Improved synthesis conditions for mixed H2/H∞ gain-scheduling control subject to uncertain scheduling parameters

ABSTRACT

The vast majority of the existing work in gain-scheduling (GS) control literature assumes perfect knowledge of scheduling parameters. Generally, this assumption is not realistic since for practical control applications measurement noises are unavoidable. In this paper, novel synthesis conditions are derived to synthesise robust GS controllers with mixed ${\mathcal{H}}_2/{\mathcal{H}}_{\infty }$ performance subject to uncertain scheduling parameters. The conditions are formulated in terms of parameterised bilinear matrix inequalities (PBMIs) that depend on varying parameters inside multi-simplex domain. The conditions provide practical GS controllers independent of the derivatives of scheduling parameters. That is, the designed controllers are feasible for implementation. Since bilinear matrix inequality problems are intractable, an iterative PBMI algorithm is developed to solve the developed synthesis conditions. By the virtue of this algorithm, conservativeness reduction is achieved with few iterations. Examples are presented to illustrate the effectiveness of the developed conditions. Compared to other design methods from literature, the developed conditions achieve better performance.

KEYWORDS:

• Linear parameter-varying systems
• gain-scheduling
• linear matrix inequality
• robust control

Acknowledgement

Ali Khudhair Al-Jiboory would like to thank the Higher Committee for Education Development (HCED) and Universityof Diyala in Iraq for their financial support during his graduate study at MSU.

Disclosure statement

No potential conflict of interest was reported by the authors.

Notes

1. To be precise, since we are dealing with LPV systems, the ${\mathcal{H}}_2$ and/or ${\mathcal{H}}_{\infty }$ performances are not well defined in the addressed problem yet. However, we use the mixed ${\mathcal{H}}_2/{\mathcal{H}}_{\infty }$ performance here with slightly abused terminology so that the reader can easily grasp our problem setting. We will postpone the strict definition of the control problem until Section 4 since necessary definitions and transformations need to be introduced in the next two sections.

2. Sometimes the dependency on t will be omitted for notational simplicity.

3. Equation (50(50) $$\begin{array}{ccc}\hfill Z_0& =& \frac{1}{2^{2q}}\sum _{j_1=1}^2\sum _{j_2=1}^2\cdots \sum _{j_q=1}^2\sum _{k_i=1}^2\sum _{k_2=1}^2\cdots \hfill \\ & & \sum _{k_q=1}^2{\mathcal{Z}}_{j_1,j_2,...,j_q,k_1,k_2,...,k_q},\hfill \\ \hfill Z_i& =& \frac{1}{2^{2q}{\bar{\theta }}_i}\sum _{j_1=1}^2\sum _{j_2=1}^2\cdots \sum _{j_q=1}^2\sum _{k_i=1}^2\sum _{k_2=1}^2\cdots \hfill \\ & & \sum _{k_q=1}^2{(-1)}^{j_i+i}{\mathcal{Z}}_{j_1,j_2,...,j_q,k_1,k_2,...,k_q}.\hfill \end{array}$$(50) ) is derived from Equation (26(26) $$\begin{array}{ccc}& & {\mathcal{Z}}_{j_1,j_2,...,j_q,k_1,k_2,...,k_q}\hfill \\ & =& Z_0+\sum _{i=1}^q\{{(-1)}^{j_i+1}{\bar{\theta }}_i+{(-1)}^{k_i+1}{\bar{\delta }}_i\}{Z}_i,\hfill \end{array}$$(26) ) using simple algebraic manipulations; however, the proof is omitted here for brevity.