CHAPTER SEVEN

NEED FOR OBSERVERS – UNKNOWN INPUT OBSERVER (UIO)

1. Most modern control laws need the full state vector for their

implementation. In general, the full state is not accessible making these laws

unimplementable. Even when the states are accessible the cost of the sensors

could be very high. This makes the use of state estimators mandatory. Another

problem that real plants present is that their models are not known accurately.

For such plants the classical Luenberger observer does not work, thus further

compounding the problem of state estimation.

2. The problem of designing an observer for a multivariable linear system

partially driven by unknown inputs is of great interest. Such a problem arises in

systems subject to disturbances or with inaccessible/unmeasurable inputs and in

many applications such as fault detection and isolation, parameter identification

and cryptography. This problem has been studied extensively for the last two

decades. One approach has been to design linear observers (both full order and

reduced order). In the literature, linear observers which are completely

independent of the unmeasurable disturbances are known as unknown input

observers (UIOs) [1–5]. In particular, easily verifiable system theoretic conditions,

which are necessary and sufficient for the existence of UIOs, have been

established (see for example [6] or [5]). One possible statement of these

conditions is that the transfer function matrix between the unmeasurable input

and the measured outputs must be minimum phase and relative degree one.

3. The concept of sliding mode control [7–9] has been extended to the

problem of state estimation by an observer, for linear systems [9], uncertain

linear systems [10, 11] and nonlinear systems [12–14]. Using the same design

principles as for variable structure control, the observer trajectories are

constrained to evolve after a finite time on a suitable sliding manifold by the use

of a discontinuous output injection signal (the sliding manifold is usually given by

the difference between the observer and the system output). Subsequently the

sliding motion provides an estimate (asymptotically or in finite time) of the system

states. Sliding mode observers have been shown to be efficient in many

applications, such as in robotics [15, 16], electrical engineering [17–19], and fault

detection [20, 21].

4. The Lyapunov based approach of Walcott and Zak [2, 3, 4] considered the

problem of state observation in presence of bounded uncertainties/UI. Slotine [5]

examined the potential use of sliding surfaces for observer design for systems in

companion form with extension to non-linear systems. Lopez in [6] formulated an

alternative form of sliding mode observer wherein the output disturbances are

transformed into state disturbances, avoiding noise amplification. Similar work

was seen in [7] wherein the UI and model uncertainties were considered as a

fictitious state added to the slotine like structure [5]. Although, the methods based

on this category [5, 7] only require the output information, they suffer drawbacks.

As, choosing gains, for eg. k1 to ensure sliding depends on the estimation error |

ex2|max which is unknown. The control requires switching terms to deal with

unknown inputs thereby introducing chatter and complicating the analysis. It can

be shown that the second order PIO could bring about the same improvements

that the switching terms used in [7] brought in while being added to the first order

PIO, thereby simplifying the analysis and design. Chen [1] proposed a

disturbance observer which required the states of the system. There are other

disturbance observers also in literature [8] which also require the states in order

to estimate the disturbances. In general, the states are not available. The state

estimator is possible using Luenberger type [9] of observers provided the system

does not have uncertainties. Thus, we have a situation that the state estimation

can be done in the absence of disturbances and disturbance estimation is

possible if the states are available.

5. Luenberger Observer

The classical Luenberger observer is the fundamental state observer and all

other observers are extensions of this basic structure. Consider a plant defined

by,

___________________________________ (7.1)

Assuming the state x to be approximated by the state ˆx of the dynamic model,

_________________________________ (7.2)

and denoting e = x − ˆx, the state error dynamics is deduced as,

_____________________________ (7.3)

Thus, the dynamic behavior of the error vector is determined by the eigen values

of (A − LC). Thus, by choosing L such that (A − LC) is stable, the error dynamics

will reduce to zero asymptotically.

6. Limitations of Luenberger Observer

The problem with this observer is that it fails when the output is sensed in

presence of model uncertainties and/or sensor noise. To examine the reason

consider the plant with lumped uncertainties d as,

__________________________ (7.4)

and the observer modeled as eqn. 2.3, the error dynamics is thus deduced as,

____________________ (7.5)

The last terms shows that the proportional gain observer tends to amplify the

measurement noise by L and that the lumped uncertainty d affects the

convergence.

7. Chen’s Disturbance Observer

In [1], the non linear disturbance observer is considered. The general description

of the plant for a SISO system is given as,

___________________ (7.6)

Where, f(x), g1(x) and g2(x) are the vector of nonlinear functions of states x, u is

the control input and d the unknown disturbance whose estimate is given by,

________________________ (7.7)

____ (7.8)

Where, z1 and are the estimates of the unknown disturbance and the internal

state of the nonlinear observer respectively and p(x) is a nonlinear function to be

designed. The nonlinear observer gain l(x) is defined as,

______________________________ (7.9)

Let, the disturbance estimation error be defined as,

___________________________ (7.10)

Thus, it can be shown that, under the assumption that the disturbances are slow

varying, z1 approaches d exponentially if p(x) is chosen such that,

_______________________ (7.11)

is globally exponentially stable for all x 2 <n. As far as the stability of the

estimation error is concerned, any nonlinear vector-valued function

such that equation (7.11) is asymptotically stable and can be chosen. After l(x)

has been chosen, p(x) is found by integration.

8. Limitations of Chen’s DO

The problem with this observer is that it fails when the assumption of is

violated. Thus, under the assumption that we can write,

_____________________________________________ (7.12)

or

___________________________________________ (7.13)

Thus, if C1 > 0, then .

10. Extended State Observer (ESO)

Many controllers need information of complete state vector. In this circumstance,

it is desired to have estimation of uncertainties as well as the complete state

vector. This requirement of obtaining the estimate of uncertainty as well as states

in an integrated manner is met by the Extended State Observer (ESO) [17], [18].

The ESO is an observer which can estimate the uncertainties along with the

states of the system enabling disturbance rejection or compensation. Unlike

traditional (linear or nonlinear) observers, the ESO estimates the effect of

uncertainties, un-modeled dynamics and external disturbances acting on the

system as an extended state of the original system. the ESO is an observer

which can estimate the uncertainty along with the states of the system enabling

disturbance rejection or compensation. The ESO regards all factors affecting the

plant, including the nonlinearities, uncertainties, the coupling effects and the

external disturbances as the total uncertainty to be observed and in that sense, it

can be viewed as an unknown input observer or disturbance observer. Since the

observer estimates the uncertainty as an extended state of the original system, it

is designated as Extended State Observer. Its merit is that it is relatively

independent of mathematical model of the plant, performs better and is simpler to

implement. The robustness is inherent in its structure as will be obvious in next

section. In Ref. [17], a comparison study of performances and characteristics of

three advanced state observers namely high gain observer, ESO and Sliding

Mode Observer is presented and it is shown that over all the ESO is much

superior in dealing with uncertainties, disturbances and sensor noise. Several

diverse applications of ESO based control strategies have appeared in literature.

Control of induction motor drive [21], aircraft attitude control [22], hydraulic

position servo system [23], and torsional vibration control of main drive system of

a rolling mill [24] are some examples to mention.

11. Proportional Integral Observer (PIO)

In literature [10, 11, 12, 13, 14, 15] we see copious results where simultaneous

state and disturbance observation is reported using proportional integral type of

observers. Most of these results are given for systems in which the UI is a

function of time and the intended application is fault diagnosis or fault tolerance.

It is seen from these results that it is the integral term that makes the estimation

of UI possible. Although, all these papers give the mathematical proof for the

convergence of error in estimation of UI to zero, but these treatments do not

appeal to explain as to why the integral term indeed gives an estimate of the UI.

Chang [16] proposed the discrete time PIO for a condition when the disturbance

input is constant for a MIMO system. The structure is firstly, discrete based and

hence requires an entirely different formulation for continuous implementation.

Secondly, the formulation fails when the disturbance is time varying, hence, is

not practical for implementation.