Abstract
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.
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T.J.H. wrote the main manuscript text being instructed by J.Z. All authors reviewed the manuscript.
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Tianjian Hou and Jun Zhou are contributed equally to this work.
Proof of Theorem 1
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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Hou, T., Zhou, J. Observer Design and State-Feedback Stabilization for Nonlinear Systems via Equilibrium Manifold Expansion Linearization. Qual. Theory Dyn. Syst. 23 (Suppl 1), 254 (2024). https://doi.org/10.1007/s12346-024-01115-8
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DOI: https://doi.org/10.1007/s12346-024-01115-8