Optimal energy compensation for disturbed systems with time-varying delays

Abstract

This paper focuses on the compensation of disturbances in linear dynamical systems with time-varying delays. The objective is to reduce the effect of any measured disturbances and restore the observations to their normal state by a final time T. The research paper presents a minimum energy problem describing the mathematical framework to obtain a convenient control ensuring the disturbance compensation. The presented application to the two-dimensional diffusion equation emphasizes the practical applicability and relevance of the proposed approach.

Appendix A: Minimum energy compensation using LQR approach

For sake of diversity, we present hereafter a general approach also widely used, where this compensation problem is examined as a Linear Quadratic Regulator minimization. In practice, the comparison of methods is classically recommended for the implementation of any analytical method. The LQR method is practically and widely used in control theory.

We consider the following cost function defined for a finite time horizon T as follows:

$$\begin{aligned} \begin{aligned} J(u)&=\int _{0}^{T}\Big \langle CHu+C{\tilde{H}}f,Q\Big ( CHu+C{\tilde{H}}f\Big ) \Big \rangle _{\textrm{Y}}\;ds\\&\quad +\displaystyle \int _{0}^{T}\Big \langle u(s),Ru(s) \Big \rangle _{\textrm{U}}\;ds; \quad u\in L^2(0,T;\textrm{U}) \end{aligned} \end{aligned}$$

(A1)

where Q end R are self-adjoint operators assumed to be positive and coercive. The operators H and ${\tilde{H}}$ are defined, respectively, in (13) and (14).

We examine thereafter the existence of $u^*\in L^2(0,T;\textrm{U})$ such that:

$$\begin{aligned} J(u^*)={\left\{ \begin{array}{ll} \textrm{min}\quad J(u)\\ u\in L^2(0,T;\textrm{U}) \end{array}\right. } \end{aligned}$$

(A2)

We have:

$$\begin{aligned} J(u)={} & {} \displaystyle \int _{0}^{T}\Big \langle CHu,QCHu\Big \rangle _{\textrm{Y}}+2\;\Big \langle CHu,QC{\tilde{H}}f\Big \rangle _{\textrm{Y}}\\{} & {} \quad +\Big \langle C{\tilde{H}}f,QC{\tilde{H}}f\Big \rangle _{\textrm{Y}}ds\\{} & {} \quad +\displaystyle \int _{0}^{T}\Big \langle u(s),Ru(s)\Big \rangle _{\textrm{U}}ds \end{aligned}$$

i.e.,

$$\begin{aligned} J(u)= \Pi (u,u)-2{\mathcal {L}}v+J(0) \end{aligned}$$

where

$$\begin{aligned}{} & {} \Pi : \textrm{U}\times \textrm{U}\rightarrow {\mathbb {R}} \\{} & {} \Pi (u,u) =\displaystyle \int _{0}^{T}\Big \langle CHu,QCHu\Big \rangle _{\textrm{Y}}+\Big \langle u(s),Ru(s)\Big \rangle _{\textrm{U}}ds\\ \end{aligned}$$

and

$$\begin{aligned}{} & {} {\mathcal {L}}: \textrm{U}\rightarrow {\mathbb {R}}\\{} & {} {\mathcal {L}}u=\displaystyle \int _{0}^{T}\Big \langle CHu,QC{\tilde{H}}f\Big \rangle _{\textrm{Y}}\;ds \end{aligned}$$

also

$$\begin{aligned} J(0)=\int _{0}^{T}\Big \langle C{\tilde{H}}f,QC{\tilde{H}}f\Big \rangle _{\textrm{Y}}ds \end{aligned}$$

According to Lax–Milgram theorem, J admits a unique minimum in $u^*$ characterized by:

$$\begin{aligned} \Pi (u^*,u)={\mathcal {L}}u;\quad \quad \forall u\in L^2(0,T;\textrm{U}) \end{aligned}$$

or equivalently

$$\begin{aligned}{} & {} \displaystyle \int _{0}^{T}\Big [\Big \langle CHu^*,QCHu\Big \rangle +\Big \langle u^*(s),Ru(s)\Big \rangle \Big ] ds=\\{} & {} -\displaystyle \int _{0}^{T}\Big \langle CHu,QC {\tilde{H}}f\Big \rangle \;ds;\\{} & {} \forall u\in L^2(0,T;\textrm{U}) \end{aligned}$$

or also

$$\begin{aligned}{} & {} \displaystyle \int _{0}^{T}\Big \langle H^*C^*QC Hu^*,u(s)\Big \rangle \\{} & {} +\Big \langle Ru^*(s),u(s)\Big \rangle +\Big \langle H^*C^*QC{\tilde{H}}f,u(s)\Big \rangle ds=0;\\{} & {} \forall u\in L^2(0,T;\textrm{U}) \end{aligned}$$

i.e.,

$$\begin{aligned}{} & {} \displaystyle \int _{0}^{T}\Big \langle H^*C^*QC Hu^*+Ru^*+ H^*C^*QC {\tilde{H}},u(s)\Big \rangle ds=0; \\{} & {} \quad \forall u\in L^2(0,T;\textrm{U}) \end{aligned}$$

Then

$$\begin{aligned} \left( H^*C^*QCH+R\right) u^*=-H^*C^*QC{\tilde{H}}f \end{aligned}$$

Consequently, the corresponding optimal control is given by:

$$\begin{aligned} u^*=-\left( H^*C^*QC H+R\right) ^{-1} H^*C^*QC {\tilde{H}}f \end{aligned}$$

Let us remark that if $Q=I$ and $R=\alpha I$, we obtain:

$$\begin{aligned} u^*=-\left( H^*C^*C H+\alpha I\right) ^{-1} H^*C^*C {\tilde{H}}f \end{aligned}$$

and if $\alpha<<1$ (i.e., $\alpha$ is small), one obtain practically the same results established using the Hilbert Uniqueness Method (22). Therefore, the two methods are sufficiently consistent.