Adaptive stabilisation of discrete LTI plant with bounded disturbances via finite capacity channel

ABSTRACT

The paper is devoted to adaptive feedback stabilisation for discrete linear time-invariant plants with bounded disturbances and quantisation. The quantisation occurs due to the finite capacity of the discrete-time two-way communication channel. A simple adaptive controller based on the Yakubovich's recursive goal inequalities method is designed. The bound for the minimum channel capacity sufficient to stabilise any stabilisable system with unknown parameters is evaluated for any prespecified compact set of unknown parameters. It is shown that if the channel capacity exceeds the above bound then the upper bound for limit output error is proportional to the disturbance intensity and inversely proportional to the exponential of the channel capacity. The design and performance of the proposed algorithm are illustrated by an example.

KEYWORDS:

• Adaptive control
• finite-converging algorithms
• quantisation
• communication channel
• channel capacity

Disclosure statement

No potential conflict of interest was reported by the authors.

Notes

1. If α(λ) and β(λ) have common stable (outside of unit circle) roots μ _j then both polynomials can be cancelled by ∏ _j (λ − μ _j ). Otherwise if the common roots are unstable then stabilisation problem is unsolvable.

2. In order to ensure this implication inequalities (18(18) $$\begin{array}{c}\hfill \nu >lo{g}_2\left(1/2+{\varkappa }^{-(2n+1)}(1+G)\left[1+{\rho }^{-1}\right]\frac{H+1}{2}\right).\end{array}$$(18) ) and (26(26) $$\begin{array}{ccc}& & {\Delta }_{k-1}>\underset{\_}{\Delta }\hfill \\ & & =C_f\frac{1+G}{\rho }{\left[M-1/2-\frac{1+G}{{\varkappa }^{2n+1}}\left(1+{\rho }^{-1}\right)\frac{H+1}{2}\right]}^{-1}.\hfill \end{array}$$(26) ) were intensionally weakened.

Additional information

Funding

This work was supported by SPbSU [grant number 6.38.230.2015]; RFBR [grant number 17-08-01728]; Government of Russian Federation [grant number 074-U01].