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## Quantized average consensus via dynamic coding/decoding schemes.(English)Zbl 1290.93005

Summary: In the average consensus a set of linear systems has to be driven to the same final state, which corresponds to the average of their initial states. This mathematical problem can be seen as the simplest example of coordination task. In fact it can be used to model both the control of multiple autonomous vehicles which all have to be driven to the centroid of the initial positions, and to model the decentralized estimation of a quantity from multiple measure coming from distributed sensors. This contribution presents a consensus strategy in which the systems can exchange information among themselves according to a fixed strongly connected digital communication network. Beside the decentralized computational aspects induced by the choice of the communication network, we here have also to face the quantization effects due to the digital links. We here present and discuss two different encoding/decoding strategies with theoretical and simulation results on their performance.

### MSC:

 93A14 Decentralized systems 68T42 Agent technology and artificial intelligence 93C55 Discrete-time control/observation systems 93C05 Linear systems in control theory 93D05 Lyapunov and other classical stabilities (Lagrange, Poisson, $$L^p, l^p$$, etc.) in control theory

### Keywords:

average consensus; quantization; coding/decoding schemes
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### References:

 [1] Olfati-Saber, Consensus and cooperation in networked multi-agent systems, IEEE Proceedings 95 (1) pp 215– (2007) · Zbl 1376.68138 [2] Carli, Communication constraints in the average consensus problem, Automatica 44 (3) pp 671– (2008) · Zbl 1283.93014 [3] Kashyap, Quantized consensus, Automatica 43 (7) pp 1192– (2007) [4] Yildiz ME Scaglione A Differential nested lattice encoding for consensus problems 89 98 [5] Xiao, Distributed average consensus with least-mean-square deviation, Journal of Parallel and Distributed Computing 67 (1) pp 33– (2007) · Zbl 1109.68019 [6] Carli R Fagnani F Frasca P Taylor T Zampieri S Average consensus on networks with transmission noise or quantization 1852 1857 [7] Aysal TC Coates M Rabbat M Distributed average consensus using probabilistic quantization 640 644 · Zbl 1390.94083 [8] Carli, Quantized coordination algorithms for rendezvous and deployment, SIAM Journal on Control and Optimization (2007) · Zbl 1192.68845 [9] Nair, Feedback control under data rate constraints: an overview, IEEE Proceedings 95 (1) pp 108– (2007) [10] Elia, Stabilization of linear systems with limited information, IEEE Transactions on Automatic Control 46 (9) pp 1384– (2001) · Zbl 1059.93521 [11] Fagnani F Zampieri S Performance evaluations of quantized stabilizers 1897 1901 [12] Silvester, Determinants of block matrices, The Mathematical Gazette 84 (501) pp 460– (2000) [13] Boyd, Linear Matrix Inequalities in System and Control Theory, 15 of Studies in Applied Mathematics (1994) · Zbl 0816.93004 [14] Xiao L Boyd S Lall S A scheme for robust distributed sensor fusion based on average consensus 63 70 [15] MartÃnez, On synchronous robotic networks part I: models, tasks and complexity, IEEE Transactions on Automatic Control 52 (12) pp 2199– (2007) [16] Boyd S Ghosh A Prabhakar B Shah D Gossip algorithms: design, analysis, and application 1653 1664 [17] Fagnani, Quantized stabilization of linear systems: Complexity versus performances, IEEE Transactions on Automatic Control 49 (9) pp 1534– (2004) · Zbl 1365.93394 [18] Carli R Bullo F Zampieri S Quantized average consensus via dynamic coding/decoding schemes · Zbl 1290.93005
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.

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# Help with finding the length of one of the sides of a triangle

\begin{align}\frac z{\sin E}&=\frac y{\sin D}\tag{1}\\ z^2 &= x^2 + y^2 - 2xy\cos E\tag{2}\\ y^2 &= z^2 + x^2 - 2zx\cos D\tag{3}\end{align}

Solve for $z$

My solution is as follow.

I stopped at the equation:

$$(2xy)^2 - (x^2 + y^2 - z^2)^2 = (2zx)^2 - (z^2 + x^2 - y^2)^2$$

Please help me solve for $z$ in the above equation.

Thank you

• I think, It's impossible. We need more given. – Michael Rozenberg May 25 '17 at 7:08
• All data are given. Also, I arrived at the equation: (2xy)^2 - (x^2 + y^2 - z^2)^2 = (2zx)^2 - (z^2 + x^2 - y^2)^2 There is only one unknown in the above equation, which is z. x and y are given, and they are x = 70.64622683 and y = 147.2977643 – Chapel Li May 25 '17 at 7:13
• are the angles given? – Dr. Sonnhard Graubner May 25 '17 at 7:14
• Angles are not given – Chapel Li May 25 '17 at 7:16
• The reason why I stopped at the equation: (2xy)^2 - (x^2 + y^2 - z^2)^2 = (2zx)^2 - (z^2 + x^2 - y^2)^2 is because I don't know how to solve a quartic function. – Chapel Li May 25 '17 at 7:21

If you expand your final equation $$(2xy)^2 - (x^2 + y^2 - z^2)^2 = (2zx)^2 - (z^2 + x^2 - y^2)^2$$ you will get an equation which has equal terms in $x^4$, $y^4$, and $z^4$ on both sides. Removing these gets you:
$$2x^2y^2 + 2x^2z^2 + 2y^2z^2 = 2x^2y^2 + 2x^2z^2 + 2y^2z^2$$