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A new methodology for the identification of the values of unknown disturbance signals acting in the input and output measurements of the dynamic linear system is presented. For the solution of this problem, the new idea of the use of two different state observers, which are coworking simultaneously in parallel, was elaborated. Special integral type observers are operating on the same finite time window of the width T and both can reconstruct the exact value of the vector state x(T) on the basis of input-output measurements in this interval [0, T]. If in the input-output signals the disturbances are absent (measurements of I/O signals are perfect) then both observers (although they are different) reconstruct the exact and the same state x(T). However, if in the measurement signals the disturbances are present then these observers will reconstruct the different values of the final states x1(T)=x(T)+e1(T) and x2(T)=x(T)+e2(T). It is because they have different norms and hence they generate different errors in both estimated states. Because of the disturbances, the real state x(T) is unknown, but it is easy to calculate the state difference: x1(T)-x2(T)=e1(T)-e2(T). It occurs that, based on this difference, the values of all the disturbances acting during the control process can be identified. In the paper, the theory of the exact state observation and application of such observers in online mode is recalled. The new methodology for disturbances identification is presented.

In the classic control theory and its applications, a common practice in the estimation of inaccessible for measurement state vector of a linear observable system is the use of Luenberger type asymptotic observers. D.G. Luenberger in [

Nowadays in many real-time control applications and fault detection, the finishing of the observation tasks in assumed and possibly short time T is an important requirement. The asymptotic state estimators may not be sufficient for this purpose. The power of the modern microcomputers makes the design of the other online observation algorithms possible. They reconstruct the value of the current state vector

The general theory of the exact reconstruction of the finite dimension state for linear systems in Hilbert spaces as well as the rules for designing of the exact state observers with minimal norm was formulated and presented by W. Byrski and S. Fuksa in 1984 [_{1}(_{2}(

In engineering sciences, commonly used is the function space L^{2}[0.T]. This space is defined as a space of functions for which the second power is integrable in Lebesgue sense, on the interval [0, T]. The existence of such integral enables definition of the norm of the function as the square root of this integral. In many cases, it can represent the square root of the signal’s energy. The space L^{2}[0.T] belongs to class of Hilbert spaces; hence the inner product of its two elements has the form of the integral operator.

The studies on the exact state observation were undertaken by various authors, although they may be considered as particular cases of the general theory of the exact state observation [

As it was stated before, the exactness in reconstruction of the state is possible by the use of the exact state observers but only under assumption that no input-output noise or disturbances occur (i.e., in the case of perfect input-output measurements). Hence, in practical case of noisy measurements the use of the exact observers gives also a reconstruction error. In the most popular version of the exact observers it was assumed that the input function (control) is perfectly known and for state calculation there is no need for its extra measurement. The last assumption however, in practical application, is not always proper because, e.g., the actuator and the control valve produce the system input signal and it may differ from the control signal u, which is generated by the computer (the number of impulses). From the general theory of the exact and optimal observation one can obtain the observer with minimal norm. It guarantees that, based on the perfect input-output measurements, the state x will be reconstructed exactly and for the measurements with bounded disturbances on y and u (disturbances with bounded norm, from unit balls) the norm of the state reconstruction error will be minimal.

In publications [^{2}[0,T]; hence the inner product is represented by the integral operator. Therefore the name “integral observers” was also frequently used to underline its contrary type to differential structure of the Kalman Filter (LQF) or Luenberger observer [

The extended results of the online exact observation and application were presented in [

The most important properties and features of the integral observers used for the exact reconstruction of the continuous state are

problem was formulated for Multi-Input Multi-Output (MIMO) continuous linear time invariant (LTI) systems given by the standard state matrix equation model,

the state observer has integral description in space L^{2}[0,T] and can be used online (in interval [t-T, t]),

the possible existence of the disturbance in both the control u(t) and output y(t) signals is assumed,

the optimal formulas of the observer are obtained by the minimization of its norm which is the function of assumed weighting coefficients,

the independence of the state observation from the initial conditions of the real state (unknown) occurs,

fixed finite observation time interval [0,T] is used.

A brief explanation regarding the first point may be useful. The theory of the exact state observation in function space L^{2}[0, T] can use the idea of continuous functions and all the mathematical proofs as well as the final results relate to continuous functions. However, of course for the real applications the term continuous measurements of such functions means that for computer calculations it is enough to have standard discrete measurements, although with frequency, according to Nyquist-Shannon sampling theorem. This establishes a sufficient condition for a sample rate that permits a discrete sequence of samples to capture all the information from a continuous-time signal of finite bandwidth. In that case numerical calculations of integrals, e.g., by Simpson’s rule, will guarantee accuracy which will correspond to continuous version.

In this paper the quite new idea of the exact state observers application will be presented. Using two different exact state observers (with different norms) working simultaneously in parallel structure on the same time interval, it is possible to calculate the unknown values of some disturbances, which affect the input-output measurements. The calculation is possible either in the batch mode or in online observation version.

If in the input-output measurements the disturbances are absent (measurements of I/O signals are perfect), then two different observers reconstruct the exact and the same state x(T). However, if in the measurement signals the disturbances are present (y+z_{1}, u+z_{2}) then the first observer reconstructs the value x_{1}(T) and the second x_{2}(T). This is because the observers have different norms and in the case of disturbances they generate various errors e_{1}(T) and e_{2}(T); i.e., they generate the estimates x_{1}(T)=x(T)+e_{1}(T) and x_{2}(T)=x(T)+e_{2}(T). The real undisturbed state x(T) is unknown, but it is easy to find the states difference _{1}(T) - x_{2}(T) = e_{1}(T) - e_{2}(T). It occurs that, based on this difference, the values of all the disturbances acting during the control process can be identified. In the next sections the first observer will be marked as the observer (a) and the second observer as the observer (b).

Such identification of disturbances cannot be performed using classical asymptotic state estimators like Kalman Filter, due to the unknown real value (even theoretical) of the state in both estimators. Such an approach, by the use of classical estimators (e.g., bank of Kalman Filters) for the disturbance isolation, was tested however in [

In Section

Let a linear state observable MIMO system be given

Assume that we perfectly measure the control u and the output y on the interval [0, T], where T is the fixed observation time.

Our purpose is to determine the state x(T). We assume the following.

The state space X = ^{2}[0,T]: _{1}(T,_{2}(T,_{1}(_{2}(

For these assumptions, the general conditions for the observation matrices G_{1}, G_{2} should be determined, in such a manner that formula (_{1} and G_{2} fulfill conditions_{1}(_{1}, the matrix G_{2} should fulfill the second constraint (_{1}, G_{2} matrix pairs which fulfill (

Any exact state observer perfectly reconstructs the state of the system (_{1}^{2}[0,T], z_{2}^{2}[0,T],

Then the state estimate is given by^{2}[0,T] assuming that disturbances are bounded and normalized to unit balls in L^{2}[0,T],

From continuity and linearity in (_{1}, G_{2}) is closed, linear manifold in the space _{1},G_{2} and

For simplification (without loss of generality) we will assume identity weighting coefficients _{2}. The task of the optimization is minimization of the norm

Because of (

where

_{1},

the symbol

From constraint (_{1} rows

Hence from optimality condition

where

apostrophe

Transposition of (_{1} and P_{2} of [n×n] dimension.

Denote by

The vectors

Two cases will be considered:

The weighting coefficients

The weighting coefficients

_{2} from (

Finally we have the matrices P_{1} and G_{2} (marked finally as

Interestingly, the same forms of G_{1}, G_{2} as in (

By the way, it means that the use of the above version of the observers in different applications of the exact state observers [

In authors’ research, it turned out that the norm (

An exemplary shape of the norm, as the function of observation time T, for the second-order system.

Presented integral form of the exact observers given on finite time interval [0,T]

To this end one can design the structure of Moving Window Observer (MWO). Equation (

One can use two possible representations of MWSO at time t:

The matrices G_{1}, G_{2} do not depend on current time t and can be calculated only once and offline in interval [0, T]. Then they may be stored in memory registers in as many samples as needed for accurate calculation of integrals, depending on discretization time of measurements of y and u.

Digital control equipment should have enough computation power for online calculation of thousands of multiplications and summations per second. For the nowadays industrial computers (IPC) this is no problem, let alone for those with digital signal processors (DSP).

During designing of the observer, the main problem is the right choice of the observation time T. The short interval T results in quick start of the online state reconstruction process and requires fewer calculations during the moving window mode but the observer is more highly sensitive to the disturbances (has the bigger norm). The longer time T results in bigger time delay in starting of observation and causes more calculations within the window but the observer is less sensitive to disturbances (has the smaller norm).

Let a linear state observable system be given^{1}, and the output y(t)^{1}, for

Assume that we perfectly measure the control u and the output y (without any disturbances) on the interval [0, T], where T is the fixed observation time.

For the exact state observation, we will use simultaneously working two observers (a), (

Both observers reconstruct the exact state

However, if during control process the disturbances d_{2} of the control and disturbances d_{1} of the output occur and the measurement noises occur (measurement disturbances on input-output z_{2}, z_{1}), then we have a situation like in Figure

The system for reconstruction of disturbances.

Our purpose is to determine the state

The equation of the disturbed system has the form^{1}, the output y(t)^{1}, disturbances d_{1}(t)^{1}, d_{2}(t)^{1}, z_{1}(t)^{1}, z_{2}(t)^{1}, for

The control signal u(t) is known, because it is generated by the deterministic control algorithm in industrial PC or Programmable Logic Controller (PLC).

The algorithm of disturbance detection is presented in Figure

The reconstructed state

_{1}, z_{1}, d_{2}, z_{2} within the interval T are constant.

Such assumption is reasonable if the interval T is small.

Let us mark two vector-matrices [nx1] for each observer, with the second indexes, which mean the numbers (items) of the state vector elements.^{2}. These values

Hence, for the system’s order n=2, D is the square matrix, D[2x2], and for the equation _{2} is given by_{1}+z_{1} at the time T and the exact calculation of the constant values d_{1}, z_{1} is not possible by this method (see Conclusions).

All the above considerations were carried out, for the one observation window [0, T] (batch mode).

If we use the idea of the Moving Window State Observer (_{1}(t) and x_{2}(t), one can also design the Moving Window Disturbances Observer, for the online identification of the disturbances for t>T.

We can have continuous equation for the online reconstruction of the disturbances:_{1}(t) and G_{2}(t) calculated offline must be stored in computer memory within [0, T], e.g., as 100 samples, with Δ=0.1 sec each (T=10 sec). The same samples time must apply to I/O signal measurement. These measurements may be delivered in online mode and final calculation of the observers integrals is performed in real time numerically in the last window [t-T, t] for each t, i.e., in numerical version [iΔ-T, iΔ] for each i (with the use of the best integration procedure, e.g., with Simpson’s rule).

There is no real-time computation problem with online calculation of such Moving Window Observer based on (

All the simulation data used to support the findings of this study are included within the article.

Assume that second-order system is given.

We will derive the exact state x(T) optimal observer formula, for two cases of weighting coefficients

^{−1} from (

The shape of the matrix functions

The shape of the matrix functions

The norm (^{−1} [2x2].

The shapes of the integral functions (differences) in the matrix D are visible in Figure

The shape of the integral functions D_{11}, D_{12}, D_{21}, D_{22.}

Assuming T=2 and using (80) and (84) equations, one can calculate the real matrix D and inv(D) from (_{1}=z_{2}, we have d_{1}=0.75, z_{1}=z_{2}=0.25. These values are the same as those assumed in simulation.

In the paper, the quite new methodology of identification of the unknown constant values of the disturbances acting in dynamical control system was presented. To this end, the theory and application of the exact state integral observers were used. The structure of the Moving Window Disturbance Observer, which consists of two MWO, is defined.

It is possible to identify exactly three values of disturbances d_{1}+z_{1}, d_{2}, z_{2}. However, there is also the possibility of identification of the disturbance d_{1} if we assume that constant values of the noises are the same; z_{1}=z_{2} (e.g., if the identified value of z_{2} represents the mean value of the noise z_{2}(t) and the noise z_{1}(t) in the interval T). Then _{1} and z_{2}, there is also possible estimation of the norm of ^{2}[0,T] (z_{2} was calculated).

Because we know the value of d_{1} + z_{1}, then we can calculate and estimate the norm _{1}, z_{1}, z_{2} we have^{2}[0,T] spaces were recalled. In this problem, there is no need for the discussion about the convergence of the method. There is no differential equation (unlike in the Kalman Filter theory). All algorithms are based on integration operations on finite windows, and even for an unstable model, because, of the finite interval of the window, integrals cannot tend to infinity. With nondisturbed y and u measurements, the state of the unstable object will be still reconstructed exactly. In the case of disturbed measurements, there is an error in the reconstruction of the state, not due to the instability of the object, but due to measurement errors. Of course, one must assume that numerical Simpson procedures of integration are correct.

The numerical example (in Matlab/Simulink) confirms the correctness of this new method for the disturbance identification.

The data used to support the findings of this study are included within the article.

The authors declare that they have no conflicts of interest.

This work was supported by the scientific research funds from the Polish Ministry of Science and Higher Education within the AGH UST Agreements no 11.11.120.396 and 11.11.120.859.