a2, L3 = β3 – a3


Ackermann's formula

Again, Ackermann's formula can be used to effectively implement

the process of:

• transformation of an observable system to observer canonical form

• design of estimator gains by inspection

• transformation back to original states

c.f.

MATLAB: Specify desired estimator poles, dep

• L = acker(A’, C’, dep)’

• L = place(A’, C’, dep)’


Relationships between canonical forms: Duality

Observercanonicalform

Controllercanonicalform

Observabilitycanonicalform

Controllabilitycanonicalform


Regulator with estimated states

Plant:

Control law:

Estimator:


State equations for closed-loop system

Plant:

Estimator:

Control law:

Combining these equations:

Thus the closed-loop state equations are:



These equations may also be written in terms of the estimation

error

We had:

Subtracting:

Substituting in the first equation yields:

With this alternative set of state variables the closed-loop system

equations are:



The closed-loop characteristic polynomial is thus:

which is block triangular. Hence:

Thus, the poles of the CL system consist of the poles obtained by

full state feedback through K, together with the estimator poles

determined by the selection of L

That is, the control law and the estimator can be designed

independently of each other Separation Principle


Ex: regulation of water level in two-tank system

The system can be described

For



Specs: Required closed-loop time constants are 1 = 0.2 s, 2 = 0.05

s; i.e., desired CL char. poly is

Control law: Demonstrate Ackermann's formula:



Estimator: Try estim. poles = 2 x controller poles;

• i.e., desired CL char. poly is α = (s +10)(s + 40) = s2 + 50s + 400

Demonstrate Ackermann's formula:



Complete system

• Equivalent feedback compensator:

• Transfer function for compensator:


Example 7.25


Reduced-order estimators

In the examples so far, we have estimated all the states

In the situation where we can measure some of the states directly,

we do not need to reconstruct them in an estimator

• note, however, that if the measurements are noisy, a full-state estimator

will ‘smooth’ these data, as well as reconstruct the unmeasured states

If the measurements are reliable, then a reduced order estimator

will be more accurate, because some of its outputs will be direct

measurements

A reduced-order estimator will be of lower order, thus requiring less

computational power



Plant model:

Partition states into measured xa and estimated xb

That is:

possibly multiplemeasurements



Hence, observer:

Now, is in principle known, but

derivative term is a problem: difficult to realize

So, introduce a change in variable:


Error dynamics

Estimation error

That is,

Hence, design Lr as before, by selecting poles for Abb − LrAab

Design: Lr = place(A_bb’, A_ab’, dep)’


Example 7.26


Selection of estimator poles

CL char. poly:

The estimator poles (roots of aL(s) = 0) are usually chosen to be

between 2 to 6 times faster than the controller poles (roots of aK(s)

= 0)

Clearly there is no direct effect of choosing faster estimator poles

on control effort. However, consider the effect of sensor noise.

Assume that the sensed output is . The estimation

error dynamics then become:

Thus, the higher estimator gains required to produce faster

estimator dynamics will further amplify the sensor noise, thereby

corrupting the state estimates


Symmetric root locus for SISO system estimator

Estimator dynamics with 'process noise' w and 'sensor noise' v :

The process noise w can represent unknown disturbances and errors

in the plant model parameters

Recall the estimator equations:

If w is large:

• our plant model is uncertain, and hence a poor predictor

• we would put greater emphasis on the sensor data to correct the model

predictions larger L, faster estimator poles

If v is large:

• our measurements are uncertain, and hence provide poor corrections

• we would put greater emphasis on the model predictions smaller L,

slower estimator poles


Symmetric root locus for SISO system estimator

In optimal estimation theory

• the process noise is modeled as white noise with variance

• the sensor noise is modeled as white noise with variance

The estimator pole locations which will minimize the variance of the

state estimation error can be found from the solution of the

symmetric root locus (SRL) equation:

where is a measure of the plant model uncertainty

relative to the measurement uncertainty, and

is the plant transfer function from the process noise input to the

measured output

If the process noise w and control input u are additive (Bw = B), the

same SRL can be used for controller and estimator pole selection

Control Systems - Chibum Lee | personal blog · Chibum Lee -Seoultech Control Systems...

Documents

Transcript of Control Systems - Chibum Lee | personal blog · Chibum Lee -Seoultech Control Systems...