A new probabilistic robust control approach for system with uncertain parameters. (English) Zbl 1338.93349
Summary: This paper addresses the issues of conservativeness and computational complexity of robust control. A new probabilistic robust control method is proposed to design a high performance controller. The key of the new method is that the uncertainty set is divided into two parts: \(r\)-subset and the complementary set of \(r\)-subset. The contributions of the new method are as follows: (i) a deterministic robust controller is designed for \(r\)-subset, so it has less conservative than those designed by using deterministic robust control method for the full set; and (ii) the probabilistic robustness of the designed controller is evaluated just for the complementary set of \(r\)-subset but not for the full set, so the computational complexity of the new method is reduced. Given expected probability robustness, a pertinent probabilistic robust controller can be designed by adjusting the norm boundary of \(r\)-subset. The effectiveness of the proposed method is verified by the simulation example.
MSC:
| 93E03 | Stochastic systems in control theory (general) |
| 93B35 | Sensitivity (robustness) |
| 93C41 | Control/observation systems with incomplete information |
| 93C15 | Control/observation systems governed by ordinary differential equations |
Keywords:
norm-bounded; parametric uncertainty; randomized algorithm; \(H_{\infty}\) control; probabilistic robustnessSoftware:
RACTReferences:
| [1] | SafonovM. G., “Origins of robust control: Early history and future speculations,” Ann. Rev. Control, Vol. 36, pp. 173-181 (2012). |
| [2] | CalafioreG. C., F.Dabbene, and R.Tempo, “Research on probabilistic method for control the system design,” Automatica, Vol. 47, No. 2, pp. 1279-1293 (2011). · Zbl 1219.93038 |
| [3] | ChenX., K.Zhou, and J.Aravena, “A new statistical approach for the analysis of uncertain systems,” J. Syst. Sci. Complex., Vol. 22, No. 1, pp. 1-34 (2009). · Zbl 1178.93082 |
| [4] | DoyleJ. C., J.Wall, and G.Stein, “Performance and robustness analysis for structured uncertainty,” Proc. IEEE Conf. Decis. Control, Orlando, FL, pp. 629-636 (1982). |
| [5] | BarmishB. R., C. M.Lagoa, and R.Tempo “Radially truncated uniform distributions for probabilistic robustness of control systems,” Proc. Amer. Control Conf., Albuquerque, NM, pp. 853-857 (1997). |
| [6] | FujisakiY., Y.Oishi, and R.Tempo, “Mixed deterministic/randomized methods for fixed order controller design,” IEEE Trans. Autom. Control, Vol. 53, No. 9, pp. 2033-2047 (2008). · Zbl 1367.93228 |
| [7] | VidyasagarM., and V.Blondel, “Probabilistic solutions to some NP‐hard matrix problems,” Automatica, Vol. 37, pp. 1397-1405 (2001). · Zbl 1031.93165 |
| [8] | LuB., and F.Wu, “Probabilistic robust linear parameter ‐ varying control of an F‐16 aircraft,” J. Guid. Control Dyn., Vol. 29, No. 6, pp. 1454-1459 (2006). |
| [9] | FujisakiY. and Y.Oishi, “Guaranteed cost regulator design: a probabilistic solution and a randomized algorithm,” Automatica, Vol. 43, pp. 317-324 (2007). · Zbl 1111.93085 |
| [10] | BaiE. W., R.Tempo, and M.Fu, “Worst‐case properties of the uniform distribution and randomized algorithms for robustness analysis,” Math. Control Signal Syst., Vol. 11, No. 3, pp. 183-196 (1998). · Zbl 0921.93007 |
| [11] | CalafioreG. C., “Random convex programs,” SIAM J. Optim., Vol. 20, No. 6, pp. 3427-3464 (2010). · Zbl 1211.90168 |
| [12] | AlamoT. and R.Tempo, “Randomized strategies for probabilistic solutions of uncertain feasibility and optimization problems,” IEEE Trans. Autom. Control, Vol. 54, No. 11, pp. 2545-2558 (2009). · Zbl 1367.90106 |
| [13] | ChenX., J.Aravena, and K.Zhou, “Risk analysis in robust control‐ making the case for probabilistic robust control,” Proc. Amer. Control Conf., Portland, OR, pp. 1533-1538 (2005). |
| [14] | PiccoliB., K.Zadamowska, and M.Gaeta, “Stochastic algorithms for robustness of control performances,” Automatica, Vol. 45, No. 6, pp. 1407-1414 (2009). · Zbl 1166.93387 |
| [15] | GahinetP. and P.Apkarian, “A linear matrix inequality approach to H_∞ control,” Int. J. Robust Nonlinear Control, Vol. 4, No. 4, pp. 421-448 (1994). · Zbl 0808.93024 |
| [16] | TrembaA., G.Calafiore, and F.Dabbeneet al., RACT: randomized algorithms control toolbox for MATLAB,” Proc 17th Int. Fed. Autom. Control, Seoul, Korea, pp. 390-395 (2010). |
| [17] | TempoR., G. C.Calafiore, and F.Dabbene, Randomized Algorithms for Analysis and Control of Uncertain the System, Springer Verlag, London, pp. 157-159 (2013). · Zbl 1280.93002 |
| [18] | AndersonB. D. O. and J. B.Moore, Optimal Control: Linear Quadratic Methods, Prentice Hall, Englewood Cliffs, NJ (1990). · Zbl 0751.49013 |
| [19] | ZhouK. and J. C.Doyle, Essentials of Robust Control, Prentice Hall, London, pp. 213-216 (1998). |
| [20] | CalafioreG. C. and F.Dabbene, “A probabilistic analytic center cutting plane method for feasibility of uncertain LMIs,” Automatica, Vol. 43, No. 12, pp. 2022-2033 (2007). · Zbl 1138.93027 |
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