Multicriteria Adaptive Observers for Singular Systems with Unknown Time-Varying Parameters

1Graduate Institute of Ferrous Technology, Pohang University of Science and Technology, 77 Cheongam-Ro, Nam-Gu, Pohang, Gyeongbuk, Republic of Korea 2Department of Electrical Engineering, Pohang University of Science and Technology, 77 Cheongam-Ro, Nam-Gu, Pohang, Gyeongbuk, Republic of Korea 3Department of Creative IT Engineering, Pohang University of Science and Technology, 77 Cheongam-Ro, Nam-Gu, Pohang, Gyeongbuk, Republic of Korea


Introduction
State estimation, or observation, has been recognized as one of the important research issues for dynamic feedback control systems since the full state information required for high performance is not available in most cases due to the high cost of sensors and limited accessibility for measurement.For state estimation, various types of observers have been developed, including Luenberger observers [1], sliding mode observers [2], and robust observers [3].
In the presence of unknown parameters encountered in most real systems, the observers designed for nominal models are hard to be applied in practical applications.For this reason, adaptive observers have been developed to estimate unknown parameters as well as state variables from input and output measurements, and hence achieve the robustness [4][5][6][7][8].Recently, the results on adaptive observers have been successfully extended even to more general singular systems [9,10].Singular systems have extensive applications in many practical systems such as electrical systems, economics, mechanics, and chemical processes.
In implementing such practical adaptive observers over general singular systems, several criteria can be taken into account in consideration of design specifications.For example, adaptive observers can be designed according to the criteria such as  ∞ [11,12],  2 , the ultimate region size, and so on.Mostly, among them, only one criterion has been employed for design of adaptive observers.However, two or more criteria could be applied to involve multiple design objectives, leading to a multicriteria optimization problem.Multicriteria based design enables us to do trade-off analysis for how much we must lose in one objective in order to do better in the other objective.For control design, the so called mixed criteria have already been adopted for practical implementation.As in control design, it would be meaningful to design adaptive observers with multiple useful criteria that can apply even to singular systems.
In this paper, we propose multicriteria adaptive observers for general singular systems with unknown time-varying parameters.For design of multicriteria adaptive observers, two criteria are employed to achieve robustness to disturbances and uncertainties.One is the  ∞ attenuation level which is an upper bound on the  ∞ -norm of the transfer function from disturbances to estimation errors.The other is the upper bound of the ultimate region.These two criteria reflect how much disturbances and unknown parameters have effects on the estimation performance.Specially, the upper bound of the ultimate region makes the magnitudes of steady-state errors guaranteed to be upper bounded, which conflicts the  ∞ criterion and hence provides an optimal trade-off curve and achievable values.
The optimal trade-off curve between the ultimate bound and the  ∞ attenuation level is presented in the form of linear matrix inequalities (LMIs).Furthermore, the integrals of the error states are added for improving robustness to disturbances.If a singular matrix and time-varying parameters of the proposed multicriteria adaptive observers are set to be an identity matrix and constants, respectively, they reduce to existing adaptive observers for linear systems [13][14][15].Simulation examples are presented to show the feasibility and the effectiveness of the proposed observers.
The paper is organized as follows: The description of multicriteria adaptive observers is given for a class of singular systems in Section 2. In Section 3, the design of multiobjective adaptive observers with integral effort is proposed.Finally, the simulation results are illustrated in Section 4 and the conclusion is drawn in Section 5.

Multicriteria Adaptive Observers
Let us consider the following singular system:  ẋ () =  () +    () +    (, , )  () where () ∈ R  is the state, () ∈ R  is the input, () ∈ R  is the unknown time-varying parameter, (, , ) is the nonlinear term depending on the input and the output, () is the disturbance signal, () is the measured output signal, and , ,   ,   , ,  1 , and  2 are the system matrices of appropriate dimensions.For a well-defined singular system, the rank of  is assumed to be  ≤ .The nonlinear term (, , ) is known and upper bounded as with a certain positive constant  max .In addition, it is assumed that uncertain parameters and their derivatives are upper bounded as with positive constants  and .Without loss of generality, the following conditions are also assumed to hold where C is the set of complex numbers.Assumption (4) implies that the singular system (1) is observable.According to assumption (4), there exist nonsingular matrices  and  such that  +  =   , where   ∈ R × denotes an identity matrix.The general solution for  and  is given as where  1 is an arbitrary matrix of appropriate dimension and the superscript † denotes pseudoinverse.To estimate both the state variables and the unknown parameters, the following functional observer can be constructed: where () is the auxiliary variable of the observer, θ is the estimated parameter value, and , , and  are constant matrices to be determined later on for guaranteeing observation.It follows then that we have the following error dynamics: where () = () − x().If , , and  in (7) are chosen to satisfy the following conditions: the error dynamics (7) becomes where   () = () − θ(),  = ( −   )  , and the arguments of (, , ) are omitted for simplicity.Substituting (8) into (9) yields ė () = ( +   )  () +  () +     () where   = − and  3 =    2 + 1 .For the estimation of the unknown time-varying parameter (), the following parameter update equation is constructed: where ,  are matrices to be designed,   is positive constant, Γ is a diagonal weight matrix for adaptation, and   is a leakage variable.The leakage term   is defined as [16] where  th ≥  is a predefined threshold and  1 is a positive constant.In the estimation of unknown time-varying parameters, the function of last term in ( 12) is to force the estimated variable θ to inside of the set ‖ θ‖ <  th .Therefore, the term is effective when the estimated θ exists outside of the set.If the estimated parameter value θ exists outside of the set,    1 determines how fast it converges.Though there is a possibility of small oscillations on the switching surface ‖ θ‖ =  th , the leakage term ensures an bounded parameter estimation error.Also, these oscillations do not occur under nominal conditions.Furthermore, the parameter estimation is assumed to be independent of disturbances.Then,  2 = 0 holds and the general solution is given as , where  2 is an arbitrary matrix.To derive an observer gain considering the effect of disturbance, the  ∞ performance from the disturbance to the estimation error is defined as sup where sup denotes supremum and  is a matrix with appropriate dimension.Now, we shall try to construct sufficient conditions for multiobjective observer based on quadratic Lyapunov functions.
Theorem 1.For given positive scalars ,   ,  1 ,  2 ,  3 , and , if there exist matrices and  1 and scalars  1 ,  2 , and  3 satisfying the following LMIs and the equality condition where and * denotes the entry of a symmetric matrix, then, the state estimation error and the parameter estimation error are uniformly ultimately bounded for an ultimate ellipsoidal set  max with  ∞ attenuation level .Moreover, the observer gain is chosen to be   =  −1  1 .
Proof.Choose the following Lyapunov function of a quadratic form: Differentiating the Lyapunov function (21) along the state trajectory yields which comes from the following well-known inequality: Putting together inequalities in (23) and ignoring the effect of disturbances (i.e.,  = 0), we have where  is defined by The right hand side of inequality (25), except for , can be converted into an LMI and upper bounded as follows: where the Schur complement is used, and  is a design parameter to be chosen to be a small positive constant.If Ξ < − ⋅  is satisfied, then, inequality (25) can be expressed as It implies that V((),   ()) < 0 for  ⋅ (‖()‖ + ‖  ()‖) > .
When the estimation error exists outside of the bound, it approaches the inside of the bound and then it stays there according to the Lyapunov stability theory.Therefore, (),   () converge to the inside of a set parameterized by ; that is, {(,   ) | ‖‖ + ‖  ‖ < /}.It means the error dynamics is uniformly ultimately bounded with the ultimate bound /.Now, the existence of disturbances is taken into account (the case of  ̸ = 0).For the  ∞ performance sup ‖‖ 2 ̸ =0 (‖‖ 2 /‖‖ 2 ) ≤ , the following inequality is considered with the derivative of Lyapunov function in (22).

Multicriteria Adaptive Observers with Integral Effort
In this section, the multiobjective adaptive observer involving integral action is presented to improve steady-state accuracy and attain the robustness to exogenous disturbances, which is of the following form: where  is the integral of the estimation error.The proposed multicriteria adaptive observer (32) with integral effort yields the following error dynamics: The following theorem tells us that the multicriteria adaptive observer (32) is guaranteed to achieve the  ∞ attenuation level and the upper bound of the ultimate invariant set if some LMI conditions are met. max ,  is used for the design of multicriteria with integral effort in order to distinguish them from  max ,  for the one without integral term.
Theorem 2. For given scalars 0 ≤  ≤ 1,  1 > 0,  2 > 0,  3 > 0,   > 0, and  > 0, if there exist matrices  =   > 0, where then, the error dynamic ( 33) is ultimately bounded with an upper bound  max and satisfies the  ∞ performance with the  attenuation level.The observer gain is computed as Proof.Choose a Lyapunov candidate function as follows: Differentiating the Lyapunov function along the state trajectory with the condition     =  results in where equality     =  is used.For now, the case of  = 0 is considered.
Using (24) and Schur complement, where Then, the right hand side of inequality (42), except for , can be converted into (37) using the Schur complement and upper bounded.If Υ < − ⋅  is satisfied, then, the inequality can be expressed as Therefore, (), (), and   () are uniformly ultimately bounded with the ultimate bound .The rest part is similar to that of Theorem 1, so it is omitted for brevity.This completes the proof.
Remark 3. Since singular systems has a complicated structure, they provide more challenging issues.The proposed adaptive observer is more generalized than existing ones [7,8,13] that can be only applied to linear systems.Choosing  = ,  = 0, it can be applied to standard linear systems.Further, the proposed one deals with time-varying parameters.
Remark 4. The proposed observer offers flexibility between two criteria, an ultimate bound and  ∞ performance, using a multiobjective approach.Until now, such design approaches for adaptive observers in singular systems with unknown time-varying parameters have not been studied at all.

Numerical Simulation
In this section, two examples for numerical simulations are considered to verify the effectiveness of the proposed multicriteria adaptive observers.

Example 1:
A Second-Order Singular System.At the first example, the following second-order singular system is considered: The time-varying parameter is chosen to be  = 0.3 sin() and  is taken to be the sinusoidal function sin(5).The parameters are chosen as   = 10,  1 =  2 =  3 = 1, and  = 0.01.Applying Theorem 2, the optimal gains of the multiobjective proportional-integral adaptive observer with  = 0.1 are computed to be (47) In the presence of external disturbance () = 0.2 sin(10), the observer state tracks along a real state.By solving the multiobjective optimization problem, the optimal  ∞ performance index is given as sup ‖‖ 2 ̸ =0 (‖ẽ‖ 2 /‖‖ 2 ) ≤ 0.01839 and the optimal upper bound  max = 0.1855 is provided.Then, the system response curves of the system with the initial values (0) = [−2, 2] are shown in Figure 1, which include the trajectories of state and estimated states.The oscillations in the estimation of states are caused by the external disturbances due to 0.2 sin(10) and nonlinearity (, , ).The parameter estimation curve is illustrated in Figure 2. has been widely considered to predict the proper level of production of several types of goods.The state  represents the production of each industry, the matrix  corresponds to the rate of production,  is the stock placement of commodities, the input   () presents the known supply rate,  is the external supply, the disturbance () represents the uncertain industrial supply, and  corresponds to the production of commodities available for evaluation.For simulations, the system matrices are considered as follows: The optimal value of multiobjective function is computed to be 0.8346. 2 = 0.4773 and  max = 1.3706 are computed. =  is chosen for the  ∞ performance index.In the presence of the disturbances, the estimation errors converge to zero.The convergence of error dynamics is presented in Figure 3. Figure 4 shows the trajectory of the parameter estimation errors.To show the trade-off between the ultimate bound  max and the  ∞ attenuation level , the optimal solutions are solved for various  values.Plotting these optimal solutions, we obtain the Pareto optimal points as described in Figure 5.As shown in Figure 5,  max seems to be inversely proportional to  2 .Figure 6 displays the comparison of the transient trajectories for different adaptive gains.To follow real parameter  as fast as possible, a high adaptive gain is needed.However, if the adaptive gain is too large, it causes oscillations in the transient period but it has a good tracking performance for parameters as the case of   = 20.Conversely, if the adaptive gain is too small, there is comparably small oscillations in the transient period but the observer provides a poor tracking performance for parameters as the case of   = 5.Therefore, the adaptive gain should be appropriately chosen.

Conclusion
Multicriteria adaptive observers were designed according to two criteria for the  ∞ attenuation level of disturbances and the upper bound of the ultimate invariant set.The corresponding cost functions are scalarized into a single one and then the Pareto optimal solutions are obtained with Lyapunov stability in order to provide a good compromising solution.It was shown through numerical simulations that the proposed multicriteria adaptive observers have the good tracking ability.For adaptive observers for general singular systems, other criteria can be easily taken into consideration by extending the proposed design scheme.It is believed that the proposed observers could be applied to fault detection, unknown input estimation, disturbance estimation, and so on.

Figure 6 :
Figure 6: The comparison of parameter estimation with different adaptive gains.