Adaptive observer for ODE-PDE cascade systems subject to nonlinear dynamics and uncertain parameters

Abstract

In this article, state and parameter estimation problems are investigated for ODE-PDE coupled systems, for which the parabolic PDE sensor includes nonlinear dynamics and parameter uncertainty. The major difficulty we face is that the link point among the ODE part and the PDE part is not convenient to measurable. For this reason, the objective of this paper is to build an adaptive observer to provide online estimates of states and unknown parameters on the basis that only boundary state is available for measurement. First of all, the observer error system is converted to a tractable target error system by applying the decoupling transformation. Then, the least-squares parameter adaptive law is built. Under an ad hoc persistent excitation condition, the exponentially decaying of the observer error system is demonstrated by applying Lyapunov–Krasovskii functional. Finally, the effectiveness of the theoretical results is confirmed by a simulation example.

Appendices

Appendix A: Properties of the matrix function ${\Omega }(\sigma )$

To the best of our knowledge, the matrix function ${\Omega }(\sigma )$ meets the ensuing properties:

$$\begin{aligned} (a)~{} & {} {\Omega }(\sigma )=I+\sum _{k=1}^{n-1}\frac{\sigma ^{2k}}{(2k)!}\Big (\frac{A}{{h}}\Big )^{k},\nonumber \\ (b)~{} & {} A{\Omega }(\sigma )={\Omega }(\sigma )A,\nonumber \\ (c)~{} & {} A{\Omega }^{-1}(\sigma )={\Omega }^{-1}(\sigma )A,\nonumber \\ (d)~{} & {} {\Omega }(\sigma )\overset{\text {def}}{=}\left( \begin{array}{cc} I &{} 0 \\ \end{array} \right) e^{\left( \begin{array}{cc} 0 &{} \frac{A}{{h}} \\ I &{} 0 \\ \end{array} \right) \sigma }\left( \begin{array}{c} I \\ 0 \\ \end{array} \right) , \quad \forall \sigma \in {\mathcal {R}} .\nonumber \end{aligned}$$

The reader can refer to Ref. [9] for more details.

Appendix B: The well-posedness of system

Step 1. We first show the well-posedness of the ODE-subsystem (2a) with the initial data X(0). By utilizing the existence theorem of ODEs, the subsystem (2a) exists a unique solution $X(t)\in {\mathcal {C}}^1([0,+\infty ):{\mathcal {R}}^n)$.

Step 2. The well-posedness of the PDE-subsystem (2b)-(2d) will be illustrated. First, introducing the ensuing transformation for any $(\sigma ,t)\in [0,\ell ]\times [0,+\infty )$

$$\begin{aligned} q(\sigma ,t){} & {} =u(\sigma ,t)-C{\Omega }(\sigma ){\Omega }^{-1}(\ell )X(t). \end{aligned}$$

(B.1)

Together with Appendix A, we rewrite the system (2b)–(2d) as

$$\begin{aligned} {\left\{ \begin{array}{ll} q_t(\sigma ,t)=hq_{{\sigma \sigma }}(\sigma ,t)+{\bar{g}}(q,\sigma ,t),\\ q_\sigma (0,t)=\phi _3(t)\rho _3,\\ q(\ell ,t)=\phi _4(t)\rho _4, \end{array}\right. } \end{aligned}$$

where ${\bar{g}}(q,\sigma ,t)=g(q+C{\Omega }(\sigma ){\Omega }^{-1}(\ell )X,\sigma ,t)+\phi _2(\sigma ,t)\rho _2 -C{\Omega }(\sigma ){\Omega }^{-1}(\ell )\phi _1(t)\rho _1$. Obviously, function ${\bar{g}}(\cdot )$ inherits all the properties of function $g(\cdot )$. The q-subsystem turns out to be a special case of the semilinear parabolic system (13)–(14) in Ref. [26]. It can be obtained that one has a strong solution $q(\sigma ,t) \in \mathcal {H}^1(0,\ell )$ for each $t\ge 0$. Due to the fact that the transformation (B.1) is inverse, the similar result holds for PDE subsystem (2b)–(2d).

Step 3. In this part, we illustrate the well-posedness of the observer system of Table 1.

First, we develop the well-posedness of the error system (27a)–(27f). Applying the usual existence theorems for ODEs, it follows that the subsystem (27a) exists a unique solution $Z(t)\in {\mathcal {C}}^1([0,+\infty ):{\mathcal {R}}^n)$. Similarly, we obtain that the subsystem consisting of (27e)–(27f) exists a unique solution ${\tilde{\rho }}\in {\mathcal {C}}^1([0,+\infty ):{\mathcal {R}}^{m}),$ $Q^{-1}\in {\mathcal {C}}^1([0,+\infty ):{\mathcal {R}}^{m\times m})$, that is, the subsystem consisting of (21a)–(21b) exists a unique solution ${{\hat{\rho }}}\in {\mathcal {C}}^1([0,+\infty ):{\mathcal {R}}^m),$ $Q\in {\mathcal {C}}^1([0,+\infty ):{\mathcal {R}}^{m\times m})$. Then, it is easy to verify that the subsystem consisting of (27b)–(27d) has a strong solution $w(\sigma ,t)\in {{\mathcal {H}}}^{1}(0,\ell )$ by means of the similar analysis of Step 2.

Second, utilizing the usual existence theorems for ODEs, we obtain the plant (10) exists a unique solution $\lambda _{11}(t)\in {\mathcal {C}}^1([0,+\infty ):{\mathcal {R}}^{n\times m_1})$, $\lambda _{12}(t)\in {\mathcal {C}}^1([0,+\infty ):{\mathcal {R}}^{n\times m_2})$, $\lambda _{13}(t)\in {\mathcal {C}}^1([0,+\infty ):{\mathcal {R}}^{n\times m_3})$, $\lambda _{14}(t)\in {\mathcal {C}}^1([0,+\infty ):{\mathcal {R}}^{n\times m_4})$. Then, it is easy to give that the first two equations of (15) and (19) have strong solution $\lambda _{21}(\sigma ,t)\in {{\mathcal {H}}}^{1}(0,\ell )$ and $\lambda _{22}(\sigma ,t)\in {{\mathcal {H}}}^{1}(0,\ell )$ applying the similar method with Step 2. Next, applying the method of separation of variables (see Ref. [27]), it follows that the last two equations of (15) and (19) have closed-loop solution.

The above analysis implies that the plant consisting of (5a)–(5d) exists a unique solution ${\tilde{X}}\in {\mathcal {C}}^1([0,+\infty ):{\mathcal {R}}^{n})$ and ${\tilde{u}}(\sigma ,t)\in {{\mathcal {H}}}^{1}(0,\ell )$, due to (6a)–(6b). Consequently, the similar results for the observer states ${\hat{X}}(t)$ and ${\hat{u}}(\sigma ,t)$ are hold.

Appendix C: The proof of Lemma 1

From (10), we found that $\lambda _{1i}(t)(i=1,2,3,4)$ are bounded if $\lambda _{2i}(0,t)(i=1,2,3,4)$ are bounded, due to the signal $\phi _1(t)$ is bounded and $A-LC{\Omega }^{-1}(\ell )$ is a Hurwitz matrix. Therefore, we only need to suggest $\lambda _{2i}(\sigma ,t)(i=1,2,3,4)$ are bounded. Consider the ensuing Lyapunov functional

$$\begin{aligned} W_{21}(t)=&~\frac{1}{2}\int _{0}^{\ell }\lambda _{21}(\sigma ,t)\lambda _{21}^T(\sigma ,t)d{\sigma }\nonumber \\&\quad +\frac{1}{2}\int _{0}^{\ell }\lambda _{21,\sigma }(\sigma ,t)\lambda _{21,\sigma }^T(\sigma ,t)d{\sigma }. \end{aligned}$$

(C.1)

Differentiating (C.1) and applying the integration by parts formula, it follows from the first equation of (15) and (19) that

$$\begin{aligned} {\dot{W}}_{21}(t)&=\int _{0}^{\ell }\lambda _{21}(\sigma ,t)\lambda _{21,t}^T(\sigma ,t)d{\sigma }\nonumber \\&\quad +\int _{0}^{\ell }\lambda _{21,\sigma }(\sigma ,t)\lambda _{21,\sigma t}^T(\sigma ,t)d{\sigma }\nonumber \\&=-h\int _{0}^{\ell }\lambda _{21,\sigma }(\sigma ,t)\lambda _{21,\sigma }^T(\sigma ,t)d{\sigma }\nonumber \\&~~~-\int _{0}^{\ell }\lambda _{21}(\sigma ,t)\big (C{\Omega }(\sigma ){\Omega }^{-1}(\ell )\phi _1(t)\big )^Td{\sigma }\nonumber \\&~~~-h\int _{0}^{\ell }\lambda _{21,{\sigma \sigma }}(\sigma ,t)\lambda _{21,{\sigma \sigma }}^T(\sigma ,t)d{\sigma }\nonumber \\&~+\int _{0}^{\ell }\lambda _{21,{\sigma \sigma }}(\sigma ,t)\big (C{\Omega }(\sigma ){\Omega }^{-1}(\ell )\phi _1(t)\big )^Td{\sigma }. \end{aligned}$$

(C.2)

Then, by means of Young’s inequality for (C.2), we get

$$\begin{aligned} {\dot{W}}_{21}(t)=&-h\int _{0}^{\ell }\Vert \lambda _{21,\sigma }(\sigma ,t)\Vert ^2d{\sigma }\\&+\frac{\eta }{2}\int _{0}^{\ell }\Vert \lambda _{21}(\sigma ,t)\Vert ^2d{\sigma }\nonumber \\&~+\frac{1}{2\eta }\int _{0}^{\ell }\Vert C{\Omega }(\sigma ){\Omega }^{-1}(\ell )\phi _1(t)\Vert ^2d{\sigma }\nonumber \\&~-\Big (h-\frac{\zeta }{2}\Big )\int _{0}^{\ell }\Vert \lambda _{21,{\sigma \sigma }}(\sigma ,t)\Vert ^2d{\sigma }\nonumber \\&~+\frac{1}{2\zeta }\int _{0}^{\ell }\Vert C{\Omega }(\sigma ){\Omega }^{-1}(\ell )\phi _1(t)\Vert ^2d{\sigma }, \end{aligned}$$

where $\eta$ and $\zeta$ are positive scalars to be determined later. Then, applying (1a) yields

$$\begin{aligned} {\dot{W}}_{21}(t)\le&~-\Big (h-\frac{2\eta \ell ^2}{\pi ^2}\Big )\int _{0}^{\ell }\Vert \lambda _{21,\sigma }(\sigma ,t)\Vert ^2d{\sigma }\nonumber \\&~{+}\Big (\frac{1}{2\eta }{+}\frac{1}{2\zeta }\Big )\int _{0}^{\ell }\Vert C{\Omega }(\sigma ){\Omega }^{{-}1}(\ell )\phi _1(t)\Vert ^2d{\sigma }\nonumber \\&~-\Big (h-\frac{\zeta }{2}\Big )\int _{0}^{\ell }\Vert \lambda _{21,{\sigma \sigma }}(\sigma ,t)\Vert ^2d{\sigma }. \end{aligned}$$

Taking positive constants $\eta <\frac{\pi ^2 h}{2\ell ^2}$ and $\zeta <2h$, clearly, we have

$$\begin{aligned} h-\frac{2\eta \ell ^2}{\pi ^2}>0, \quad h-\frac{\zeta }{2}>0. \end{aligned}$$

Then, by means of Wirtinger’s inequality (1a) again, we obtain

$$\begin{aligned} {\dot{W}}_{21}(t)\le&~-\Big (h-\frac{2\eta \ell ^2}{\pi ^2}\Big )\frac{\pi ^2}{4\ell ^2}\int _{0}^{\ell }\Vert \lambda _{21}(\sigma ,t)\Vert ^2d{\sigma }\nonumber \\&~{+}\Big (\frac{1}{2\eta }{+}\frac{1}{2\zeta }\Big )\int _{0}^{\ell }\Vert C{\Omega }(\sigma ){\Omega }^{{-}1}(\ell )\phi _1(t)\Vert ^2d{\sigma }\nonumber \\&~-\Big (h-\frac{\zeta }{2}\Big )\frac{\pi ^2}{4\ell ^2}\int _{0}^{\ell }\Vert \lambda _{21,\sigma }(\sigma ,t)\Vert ^2d{\sigma }\nonumber \\ \le&~\Big (\frac{1}{2\eta }+\frac{1}{2\zeta }\Big )\int _{0}^{\ell }\Vert C{\Omega }(\sigma ){\Omega }^{-1}(\ell )\phi _1(t)\Vert ^2d{\sigma }\nonumber \\&~-\frac{\pi ^2}{2\ell ^2}\textrm{min}\Big \{h-\frac{2\eta \ell ^2}{\pi ^2},h-\frac{\zeta }{2}\Big \}W_{21}(t). \end{aligned}$$

It can be obtained that ${W}_{21}(t)$ is bounded, due to the signal $\phi _1(t)$ and matrix function ${\Omega }(\sigma )$ are bounded. (C.1) implies that $\int _{0}^{\ell }\Vert \lambda _{21}(\sigma ,t)\Vert ^2d{\sigma }$ and $\int _{0}^{\ell }\Vert \lambda _{21,\sigma }(\sigma ,t)\Vert ^2d{\sigma }$ are bounded. From (1b), we infer that $\Vert \lambda _{21}(\sigma ,t)\Vert$ is bounded, so is $\Vert \lambda _{11}(t)\Vert$.

In the same manner, we can infer that $\Vert \lambda _{22}(\sigma ,t)\Vert$, $\Vert \lambda _{23}(\sigma ,t)\Vert$ and $\Vert \lambda _{24}(\sigma ,t)\Vert$ are bounded, which suggest that $\Vert \lambda _{12}(t)\Vert$, $\Vert \lambda _{13}(t)\Vert$ and $\Vert \lambda _{14}(t)\Vert$ are also bounded.