Restricted accessResearch articleFirst published online 2007-6
Parameter-dependent adaptive H ∞ control design for a class of linear parameter-varying systems using polynomially parameter-dependent quadratic functions
This paper presents a new technique to design a parameter-dependent adaptive H∞ control for a class of linear parameter-varying (LPV) systems. It is assumed that the statespace matrices affinely depend on parameters that are not measurable in real-time for the control process. By introducing a Hamiltonian-Jacobi-Isaac (HJI) function and using the vector projection method, a sufficient condition is first established for the stability analysis problem in terms of a parameter-dependent linear matrix inequality (LMI). Then, by means of the polynomially parameter-dependent quadratic (PPDQ) functions, a parameter-independent LMI-based condition is derived, which enables an explicit expression to be found of the parameter-dependent adaptive H∞ control that guarantees both robust asymptotic stability and a prescribed level of disturbance attenuation for the system. Two numerical examples are given to illustrate the applicability of the proposed design approach.
ApkarianP.AdamsR. J.Advanced gain-scheduling techniques for uncertain systemsIEEE Trans. Control Systems Technol., 1998, 6, 21–32.
2.
ApkarianP.GahinetP. A.BeckerG.Self-scheduled H∞ control of linear parameter-varying systems: a design exampleAutomatica, 1995, 31 (7), 1251–1261.
3.
TanK.GrigoriadisK. M.WuF.H∞ and L2-to-L∞ gain control of linear parameter-varying systems with parameter-varying delaysIEE Proc. Control Theory Applic., 2003, 150, 509–517.
4.
ZhangX. P.TsiotrasP.KnospeC.Stability analysis of LPV time-delayed systemsInt. J. Control, 2002, 75, 538–558.
5.
BlimanP. A.Stabilization of LPV systems. In Proceedings of the 42nd IEEE CDC, Hawaii, 2003, pp. 6103–6108.
6.
ChesiG.GarulliA.TesiA.VicinoA.Polynomially parameter-dependent Lyapunov functions for robust stability of polytopic systems: An LMI approachIEEE Trans. Autom. Control, 2005, 59 (3), 365–370.
7.
FeronE.ApkarianP.GahinetP.Analysis and synthesis of robust control systems via parameter-dependent Lyapunov functionsIEEE Trans. Autom. Control, 1996, 41 (7), 1041–1046.
8.
GahinetP.ApkarianP.ChilaliM.Affine parameter-dependent Lyapunov functions and real parametric uncertaintyIEEE Trans. Autom. Control, 1996, 41 (3), 436–442.
9.
KarimiH. R.Robust stabilization with H∞ performance for a class of parameter-dependent systemsMath. Problems in Engng, 2006, 2006, Article ID 59867, 15 pages.
10.
ZhangX.TsiotrasP.IwasakiT.Parameter-dependent Lyapunov functions for exact stability analysis of single-parameter dependent LTI systems. In Proceedings of the 42nd IEEE CDC, Hawaii, 2003, Vol. 5, pp. 5168–5173.
11.
BlimanP. A.A convex approach to robust stability for linear systems with uncertain scalar parametersSIAM J. Control Optimization, 2004, 42 (6), 2016–2042.
12.
BlimanP. A.An existence result for polynomial solutions of parameter-dependent LMIsSystems and Control Lett., 2004, 51, 165–169.
13.
FrancisB. A.A course in H∞ control theory, 1986 (Springer-Verlag, Berlin).
14.
SchererC. W.Mixed H2/H∞ control for time-varying and linear parametrically varying systemsInt. J. Robust and Nonlinear Control, 1996, 6, 929–952.
15.
KarimiH. R.Robust dynamic parameter-dependent output feedback control of uncertain parameter-dependent state-delayed systemsNon-linear Dynamics and Systems Theory, 2006, 6 (2), 143–158.
16.
KarimiH. R.MaralaniP. J.LohmannB.MoshiriB.H∞ control of linear parameter-dependent state-delayed systems using PPDQ functionsInt. J. Control, 2005, 78 (4), 254–263.
17.
KarimiH. R.MoshiriB.MaralaniP. J.LohmannB.H∞ Control of linear parameter-dependent state-delayed systems. In Proceedings of the IEEE CCA, Toronto, Canada, 2005, pp. 1540–1545.
18.
DidinskyG.BasarT.Minimax adaptive control of uncertain plants In Proceedings of the 33rd IEEE CDC, Orlando, Florida, 1994, Vol. 3, pp. 2839–2844.
PanZ.BasarT.Adaptive controller design for tracking and disturbance attenuation in parametric strict-feedback nonlinear systemsIEEE Trans. Autom. Control, 1996, 43 (8), 1066–1083.
22.
YangX. H.WuF.PackardA.Adaptive control of full information In Proceedings of the ACC, Seattle, Washington, 1995, pp. 3371–3372.
23.
WangL. Y.ZhanW.Robust disturbance attenuation with stability for linear systems with norm-bounded nonlinear uncertaintiesIEEE Trans. Autom. Control, 1996, 41, 886–888.
24.
ZhouK.KhargonekarP. P.Robust stabilization of linear systems with norm-bounded time-varying uncertaintySystem Control Lett., 1988, 10, 17–20.
25.
ApkarianP.TuanH. D.Parameterized LMIs in control theorySIAM J. Control Optimization, 2000, 38 (4), 1241–1264.
26.
WuF.PrajnaS.A new approach to polynomial LPV system analysis and synthesis. In Proceedings of the ACC, Boston, 2004, Vol. 2, pp. 1362–1367.
27.
GahinetP.NemirovskyA.LaubA. J.ChilaliM.LMI control toolbox: for use with Matlab, 1995 (MathWorks Inc., Natik, Massachusetts).
28.
BoydS.GhaouiE. L.FeronE.BalakrishnanV.Linear matrix inequalities in systems and control theoryStudies in Applied Mathematics, 1994, Vol. 15 (SIAM, Philadelphia, Pennsylvania).
29.
GatermanK.ParilloP.Symmetry groups, semidefinite programs, and sums of squaresJ. Pure and Appl. Algebra, 2004, 192 (1-3), 95–128.
30.
BouaziziM.KochbatiA.KsouriM.H∞ control of LPV systems with dynamic output feedback. In 9th Mediterranean Conference on Control and automation, Dubrovnik, Croatia, 2001.
