Design of Sampled Data Nonlinear Observers Alternative Approaches in the Context of Observer Error

www.ijape.org International Journal of Automation and Power Engineering (IJAPE) Volume 3 Issue 1, January 2014 DOI: 10.14355/ijape.2014.0301.03

Design of Sampled Data Nonlinear Observers Alternative Approaches in the Context of Observer Error Linearization Costas Kravaris*1 Department of Chemical Engineering, University of Patras 26500 Patras, Greece *1

kravaris@chemeng.upatras.gr

Abstract This paper has explored alternative design approaches for sampled data nonlinear observers in the context of the observer error linearization design method. After a brief and necessary review of observer error linearization, two alternative time‐discretization approaches were examined, one involving discretization at the beginning, and the other at the end. For the case of large sampling period, a sampled data observer inter‐sample prediction was proposed.

Keywords Nonlinear Observer; Sampled Data Observer; Observer Error Linearization

Introduction In many engineering applications, system dynamics evolves continuously in time, but the measurements are discrete in time and the implementation of the observer is realized in discrete time using the sampled data. Therefore, discretization issues naturally arise in engineering applications. If the sampling period is relatively small, one possible approach is to design the observer in continuous time and discretize the observer equations at the implementation stage.

Another possibility is to discretize the system dynamics at the beginning and do the observer design in discrete time. Both approaches are reasonable and there is no clear superiority of one over the other in the case of linear systems.

In the presence of large sampling periods, special care is needed, to account for system dynamics in between sampling instants. In particular, intersample behavior of the system states should be predicted and accounted for in the observer calculations.

In the present paper, the goal is to explore all three approaches in the design of sampled data observers for nonlinear systems. Because there is a large number of nonlinear observer design methods available, the present study will focus on a particular nonlinear observer design method: observer error linearization with eigenvalue assignment.

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Observer Error Linearization In exact linearization design methods (see Andrieu and Praly, 2006; Astolfi and Praly, 2006; Kazantzis and Kravaris, 1998; Kreisselmeier and Engel, 2003; Krener and Isidori, 1983; Krener and Respondek, 1985; Krener and Xiao, 2002; Krener and Xiao, 2005; Xia and Gao, 1989 for continuous‐time systems, and Califano et al., 2003; Chung and Grizzle, 1990; Kazantzis and Kravaris, 2001; Lee and Nam, 1991; Lin and Byrnes, 1995; Xiao et al., 2003 for discrete‐time systems), an observer is built so that the resulting error dynamics is linear in curvilinear coordinates, and with pre‐specified rate of decay of the error.

Observer Error Linearization in Continuous Time Consider a continuous‐time nonlinear system of the form:

dx

dt  f ( x) (1)

y  h( x)

with x R

the state vector, y R the measured

n

output, f : R

n

 R n , h : R n  R nonlinear functions.

Let T ( x) be an invertible function from R n to R n that satisfies the linear partial differential equations: T ( x) f ( x)  AT ( x)  bh( x) (2)

x

where the matrices A and b are design parameters.

Then, the dynamic system:

dz

dt  Az  by (3)

xˆ  T 1 ( z)

driven by the measurement y, is an observer for system (1), with xˆ R n representing an estimate of the state of (1).

Notice that system (3) has linear dynamics and nonlinear output map. Moreover, it has the property that d T( xˆ )  T ( x)   A T ( xˆ )  T ( x)  (4)

dt

International Journal of Automation and Power Engineering (IJAPE) Volume 3 Issue 1, January 2014 www.ijape.org

that

which says that the error dynamics is linear when described in appropriate curvilinear coordinates, and it is governed by the eigenvalues of the matrix A. Thus, by appropriately selecting the matrix A, a prespecified rate of decay of the error can be guaranteed.



which says that the error dynamics is linear when described in appropriate curvilinear coordinates, and it is governed by the eigenvalues of the matrix Ad .

It should be noted that when the observer (3) is transformed via the transformation z  T ( xˆ ) , it takes

Thus, through appropriate selection of the matrix Ad , a given rate of decay of the error can be guaranteed.

the form:

ˆ dt

dx  f ( xˆ )  L( xˆ )( y  h( xˆ )) (5)

It is important to notice the mathematical similarity between continuous‐time and discrete time designs.

When the system dynamics is described by a set of difference equations instead of differential equations, the nonlinear observer has exactly the same structure, with linear dynamics, and the design equations become a set of linear functional equations (8) instead of linear partial differential equations (2), with function composition in the left hand side in place of the directional derivative. Existence, uniqueness and invertibility conditions for (8) are similar to (2), and the calculation method is the same (power series). See Kazantzis and Kravaris (2001) and Xiao et al. (2003).

where L( x)   T ( x)  x



1

b (6)

In other words, observer (3) is a Luenberger‐type observer, with state‐dependent gain L( xˆ ) .

The key to the application of the exact linearization method is the linear partial differential equation (2).

Existence, uniqueness and invertibility of the solution can be guaranteed under appropriate non‐resonance conditions between A and the Jacobian of f(x), observability of (1) and controllability of (A,b) (see Kazantzis and Kravaris (1998) and Krener and Xiao (2002) for details on local results, Andrieu and Praly (2006) and Astolfi and Praly (2006) for global results).

Computationally, equation (2) can be handled via a power series solution (Kazantzis and Kravaris, 1998).

Discretization of Continuous-Time Observer versus Discrete-Time Design When the sampling period is relatively small, the sampled data observer can be based upon a time‐ discretization of the continuous‐time observer.

Alternatively, the system (1) can be time‐discretized from the beginning, and a discrete‐time design of the observer can be performed.

Observer Error Linearization in Discrete Time Consider a discrete‐time nonlinear system of the form:

with x R

n

output,  : R

x( k  1)  ( x( k)) (7) y( k)  h( x( k))

Exact Discretization of the Continuous‐Time Observer Let P be the sampling period. Considering the measument to be approximately constant in between samples, the observer (3) can be discretized as follows:

the state vector, y R the measured n

R n,

h : R n  R nonlinear functions.

Let Td ( x) be an invertible function from R n to R n that satisfies the linear functional equations:

T (( x))  A T ( x)  b h( x) (8) d d d d where the matrices Ad and bd are design parameters.

Then, the dynamic system:

z( k  1)  Ad z( k)  bd y( k ) xˆ ( k )  Td1 ( z( k))

z( k  1)  Ad z( k)  bd y( k) xˆ ( k)  T 1( z( k))

(11)

P

where Ad  e AP , bd   e Abd and T ( x) is the solution 0

of (2). Notice that (11) is an exact discretization of (3) when y(t) is piecewise constant, because of the linearity of the dynamics of the continuous‐time observer.

(9)

driven by the measurement y, is an observer for system (7), with xˆ R n representing an estimate of

Discrete‐Time Design To perform a discrete‐time observer design, system (1) can be discretized in time, using any standard numerical algorithm, and thus be converted in the form (7). For example, if one uses Euler’s method, with

the state of (7).

Notice that system (9) has linear dynamics and nonlinear output map. Moreover, it has the property



Td ( xˆ ( k  1))  Td ( x( k  1))  Ad Td ( xˆ ( k ))  Td ( x( k )) (10)

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the most meaningful approach in the presence of relatively fast sampling. The reason is that because of the linearity of the observer dynamics, exact discretization of the continuous‐time observer is possible and therefore, stability of the error dynamics and rate of decay of the error will be preserved after discretization. This important feature is indeed a key advantage of the observer error linearization method.

time step equal to the sampling period P, system (1) can be approximately represented as

x( k  1)  x( k)  Pf ( x( k)) (12) y( k)  h( x( k ))

and the corresponding discrete‐time observer is given by (9) with Td ( x) being the solution of (8), which now takes the form: T ( x  P f ( x))  A T ( x)  b h( x) (13) d d d d

Sampled Data Nonlinear Observer with Intersample Prediction

It is important to note here that the representation (12) involves approximation error, which induces error in the calculation of Td ( x) from (13). Of course, it is

When sampling is not performed at a fast rate, inter‐ sample behavior becomes important and needs to be accounted for by the observer. For this purpose, the model (1) could be used to predict the evolution of the output during the time period in between two consecutive measurements. In particular, (1) can be used to estimate of the rate of change of the output, in order to be able to continuously apply a correction on the most recent measurement during this period.

possible to use a higher‐order, more accurate numerical method than Euler’s, but the approximation error will still be present and its size needs to be evaluated in a particular application.

Consistency of Discrete‐Time and Continuous‐Time Observers in the Limit of Very Fast Sampling Consider the situation of very small sampling period P. Taking the discrete‐time design matrices from the Euler discretization of the continuous‐time observer (3), i.e. Ad  I  PA and bd  Pb , functional equation

Since the rate of change of the output is

dy  L f h  x  , dt

n h ( x) denotes the Lie deriva‐ where L f h  x    f j ( x) x j 1

(13) takes the form:

j

tive, it is possible to estimate the rate of change of the output from the Lie derivative, evaluated at the state estimate xˆ :

T ( x  Pf ( x))  T ( x)  PAT ( x)  Pbh( x) (14) d d d But for small P,

T

T ( x  Pf ( x))  T ( x)  P d ( x) f ( x)  O( P2 ) d d x and thus, neglecting the higher‐order terms in P, (14)

dyˆ  L f h( xˆ ) , t  [ti , ti 1 ) dt

T collapses to d ( x) f ( x)  AT ( x)  bh( x) , from which d x Td ( x)  T ( x) , where T ( x) is the solution of (2).

where ti , ti 1 denote consecutive sampling instants.

Therefore the discrete‐time design is consistent with the continuous‐time observer in the sense that it converges to it, in the limit as the sampling period goes to zero.

d  L f h  xˆ  , t [ti , ti1 ) dt (15)  (ti )  y(ti )

The above can be integrated on line, with the most recent measurement as initial condition. This leads to the following inter‐sample output predictor:

with  representing the output prediction.

Comparison of the Design Approaches

When the continuous‐time observer (5) with L( x) given

Even though discrete‐time design converges to the continuous‐time observer in the limit as the sampling period goes to zero, in practice, there will be numerical error in the discrete‐time representation of the system dynamics (like e.g. in (12)), whose effect on the observer design and its propagation during observer implementation could be an important issue in a particular application.

by (6) is driven by the output predictor (15), this results in the following sampled‐data observer:



Fig. 1 depicts the construction of the sampled‐data observer, as the coupling of the continuous‐time observer with the inter‐sample output predictor.

On the other hand, a direct discretization of the continuous‐time design according to (11) seems to be

16



dxˆ  f ( xˆ )  L( xˆ )   h( xˆ ) , t [ti , ti 1 ) dt d  L h( xˆ ) , t [t , t ) (16) i i 1 f dt  (ti )  y(ti )

International Journal of Automation and Power Engineering (IJAPE) Volume 3 Issue 1, January 2014 www.ijape.org

Transactions on Automatic Control, 54 (2009): 2169‐2174.

Kazantzis, Nikolaos, and Costas Kravaris. “Nonlinear observer design using Lyapunovʹs auxiliary theorem.”

Systems & Control Letters, 34 (1998): 241‐247.

Kazantzis, Nikolaos, and Costas Kravaris. “Discrete‐time

FIG. 1 STRUCTURE OF SAMPLED DATA OBSERVER

nonlinear observer design using functional equations.”

It was shown in Karafyllis and Kravaris (2009) that as long as the sampling period does not exceed a certain limit, stability of the error dynamics and robustness with respect to measurement error for a continuous‐ time observer of the form (5) imply stability of the error dynamics and robustness with respect to measurement error for (16). In other words, the proposed sampled‐data implementation inherits the key properties of the continuous‐time observer design.

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Kreisselmeier, Gerhard, and Robert Engel. “Nonlinear observers

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Conclusions

Krener, Arthur J., and Witold Respondek. “Nonlinear observers with linearizable error dynamics.” SIAM

Three alternative design approaches for sampled data nonlinear observers have been proposed in the context of observer error linearization, the first two of which are appropriate under relatively fast sampling, and the third under slow sampling. All three approaches have solid theoretical properties. The selection of the best approach is application‐dependent.

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