Interval and linear matrix inequality techniques for reliable control of linear continuous-time cooperative systems with applications to heat transfer

ABSTRACT

Lower and upper state bounds can be computed independently for cooperative ordinary differential equations (ODEs) with interval-valued initial conditions. Then, all reachable states are enclosed by two decoupled, point-valued initial value problems (IVPs). This evaluation, however, becomes more challenging if the IVPs are, furthermore, subject to uncertain parameters. In the simplest case, to which this paper is restricted, the ODEs are linear with uncertain system and input matrices. Besides actually linear dynamics, also nonlinear input-affine state-space representations can be accounted for after embedded them into a polytopic uncertainty model representing a conservative convex combination of extremal system realisations. To perform the reachability analysis for closed-loop control structures without significant computational effort, it is reasonable to impose constraints during control synthesis so that the closed-loop ODEs remain cooperative. Suitable design procedures based on linear matrix inequalities are derived in this paper together with a validation for a prototypical heat transfer process.

Keywords:

• Cooperative dynamic systems
• interval-based control and observer design
• linear matrix inequalities
• heat transfer
• experimental validation

Disclosure statement

No potential conflict of interest was reported by the authors.

Notes

1 If the stabilisation of the upper bounding system $\overline{\mathbf{x}}\left(t\right)$ with the system matrix $\overline{\mathbf{A}}$ leads to excessively conservative controller gains, or – in the extreme case – to an infeasible set of LMIs, see also the illustrating example in Section 3.2, the respective entries in the ODEs (25(25) $\begin{array}{rl}\dot{\mathit{\zeta }}\left(t\right)& =\left[\begin{array}{cc}\underset{\_}{\mathbf{A}}& \mathbf{0}\\ \mathbf{0}& \overline{\mathbf{A}}\end{array}\right]\cdot \mathit{\zeta }\left(t\right)+L\cdot \left[\begin{array}{c}\underset{\_}{\mathbf{B}}\\ \overline{\mathbf{B}}\end{array}\right]\cdot \mathbf{w}\left(t\right)+\left[\begin{array}{c}\underset{\_}{\mathbf{B}}\\ \overline{\mathbf{B}}\end{array}\right]\cdot \mathit{\nu }\left(t\right),\\ \mathit{\zeta }\left(t\right)& =\left[\begin{array}{c}\underset{\_}{\mathbf{x}}\left(t\right)\\ \overline{\mathbf{x}}\left(t\right)\end{array}\right],\end{array}$(25) ) can be replaced by all dynamic system models for the upper state estimates ${\hat{\mathbf{w}}}_{\varsigma }\left(t\right)$ with the respective dynamics matrices ${\overline{\mathbf{A}}}_{\varsigma }$, $\varsigma \in \{1,\dots ,L\}$.

2 In most practical applications, these bounded disturbances result from state estimation errors during transient phases and from inaccurate sensor measurements, unmodeled dynamics, and bounded measurement noise.

3 To avoid excessively different magnitudes of the entries in ${\underset{\_}{\mathbf{K}}}_{\varsigma }$ and ${\overline{\mathbf{K}}}_{\varsigma }$, the matrix norm $\sum _{\varsigma =1}^L\parallel {\overline{\mathbf{Y}}}_{\varsigma }-{\underset{\_}{\mathbf{Y}}}_{\varsigma }\parallel$ can be added in a weighted manner to the cost function ${\gamma }_{\mathrm{\infty }}>0$ as a further relaxation of the classical $H_{\mathrm{\infty }}$ synthesis.

4 Note, this regularisation term does not have any implication on the stability of the closed-loop control system because the LMI constraints (47(47) $\begin{array}{rl}& \left({\mathbf{I}}_{2\times 2}\otimes \mathbf{Q}\right){\left[\begin{array}{cc}{\underset{\_}{\mathbf{A}}}_i& \mathbf{0}\\ \mathbf{0}& {\overline{\mathbf{A}}}_i\end{array}\right]}^{\mathrm{T}}-{\left[\sum _{\varsigma =1}^L{\underset{\_}{\mathbf{Y}}}_{\varsigma }\sum _{\varsigma =1}^L{\overline{\mathbf{Y}}}_{\varsigma }\right]}^{\mathrm{T}}\cdot {\mathcal{B}}_{2,i}^{\mathrm{T}}\\ & +\left[\begin{array}{cc}{\underset{\_}{\mathbf{A}}}_i& \mathbf{0}\\ \mathbf{0}& {\overline{\mathbf{A}}}_i\end{array}\right]\left({\mathbf{I}}_{2\times 2}\otimes \mathbf{Q}\right)-{\mathcal{B}}_{2,i}\cdot \left[\sum _{\varsigma =1}^L{\underset{\_}{\mathbf{Y}}}_{\varsigma }\sum _{\varsigma =1}^L{\overline{\mathbf{Y}}}_{\varsigma }\right]\prec 0,\\ & {\mathcal{B}}_{2,i}=\left[\begin{array}{c}{\underset{\_}{\mathbf{B}}}_i\\ {\overline{\mathbf{B}}}_i\end{array}\right]\end{array}$(47) ) are explicitly included in the design procedure to ensure asymptotic, respectively input-to-state, stability of the closed-loop system dynamics despite uncertain but bounded system parameters.

5 Non-negative real parts of the eigenvalues of ${\mathbb{A}}_{\mathrm{Q}}$ may arise, for example, if $\overline{\mathbf{B}}-\underset{\_}{\mathbf{B}}=\mathbf{0}$ holds.