Observer Design and State-Feedback Stabilization for Nonlinear Systems via Equilibrium Manifold Expansion Linearization

Linearization remodeling and state-feedback control for a class of autonomous nonlinear systems based on equilibrium manifold expansion (EME) are visited and explicated in this paper, including linearization approximation, state-feedback stabilization and state estimation. More precisely, firstly, EME linearized remodels of nonlinear systems are explained and their existence is validated rigorously; secondly, EME-based state-feedback control and observer design are developed analytically with EME remodeling and gain scheduling; thirdly, stabilization under EME-based state feedback and observers are tackled, respectively; finally, feasibility and efficiency of the EME approach are illustrated by numerical simulations.

No datasets were generated or analysed during the current study.

Authors and Affiliations

1. School of Energy and Electrical Engineering, Hohai University, Focheng West Road, Nanjing, 211100, Jiangsu Province, China

   Tianjian Hou

2. School of Energy and Electrical Engineering, School of Artificial Intelligence and Automation, Hohai University, Focheng West Road, Nanjing, 211100, Jiangsu Province, China

   Jun Zhou

Contributions

T.J.H. wrote the main manuscript text being instructed by J.Z. All authors reviewed the manuscript.

Corresponding author

Correspondence to Tianjian Hou.

Conflict of interest

The authors declare no Conflict of interest.

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Tianjian Hou and Jun Zhou are contributed equally to this work.

Proof of Theorem 1

Since it is difficult to solve (31) for \(r_i(\alpha )\) directly, a specific form of \(K(\alpha )\) will be given such that (31) is transformed to fit the Cauchy-Kowalewski theorem [19], by which the linear feedback gain \(K(\alpha )\) can be determined row-by-row, meanwhile the desired eigenvalues are specified. To this end, we recall the definition for Kronecker indices.

Definition 3

(Kronecker indices [17]) For an LTI system \(\dot{x}=Ax+Bu\), where \(x\in \mathbb {R}^n\) and \(u\in \mathbb {R}^m\), respectively; and \(A\in \mathbb {R}^{n\times n}\), \(B=\left[ b_1,\ldots ,b_m\right] \in \mathbb {R}^{n\times m}\). Assume that the matrix pair \(\left( A,B\right) \) is controllable. The ith Kronecker index \(v_i\) is the minimum positive integer such that the vectors group \(\{b_i,Ab_i,\ldots ,A^{v_i-1}b_i\}\) is linearly independent. Accordingly, \({\sum }_{i=1}^m{v_i}=n\).

Now we are ready to state Lemma 6.

Lemma 6

([5]) Suppose that the nonlinear system (1) satisfies the assumptions of Theorem 1. For each row \(r_i(\alpha )=[k_{i1}(\alpha ),\ldots ,k_{in}(\alpha )]\) of \(K(\alpha )\in \mathbb {R}^{m\times n}\), we partition it by the ith Kronecker indices of the matrix pair \((A(\alpha ),B(\alpha ))_{\alpha =0}\), denoted by \(v_i\), in form of \(r_i(\alpha )=[r_{i+}(\alpha ),r_{i-}(\alpha )]\) with

Based on \(\{r_1(\alpha ),\ldots ,r_m(\alpha )\}\) and their partitioning, we can re-write that

in a neighborhood of \(\alpha =0\) and \(r_{i-}(0)=0\). Then, given m analytic functions \(r_{i+}(\alpha ):\mathbb {R}^m\rightarrow \mathbb {R}^{n-v_i}\) with \(r_{i+}(0)\), there exist an analytic function \(K(\alpha )\) such that the closed-loop EME model (12) has the specified eigenvalues that are analytic in \(\alpha \).

In Lemma 6, the Kronecker indices are arranged as \(v_1\le v_2\le \ldots \le v_m\) by ordering the column subscripts of \(B(\alpha )_{\alpha =0}\) appropriately, as explained by Definition 3.

Remark 2

Since each and all \(r_{i-}(\alpha )\) can be parameterized by \(r_{1+}(\alpha )\), \(\ldots \), \(r_{m+}(\alpha )\) and \(\alpha \), Lemma 6 actually gives a standard form of \(K(\alpha )\) such that the eigenvalues of the EME state matrix \(F(\alpha )\) can be assigned arbitrarily. Indeed, such parametrization is achieved by solving

where \({\lambda _i}_{i=1}^n\) are the expected eigenvalues for the closed-loop EME model \(\Sigma _\textrm{FB}\).

\(K(\alpha )\)-parametrization can be briefly explained as follows. We define

It can be proved that \(\frac{\partial p(\alpha ,K(\alpha ))}{\partial r_-(\alpha )}|_{\alpha =0}\) is invertible, thus the implicit function theorem says that each and all \(r_{i-}(\alpha )\) can be parameterized.

If we set the specified eigenvalues of \(F(\alpha )\) as \(\lambda _1\), \(\ldots \), \(\lambda _n\), correspondingly we have

By solving this equation for \(r_{i-}(\alpha )\) in terms of \(r_{i+}(\alpha )\), we obtain m analytic functions \(r_{i-}(\alpha )\) in form of \(\alpha \), \(r_{1+}(\alpha )\), \(\ldots \), \(r_{m+}(\alpha )\), while eigenvalues specification of \(F(\alpha )\) are considered.

By Lemma 6 and Remark 2, each row \(r_{i}(\alpha )\) will be determined in terms of \(r_{i+}(\alpha )\), whenever the latter is provided. In the sequel, we summarize the major steps for \(K(\alpha )\)-parametrization.

Firstly, we partition the matrix \(\Phi _j(\alpha )\in \mathbb {R}^{(m-j)\times n}\) in (31) also by the Kronecker indices \(\{v_1.\ldots ,v_m\}\) into \((m-j)\times (n-v_i)\) and \({(m-j)\times v_i}\) blocks as

where

and

Secondly, we take the derivative of \(r_{i-}(\alpha )\) with respect to \(\alpha _j\), where \(\alpha =[\alpha _1,\ldots ,\alpha _m]^T\).

Note that \(\alpha _j\) exists not only in \(r_{i+}(\alpha )\), \(\ldots \), \(r_{m+}(\alpha )\) implicitly, but also appears explicitly as the jth entry of \(r_{i-}(\alpha )\). In the above, a differential operator \(D_i f\), is introduced to denote the partial derivative of a function \(f=f(x_1,\ldots ,x_m)\) with respect to \(x_i\), \(i=1,\ldots ,m\). Thus,

denotes the partial derivative of \(r_{i-}(\alpha )\) with respect to \(\alpha _j\).

Thirdly, similar to \(\Phi _{j}^{i-}(\alpha )\), we define the derivative of \(r_{i-}(\alpha )\) with respect to \(\alpha _{j+1}\), \(\ldots \), \(\alpha _{m}\) as

where \(R_{j}^{i-}(\alpha )\) is analytic in \(\alpha \).

Then, substituting (A1) into (31), the relations of (31) can be expressed in \(r_{i+}(\alpha )\) as

where \(j=1\ldots ,m-1\), and

• \(G_j(\alpha )\) is defined as

  $$\begin{aligned} \begin{aligned} G_j(\alpha )&=G_j^+(\alpha )+G_j^-(\alpha )\\&\quad \triangleq \left[ \begin{array}{*{20}{c}} \Phi ^{1+}_j(\alpha )&{}\cdots &{}0\\ \vdots &{}\ddots &{}\vdots \\ 0&{}\cdots &{}\Phi ^{m+}_j(\alpha )\\ \end{array}\right] +\left[ \begin{array}{*{20}{c}} \Phi ^{1-}_j(\alpha )\frac{\partial r_{1-}}{\partial r_{1+}}&{}\cdots &{} \Phi ^{1-}_j(\alpha )\frac{\partial r_{1-}}{\partial r_{m+}}\\ \vdots &{}\ddots &{}\vdots \\ \Phi ^{m-}_j(\alpha )\frac{\partial r_{m-}}{\partial r_{1+}}&{}\cdots &{} \Phi ^{m-}_j(\alpha )\frac{\partial r_{m-}}{\partial r_{m+}}\\ \end{array}\right] \end{aligned} \end{aligned}$$

  Since \(G_j(\alpha )\) is of full rank [5], m independent equations can be derived to fix all m analytic functions \(r_{i+}(\alpha )\). By Cauchy–Kowalewski Theorem [18], there is an analytic solution to (A2) for each fixed j by choosing initial \(r_{i+}(\alpha )\). Thus, there exists an solution to (31). When \(r_{i+}(\alpha )\) is available, we determine \(r_{i-}(\alpha )\) since it follows from \(r_i(\alpha )\).

• \(H_j(\alpha )\) denotes the left-hand side of (A2), which can be expanded as

  $$\begin{aligned} \begin{aligned} H_j(\alpha )&=[H^T_{j1}(\alpha ),\ldots ,H^T_{jm}(\alpha )]^T\\&= \left[ \begin{array}{*{20}{c}} \Phi _j^{1+}(\alpha )\frac{\partial r_{1+}(\alpha )}{\partial \alpha _j} +\Phi _j^{1-}(\alpha )\frac{\partial r_{1-}}{\partial r_{1+}}\frac{\partial r_{1+}}{\partial \alpha _j}(\alpha ) +\cdots +\Phi _j^{1-}(\alpha )\frac{\partial r_{1-}}{\partial r_{m+}}\frac{\partial r_{m+}}{\partial \alpha _j}(\alpha )\\ \vdots \\ \Phi _j^{m+}(\alpha )\frac{\partial r_{m+}(\alpha )}{\partial \alpha _j} +\Phi _j^{m-}(\alpha )\frac{\partial r_{m-}}{\partial r_{1+}}\frac{\partial r_{1+}}{\partial \alpha _j}(\alpha ) +\cdots +\Phi _j^{m-}(\alpha )\frac{\partial r_{m-}}{\partial r_{m+}}\frac{\partial r_{m+}}{\partial \alpha _j}(\alpha )\\ \end{array}\right] \end{aligned} \end{aligned}$$

  Further expanding \(H_{ji}(\alpha )\), we obtain that

  $$\begin{aligned}{} & {} H_{ji}(\alpha )\\{} & {} \quad = \Phi _j^{i+}(\alpha )\frac{\partial r_{i+}(\alpha )}{\partial \alpha _j} +\left( \Phi _j^{i-}(\alpha )\frac{\partial r_{i-}}{\partial r_{1+}}\frac{\partial r_{1+}}{\partial \alpha _j}(\alpha ) +\cdots +\Phi _j^{i-}(\alpha )\frac{\partial r_{i-}}{\partial r_{m+}}\frac{\partial r_{m+}}{\partial \alpha _j}(\alpha )\right) \\{} & {} \quad = \left[ \begin{array}{*{20}{c}} \frac{\partial \phi _{x_{e1}}(\alpha )}{\partial \alpha _{j+1}}&{}\cdots &{}\frac{\partial \phi _{x_{e(n-v_i)}}(\alpha )}{\partial \alpha _{j+1}}\\ \vdots &{}\ddots &{}\vdots \\ \frac{\partial \phi _{x_{e1}}(\alpha )}{\partial \alpha _m}&{}\cdots &{}\frac{\partial \phi _{x_{e(n-v_i)}}(\alpha )}{\partial \alpha _m} \end{array}\right] \left[ \begin{array}{*{20}{c}} \frac{\partial k_{i1}(\alpha )}{\partial \alpha _j}\\ \vdots \\ \frac{\partial k_{i(n-v_i)}(\alpha )}{\partial \alpha _j} \end{array}\right] \\{} & {} \qquad +\left[ \begin{array}{*{20}{c}} \frac{\partial \phi _{x_{e(n-v_i+1)}}(\alpha )}{\partial \alpha _{j+1}}&{}\cdots &{}\frac{\partial \phi _{x_{en}}(\alpha )}{\partial \alpha _{j+1}}\\ \vdots &{}\ddots &{}\vdots \\ \frac{\partial \phi _{x_{e(n-v_i+1)}}(\alpha )}{\partial \alpha _m}&{}\cdots &{}\frac{\partial \phi _{x_{en}}(\alpha )}{\partial \alpha _m} \end{array}\right] \\{} & {} \qquad \cdot \left( \frac{\partial r_{i-}}{\partial r_{1+}} \left[ \begin{array}{*{20}{c}} \frac{\partial k_{11}(\alpha )}{\partial \alpha _j}\\ \vdots \\ \frac{\partial k_{1(n-v_1)}(\alpha )}{\partial \alpha _j} \end{array}\right] +\cdots +\frac{\partial r_{i-}}{\partial r_{m+}} \left[ \begin{array}{*{20}{c}} \frac{\partial k_{m1}(\alpha )}{\partial \alpha _j}\\ \vdots \\ \frac{\partial k_{m(n-v_m)}(\alpha )}{\partial \alpha _j} \end{array}\right] \right) \end{aligned}$$

  where the Jacobian matrix \(\frac{\partial r_{i-}}{\partial r_{i+}}\) is given by

  $$\begin{aligned} \frac{\partial r_{i-}}{\partial r_{i+}} =\left[ \begin{array}{*{20}{c}} \frac{\partial k_{i(n-v_1+1)}}{\partial k_{i1}}&{}\cdots &{}\frac{\partial k_{i(n-v_1+1)}}{\partial k_{i(n-v_i)}}\\ \vdots &{}\ddots &{}\vdots \\ \frac{\partial k_{in}}{\partial k_{i1}}&{}\cdots &{}\frac{\partial k_{in}}{\partial k_{i(n-v_i)}} \end{array}\right] \in \mathbb {R}^{v_i\times (n-v_i)} \end{aligned}$$

• \(F_j(\alpha )=[F^T_{j1}(\alpha ),\ldots ,F^T_{jm}(\alpha )]^T\) is a known analytic function, where the ith row \(F_{ji}(\alpha )\) of \(F_j(\alpha )\) can be given by substituting (30) into \(H_{ji}(\alpha )\) as

  $$\begin{aligned} \begin{aligned} F_{ji}(\alpha )&=\left[ \begin{array}{*{20}{c}} \frac{\partial k_{i1}(\alpha )}{\partial \alpha _{j+1}}&{}\cdots &{}\frac{\partial k_{i(n-v_i)}(\alpha )}{\partial \alpha _{j+1}}\\ \vdots &{}\ddots &{}\vdots \\ \frac{\partial k_{i1}(\alpha )}{\partial \alpha _m}&{}\cdots &{}\frac{\partial k_{i(n-v_i)}(\alpha )}{\partial \alpha _m} \end{array}\right] \left[ \begin{array}{*{20}{c}} \frac{\partial \phi _{x_{e1}}(\alpha )}{\partial \alpha _j}\\ \vdots \\ \frac{\partial \phi _{x_{e(n-v_i)}}(\alpha )}{\partial \alpha _j} \end{array}\right] +\left[ \begin{array}{*{20}{c}} \frac{\partial k_{i1}(\alpha )}{\partial \alpha _{j+1}}&{}\cdots &{}\frac{\partial k_{i(n-v_i)}(\alpha )}{\partial \alpha _{j+1}}\\ \vdots &{}\ddots &{}\vdots \\ \frac{\partial k_{i1}(\alpha )}{\partial \alpha _m}&{}\cdots &{}\frac{\partial k_{i(n-v_i)}(\alpha )}{\partial \alpha _m} \end{array}\right] \\&\qquad \cdot \left( \frac{\partial r_{i-}}{\partial r_{1+}} \left[ \begin{array}{*{20}{c}} \frac{\partial \phi _{x_{e(n-v_i+1)}}(\alpha )}{\partial \alpha _j}\\ \vdots \\ \frac{\partial \phi _{x_{en}}(\alpha )}{\partial \alpha _j} \end{array}\right] +\cdots +\frac{\partial r_{i-}}{\partial r_{m+}} \left[ \begin{array}{*{20}{c}} \frac{\partial \phi _{x_{e(n-v_i+1)}}(\alpha )}{\partial \alpha _j}\\ \vdots \\ \frac{\partial \phi _{x_{en}}(\alpha )}{\partial \alpha _j} \end{array}\right] \right) \end{aligned} \end{aligned}$$

Finally, we obtain the feedback gain \(K(\alpha )\) row-by-row. Since \(K(\alpha )\) is available, \(\hat{k}(\alpha )\) can be integrated through (27), and thus the equation (25) can be written as

by which we can determine each scalar function \(k_i(x)\) in k(x).

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