Optimal design of Luenberger reduced-order observer with low sensitivity for linear multivariable systems

Abstract

For the linear multivariable systems, by combining both merits of orthogonal-function approach and evolutionary optimization, in this paper, a new method is presented for designing a Luenberger reduced-order observer to solve the low-sensitivity design issue for physical system parameter deviation and simultaneously to minimize a measurement of the quadratic performance for reducing state transient estimation error. Two given examples illustrate the effectiveness of the presented new low-sensitivity design approach on state estimation performance. From the given examples, it shows that the estimated state errors are not sensitive to system parameter deviation and have the asymptotical convergence property. Besides, the performances are apparently superior to those without considering low-sensitivity design means.

Introduction

State feedback control system is based on the available state variables. For dynamic control systems in which state variables cannot be obtained by direct measurement, the system states must be estimated. Therefore, designing state observers is one of the most important problems for state-feedback control systems. Many researchers have dedicated to the problem for designing state observer.^1–17 In practice, some states are known. Estimating available states is meaningless. In this case, a reduced-order observer can be used to estimate the unknown states. The large transient estimator error (at the beginning, the estimated trajectory deviates from the desired one) leads to a peaking phenomenon which may be impractical or unsafe to use. ¹⁰ Under the best knowledge of the author, for linear multivariable control systems, the performance improvement problem for transient estimation of Luenberger “reduced-order observer” was studied only by a small number of researchers.^1,3,4,14 That is, the original research works, which are about the performance improvement problem for transient estimation of Luenberger reduced-order observer are presented by Kung and Yeh, ¹ Horng and Chou,^3,4 as well as Chou and Cheng. ¹⁴ But, the approaches of Kung and Yeh ¹ as well as Horng and Chou^3,4 cannot be applied to designing the optimal Luenberger reduced-order observer when the number of unmeasured states is smaller than the number of outputs. The optimal design means proposed by Chou and Cheng ¹⁴ can not only exhibit better estimation accuracy than the methods of Kung and Yeh as well as Horng and Chou, but also can overcome the shortcoming of the means presented by Kung and Yeh ¹ as well as Horng and Chou.^3,4 The method of Chou and Cheng ¹⁴ is used to design a Luenberger reduced-order observer such that (i) the measurement of quadratic performance for state estimation error is minimized to suppressing the error of transient estimation, and (ii) the eigenvalues of the reduced-order observer have the asymptotical convergence property.

The parameters of control system are subject to variations due to several factors. ¹⁸ To accomplish successful system study, it is essential to consider the deviation of a system from its nominal behavior caused by the deviation of its parameters from their normal performance characteristics. In practice, the observer is designed to have optimal estimation for one set of system parameter values, but the designed observer no longer remains having optimal estimation for a different of values when the system parameters may deviate from their nominal values. Hence it is necessary to consider low sensitivity issue for designing the observer. The approach of Chou and Cheng ¹⁴ can be used to design a Luenberger reduced-order observer such to have asymptotical convergence property and effectively suppress the transient estimation error. But, Chou and Cheng ¹⁴ do not consider the low-sensitivity issue in designing the reduced-order observer. In addition, under the best of knowledge of the author, there is no existing literature to study the problem of designing reduced-order observers simultaneously considering both transient estimation performance improvements and low sensitivity. So, in this paper, the purpose is to investigate the design problem for Luenberger reduced-order observer with considering three design issues simultaneously as following: (i) guaranteeing eigenvalues to have asymptotically convergent performance, (ii) reducing the sensitivity of state estimation error trajectories to system parameter changes, and (ii) minimizing the quadratic performance measure for reducing transient estimation error.

This paper is organized as follows. The problem statement is described in Section 2. Section 3 proposes the design approach of optimal linear Luenberger reduced-order observer. Two demonstrative examples are presented in Section 4. The final section, Section 5, presents the conclusions.

Problem statement

A linear time-invariant system is considered as:

$$\stackrel{·}{x}\left(t\right)=\tilde{A}(\epsilon )x\left(t\right)+\tilde{B}(\epsilon )u\left(t\right),$$

(1)

$$y=\tilde{C}x(t),$$

(2)

in which the system $(\tilde{A}(\epsilon ),\tilde{C})$ is observable, $x(t)$ is the $n\times 1$ state vector, $y(t)$ is the $m\times 1$ output vector, $u(t)$ is the $r\times 1$ control vector, C is an $m\times n$ constant matrix of full rank m, $\tilde{A}(\epsilon )$ and $\tilde{B}(\epsilon )$ are, respectively, $n\times n$ and $n\times r$ constant matrices which are functions of scalar parameter $\epsilon ,$ where the nominal value of $\epsilon$ is ${\epsilon }_o.$ In equation (1), it is assumed that $\tilde{A}(\epsilon )$ and $\tilde{B}(\epsilon )$ are differentiable in $\epsilon .$ By performing appropriate coordinate transformation,^2,19 equations (1) and (2) of the system can be reformulated as:

$$\stackrel{·}{x}\left(t\right)=A(\epsilon )x\left(t\right)+B(\epsilon )u\left(t\right),$$

(3)

$$y\left(t\right)=\left[\begin{array}{cc}{I}_m\hfill & 0\hfill \end{array}\right]x(t),$$

(4)

where $I_m$ denotes the identity matrix of $m\times m$ . The state vector $x(t)$ in equation (3) is divided into:

$$x\left(t\right)=\left[\begin{array}{c}y(t)\hfill \\ \multicolumn{1}{c}{w(t)}\end{array}\right],$$

(5)

in which $w(t)$ is the unmeasurable $(n-m)\times 1$ vector of the state vector $x\left(t\right)$ . Therefore, according to the division of $x\left(t\right)$ , in equation (3), both matrices $A(\epsilon )$ and $B(\epsilon )$ are further divided into:

$$A(\epsilon )=\left[\begin{array}{cc}{A}_{11}\left(\epsilon \right)\hfill & A_{12}\left(\epsilon \right)\hfill \\ \multicolumn{1}{c}{A_{21}\left(\epsilon \right)}& A_{22}\left(\epsilon \right)\hfill \end{array}\right]\mathrm{and}B(\epsilon )=\left[\begin{array}{c}{B}_1(\epsilon )\hfill \\ \multicolumn{1}{c}{B_2(\epsilon )}\end{array}\right].$$

Therefore, in order to estimate $w(t)$ , a linear Luenberger $(n-m)$ -dimensional reduced-order observer is given as following^2,20:

$$\stackrel{·}{z}\left(t\right)=\tilde{F}(\epsilon )z\left(t\right)+\tilde{G}(\epsilon )y\left(t\right)+\tilde{H}(\epsilon )u(t),$$

(6)

$$\hat{w}(t)=z(t)-\mathrm{Ly}(t),$$

(7)

in which $\hat{w}(t)$ denotes the estimated vector of $w(t)$ ,

$$\tilde{F}(\epsilon )=A_{22}(\epsilon )+L{A}_{12}(\epsilon ),$$

(8)

$$\tilde{G}(\epsilon )=-\tilde{F}(\epsilon )L+L{A}_{11}(\epsilon )+A_{21}(\epsilon ),$$

(9)

$$\tilde{H}(\epsilon )=B_2(\epsilon )+L{B}_1(\epsilon ),$$

(10)

where L denotes the $(n-m)\times m$ -dimensional gain matrix of observer to being designed.

The estimated error vector is given to be

$$e\left(t\right)=w\left(t\right)-\hat{w}(t)$$

(11)

Thus, it is obtained that^14,20

$$\stackrel{·}{e}\left(t\right)=\tilde{F}\left(\epsilon \right)e\left(t\right)=(A_{22}\left(\epsilon \right)+L{A}_{12}\left(\epsilon \right))e\left(t\right).$$

(12)

By setting $\hat{w}(0)$ to zero, $e\left(0\right)=w(0)$ . Thus, from the equation (7), $z\left(0\right)=\mathrm{Ly}\left(0\right)$ can be obtained.

Taking the partial derivative equation (12) with respect to the parameter $\epsilon$ , the dynamic equation of the trajectory sensitivity of the estimated error vector can be obtained as follows:

$$\frac{d}{\mathrm{dt}}\left(\frac{\partial e\left(t\right)}{\partial \epsilon }\right)=\frac{\partial \tilde{F}\left(\epsilon \right)}{\partial \epsilon }e(t)+\tilde{F}({\epsilon }_o)\frac{\partial e(t)}{\partial \epsilon },$$

(13)

where $\tilde{F}\left({\epsilon }_o\right)=A_{22}\left({\epsilon }_o\right)+L{A}_{12}\left({\epsilon }_o\right).$ Let $e_{\epsilon }\left(t\right)=\partial e\left(t\right)/\partial \epsilon$ and ${\tilde{F}}_{\epsilon }=\partial \tilde{F}\left(\epsilon \right)/\partial \epsilon ,$ then equation (13) can be represented as

$${\stackrel{·}{e}}_{\epsilon }(t)={\tilde{F}}_{\epsilon }e(t)+\tilde{F}({\epsilon }_o)e_{\epsilon }\left(t\right),$$

(14)

in which $e_{\epsilon }\left(0\right)=0$

Therefore, by augmenting the original estimated error system equation (12) with its sensitivity model (14), the overall dynamic equation of the estimated error system considering sensitivity can be obtained as follows:

$$\begin{array}{c}\stackrel{·}{\overline{e}}\left(t\right)=\left[\begin{array}{c}\stackrel{·}{e}\left(t\right)\hfill \\ \multicolumn{1}{c}{{\stackrel{·}{e}}_{\epsilon }\left(t\right)}\end{array}\right]=\left[\begin{array}{cc}\tilde{F}\left({\epsilon }_o\right)\hfill & 0\hfill \\ \multicolumn{1}{c}{{\tilde{F}}_{\epsilon }}& \tilde{F}\left({\epsilon }_o\right)\hfill \end{array}\right]\left[\begin{array}{c}e\left(t\right)\hfill \\ \multicolumn{1}{c}{e_{\epsilon }\left(t\right)}\end{array}\right]\hfill \\ =\left[\begin{array}{cc}\tilde{F}\left({\epsilon }_o\right)\hfill & 0\hfill \\ \multicolumn{1}{c}{{\tilde{F}}_{\epsilon }}& \tilde{F}\left({\epsilon }_o\right)\hfill \end{array}\right]\overline{e}\left(t\right)=\overline{F}\overline{e}(t),\hfill \end{array}$$

(15)

where $\overline{e}\left(0\right)=\left[\begin{array}{c}w\left(0\right)\hfill \\ \multicolumn{1}{c}{0}\end{array}\right].$

A measurement of quadratic performance for estimation error of equation (15) is given as:

$$J=\underset{0}{\overset{\beta }{\int }}{\overline{e}}^T\left(t\right)Q\overline{e}(t)\mathrm{dt},$$

(16)

where ${\overline{e}}^T\left(t\right)$ denotes the transpose of vector $\overline{e}(t)$ , the matrix Q is a specified positive definite weighting matrix, the performance criterion J is responsible to reducing the sensitivity of state estimation error trajectories to system parameter changes and suppressing the transient estimation errors, and $\beta$ is the prescribed terminal time for reducing $\overline{e}(t)$ to nearly zero.

The design problem for a low-sensitivity linear Luenberger reduced-order observer in this paper is to direct finding the matrix L of observer gain so that the eigenvalues for the matrix $\tilde{F}\left({\epsilon }_o\right)=A_{22}\left({\epsilon }_o\right)+L{A}_{12}\left({\epsilon }_o\right)$ have asymptotically convergent performance and simultaneously the performance criterion J in equation (16) is minimized.

Optimal design of low-sensitivity Luenberger reduced-order observer

The orthogonal-function method (OFM) has been favorably used to investigate various systems and control problems.^14,21–23 The main feature of OFM is the conversion of integral and differential equations to algebraic equations. Thus, OFM has become very computationally prevalent because the dynamical system equations can be changed into a set of algebraic equations, which facilitates to solve the original system and control problem.^14,21–23 Therefore, the OFM is applied in this paper. All elements of $\overline{e}\left(t\right)$ are assumed to be entirely integrable over the $[0,\beta ]$ of time interval. Thus, for the $[0,\beta ]$ of time interval, the $\overline{e}\left(t\right)$ of error vector in equation (15) can be approximated by the truncated orthogonal functions as:

$$\overline{e}\left(t\right)=\sum _{i=0}^{q-1}{\overline{e}}_i{\overline{T}}_i\left(t\right)=\overline{E}\overline{T}(t),$$

(17)

in which ${\overline{T}}_i\left(t\right)(i=0,1,\cdots ,q-1)$ denote the orthogonal functions, and $\overline{T}(t)={[{\overline{T}}_0\left(t\right),{\overline{T}}_1\left(t\right),\cdots ,{\overline{T}}_{q-1}(t)]}^T$ are the $q\times 1$ orthogonal-function basis vector; q is the used orthogonal-function number to approximate the error vector $\overline{e}\left(t\right);$ ${\overline{T}}_i\left(t\right)(i=0,1,\cdots ,q-1)$ denote the orthogonal functions; ${\overline{e}}_i(i=0,1,\cdots ,q-1)$ represents the $2(n-m)\times 1$ -dimensional vector of coefficient, and $\overline{E}=[{\overline{e}}_0,{\overline{e}}_1,\cdots ,{\overline{e}}_{q-1}]$ expresses the $2(n-m)\times q$ -dimensional matrix of coefficient.

In equation (17), any orthogonal set of functions may be used. Only a truncation is considered in equation (17) by using q term number to approximate the error vector $\overline{e}\left(t\right)$ . That is, equation (17) does not represent $\overline{e}\left(t\right)$ being equal to $\overline{E}\overline{T}\left(t\right).$ According to the experiences of previous researches including the author’s works,^14,23–25 both Chebyshev and Legendre series are good choice of orthogonal functions for approximation with using only small number of orthogonal functions.

Adopting the orthogonal-function expression for $\overline{e}\left(t\right)$ in equation (17), using the integral property for the orthogonal functions as follows^23–25:

$$\underset{0}{\overset{t}{\int }}\overline{T}(\tau )d\tau =P\overline{T}(t),$$

(18)

and applying the Kronecker product, the $\tilde{E}^$ of solution for equation (15) can be gotten as:

$${\tilde{E}}^{=}{[I_{2q(n-m)}-P^T\otimes \overline{F}]}^{-1}{\tilde{E}}_0,$$

(19)

where $I_{2q(n-m)}$ denotes the identity matrix of $2\mathrm{q}(n-m)\times 2q(n-m),$ ⊗ represents the Kronecker product, ²⁶ ${\tilde{E}}^{=}{[{\overline{e}}_0^T,{\overline{e}}_1^T,\cdots ,{\overline{e}}_{q-1}^T]}^T,$ ${\tilde{E}}_0={\left[{\overline{e}}^T\left(0\right),0^T,\dots ,0^T\right]}^T,$ P expresses the orthogonal-function integrational operational matrix, and the elements of matrix P rely on the special choice of the orthogonal-function basis vector $\overline{T}\left(t\right).$ ^14,23–25 The detailed derivation of equation (18) has been given in the some research works, so, more details can refer to the papers.^24,25 Referring to the derivation process in the previous work ¹⁴ by the author of this paper, equation (19) can be obtain by using a similar derivation method.

Replacing the truncated orthogonal-function expression for the $\overline{e}(t)$ of error vector given by equation (17) into equation (16) yields

$$\begin{array}{c}J=\underset{0}{\overset{\beta }{\int }}{\overline{T}}^T\left(t\right){\overline{E}}^{T}Q\overline{E}\overline{T}\left(t\right)\mathrm{dt}=\underset{0}{\overset{\beta }{\int }}\mathrm{trace}\left(\overline{T}\left(t\right){\overline{T}}^T\left(t\right)R\right)\mathrm{dt}\hfill \\ =\mathrm{trace}\left(\underset{0}{\overset{\beta }{\int }}\overline{T}\left(t\right){\overline{T}}^T\left(t\right)\mathrm{dtR}\right)=\mathrm{trace}\left(\mathrm{WR}\right),\hfill \end{array}$$

(20)

in which $R={\overline{E}}^{T}Q\overline{E}$ and the product-integration-matrix of two orthogonal-function basis vectors is denoted by the constant matrix $W.$ ^23,27

In order to guarantee the eigenvalues of the matrix $\tilde{F}\left({\epsilon }_o\right)=A_{22}\left({\epsilon }_o\right)+L{A}_{12}\left({\epsilon }_o\right)$ having asymptotically convergent performance, the eigenvalues for the matrix $\tilde{F}\left({\epsilon }_o\right)$ must satisfy the following condition:

$$\mathrm{Re}(\lambda (\tilde{F}\left({\epsilon }_o\right)))\le \mu \left(\tilde{F}\left({\epsilon }_o\right)\right)<0,$$

(21)

where $\mathrm{Re}(\lambda (\tilde{F}\left({\epsilon }_o\right)))$ denotes the real part of eigenvalues of $\tilde{F}\left({\epsilon }_o\right),$ and $\mu \left(\tilde{F}\left({\epsilon }_o\right)\right)$ denotes the matrix measure of $\tilde{F}\left({\epsilon }_o\right).$ ²⁸ As previously mentioned, the design problem for a low-sensitivity linear reduced-order Luenberger observer is to direct finding the matrix L of observer gain so that equation (21) is satisfied and simultaneously the performance criterion J in equation (20) is minimized. Because the hybrid Taguchi-genetic algorithm (HTGA) gives more robust results and less computational time than the existing methods^24,29–31 for handling the optimization problems of parameters, the HTGA method is thus used here to designing an optimal low-sensitivity linear Luenberger reduced-order observer. The works presented by Ho and Chou, ²⁴ Tsai et al., ²⁹ Ho et al., ³⁰ as well as Ho et al. ³¹ give the details about the HTGA. In addition, in the HTGA approach, the bound of parameters of the observer gain matrix L can be specified to avoid the case of a high gain value.

Optimal design procedures of low-sensitivity Luenberger reduced-order Observer

• Step 1: Parameter setting. Input: number of generations, population size G, crossover rate $p_c$ , and mutation rate $p_m$ . Output: the J of value in equation (20), and the $(n-m)\times m$ -dimensional gain matrix L of observer.

• Step 2: Initialization. The initial chromosome population of form $\theta =[l_1,l_2,\cdots ,l_{m(n-m)}]$ is first randomly generated for executing the HTGA, where $l_i$ $(i=1,2,\cdots ,m(n-m))$ represent the entries of matrix $L.$ Next, applying equation (19) to compute the solutions of $\tilde{E}$ , and calculateJ by adopting equation (20) which is defined to be the fitness function for HTGA. Finally, the fitness values of initial population feasible for the equation (21) of matrix-measure-based constraint are computed.

• Step 3: Executing the HTGA method. ²⁹ With integrating both equations (19) and (20), the HTGA method is applying to searching for the optimal matrix L of observer gain, in which a penalty for the fitness value is rendered for the chromosome that violates the matrix-measure-based constraint of equation (21), which guarantees the matrix $\tilde{F}\left({\epsilon }_o\right)$ having asymptotically convergent performance.

• Step 4: If the stopping criterion has come upon, the optimal matrix L of observer gain and the optimal performance criterion value of J are obtained. Otherwise, return to Step 3.

Demonstrative examples

In this section, two example are presented to demonstrate the efficiency of the proposed optimization method for the design problem of Luenberger reduced-order observer with considering three design issues simultaneously as following: (i) guaranteeing eigenvalues to have the performance of asymptotical convergence, (ii) reducing the sensitivity of state estimation error trajectories to system parameter changes, and (ii) minimizing the quadratic performance measure for reducing transient estimation error. The used evolutionary environments of HTGA are as following: the maximum generation is 50, the size of population is 50, the rate of crossover is 0.75, and the rate of mutation is 0.02. The shifted Chebyshev series^23,25 is the class of orthogonal function used in this section.

• Example 1

A double-inverted system of sixth-order with four outputs is considered here and described as following ³² :

$$\begin{array}{c}A=\left[\begin{array}{cccccc}0\hfill & 0\hfill & 0\hfill & 0\hfill & 1\hfill & 0\hfill \\ \multicolumn{1}{c}{0}& 0\hfill & 0\hfill & 0\hfill & 0\hfill & \frac{1}{\epsilon }\hfill \\ \multicolumn{1}{c}{0}& 0\hfill & 0\hfill & 1\hfill & 0\hfill & 0\hfill \\ \multicolumn{1}{c}{0}& 0\hfill & 0\hfill & 0\hfill & 0\hfill & 0\hfill \\ \multicolumn{1}{c}{16}& -8\hfill & 0\hfill & 0\hfill & 0\hfill & 0\hfill \\ \multicolumn{1}{c}{-16}& 16\hfill & 0\hfill & 0\hfill & 0\hfill & 0\hfill \end{array}\right],B=\left[\begin{array}{c}0\hfill \\ \multicolumn{1}{c}{0}\\ \multicolumn{1}{c}{0}\\ \multicolumn{1}{c}{1}\\ \multicolumn{1}{c}{-1}\\ \multicolumn{1}{c}{0}\end{array}\right],\mathrm{and}\hfill \\ C=\left[\begin{array}{cccccc}1\hfill & 0\hfill & 0\hfill & 0\hfill & 0\hfill & 0\hfill \\ \multicolumn{1}{c}{0}& 1\hfill & 0\hfill & 0\hfill & 0\hfill & 0\hfill \\ \multicolumn{1}{c}{0}& 0\hfill & 1\hfill & 0\hfill & 0\hfill & 0\hfill \\ \multicolumn{1}{c}{0}& 0\hfill & 0\hfill & 1\hfill & 0\hfill & 0\hfill \end{array}\right].\hfill \end{array}$$

(22)

In this example, the output number is four. The nominal parameter value of $\epsilon$ is ${\epsilon }_0=1.$ Two states $x_5\left(t\right)$ and $x_6\left(t\right)$ are unmeasurable states, and the first four states $x_1\left(t\right),$ $x_2\left(t\right),$ $x_3\left(t\right)$ and $x_4\left(t\right)$ can be directly gotten from the output $y\left(t\right)$ . In the double-inverted system given by Golnaraghi and Kuo, ³² the states $x_1\left(t\right),$ $x_2\left(t\right),$ $x_3\left(t\right)$ and $x_4\left(t\right)$ , which can be directly obtained, are, respectively, the angles ${\theta }_1\left(t\right)$ and ${\theta }_2\left(t\right),$ the displacement and the velocity. In this example, $e\left(0\right)={[0.5,-0.5]}^T,\overline{e}\left(0\right)={[0.5,-0.5,0,0]}^T,$ $Q=I_4,$ and $\beta =2.$ By applying the proposed optimal design method for low-sensitivity reduced-order observer, it can obtain the optimal matrix L of low-sensitivity observer gain as:

$$L=\left[\begin{array}{cccc}-14.7105\hfill & 16.9136\hfill & -24.2165\hfill & 1.0307\hfill \\ \multicolumn{1}{c}{13.9722}& -17.6852\hfill & 28.6326\hfill & -35.8656\hfill \end{array}\right].$$

(23)

Here, for comparison, an optimal reduced-order observer under non-considering the design condition of low-sensitivity to parameter change is designed. The optimal matrix L of non-low-sensitivity reduced-order observer gain can be acquired as

$$L=\left[\begin{array}{cccc}-62.0676\hfill & -25.9752\hfill & 19.2772\hfill & 18.2523\hfill \\ \multicolumn{1}{c}{62.6058}& 13.8064\hfill & 100.981\hfill & -16.1219\hfill \end{array}\right].$$

(24)

The comparisons of state estimation errors for the to-be-estimated states $x_5\left(t\right)$ and $x_6\left(t\right)$ of optimal low-sensitivity estimation and optimal non-low-sensitivity estimation are given in Figures 1 and 2 with $q=8,$ respectively, where the parameter value of $\epsilon$ deviates from $1$ into $0.9.$ By adopting the proposed optimal low-sensitivity design method, it can be seen that clearly the errors of transient estimation have been substantially decreased and the estimated errors of state can converge to zero. It can also be seen that, when the parameter value of $\epsilon$ has deviation, the estimates of the optimal low-sensitivity observer are indeed less sensitive to parameter deviation and more accurate than those of the optimal non-low-sensitivity observer.

Figure 1.

A comparison of state estimation errors for the state $x_5(t)$ of optimal low-sensitivity estimation () and optimal non-low-sensitivity estimation (), respectively.

Figure 2.

A comparison of state estimation errors for the state $x_6(t)$ of optimal low-sensitivity estimation () and optimal non-low-sensitivity estimation (), respectively.

• Example 2

Consider a DC motor which converts an electrical input voltage signal into an output torque here and described as following ³³ :

$$A=\left[\begin{array}{cc}0\hfill & 1\hfill \\ \multicolumn{1}{c}{0}& -1/\epsilon \hfill \end{array}\right],B=\left[\begin{array}{c}0\hfill \\ \multicolumn{1}{c}{1/\epsilon }\end{array}\right],\mathrm{and}C=\left[\begin{array}{cc}1\hfill & 0\hfill \end{array}\right].$$

(25)

In this example, the nominal parameter value of $\epsilon$ is ${\epsilon }_0=1.$ One state $x_2\left(t\right)$ is unmeasurable state, and the first state $x_1\left(t\right)$ can be directly obtained from the output $y\left(t\right)$ . In this example, $e\left(0\right)=0.2,\overline{e}\left(0\right)={\left[0.2,0\right]}^T,$ $Q=I_2,$ and $\beta =1.$ By applying the proposed optimal design method for low-sensitivity reduced-order observer, it can obtain the optimal matrix L of low-sensitivity observer gain as: $L=-13.74$ . Here, for comparison, an optimal reduced-order observer under non-considering the design condition of low-sensitivity to parameter change is designed. The optimal matrix L of non-low-sensitivity reduced-order observer gain can be acquired as $L=-56.9$ . The comparison of state estimation error for the to-be-estimated state $x_2\left(t\right)$ of optimal low-sensitivity estimation and optimal non-low-sensitivity estimation is given in Figure 3 with $q=8,$ where the parameter value of $\epsilon$ deviates from $1$ into $0.9.$ By adopting the proposed optimal low-sensitivity design method, it can be seen that clearly the estimated error of state can converge to zero. When the parameter value of $\epsilon$ deviates from $1$ into $0.9,$ the estimation error generated from the optimal observer without considering low sensitivity design leads to oscillation and failure to converge to zero. That is, it can also be seen that, when the parameter value of $\epsilon$ has deviation, the estimate of the optimal low-sensitivity observer is indeed less sensitive to parameter deviation and more accurate than that of the optimal non-low-sensitivity observer.

Figure 3.

A comparison of state estimation errors for the state $x_2(t)$ of optimal low-sensitivity estimation () and optimal non-low-sensitivity estimation (), respectively.

Conclusions

By using both orthogonal function method and evolutionary optimization, in this paper, a new method has been presented for designing the low-sensitivity Luenberger reduced-order observer to implement three design requirements: guaranteeing the estimation errors to have asymptotically convergent performance, suppressing the estimation error of transient response, and reducing the sensitivity of state estimation error trajectories to system parameter changes. The results of two demonstrative examples show that the proposed new approach indeed facilitates to achieve the mentioned-above three design requirements. The constraint of the matrix-measure-based condition in equation (21) makes the error vector of state estimation asymptotically converging to zero with reducing the steady-state errors. In addition, the presented design method of observer with the measurement for quadratic performance of equation (16) supplies a penalty for both transient error and sensitivity to system parameter changes. This paper has shown that the existence of a physically achievable, optimally designed matrix L of observer gain can improve the transient error performance without being sensitive to change in system parameters. That is, this paper presents a pioneering work in handling the issue of designing reduced-order observer under considering both low-sensitivity to system parameter changes and transient estimation performance improvement. The future research direction is to combine the proposed low-sensitivity reduced-order observer design method with the Takagi-Sugeno fuzzy modeling approach for designing the low-sensitivity reduced-order observer of nonlinear systems.