Research On Robust Control Of Polynomial Parameter-dependent Systems

At present, after decades of development, the robust control theory has made very rich research achievements. In the course of research for uncertain system, most of the domestic and overseas scholars solve the problem based on the quadratic stability. But the quadratic stability requirements a unity of Lyapunov function for all allow uncertain parameters, which the results will inevitably be introduced large conservatism. This issue has caused more and more attention from scholars.To solve this problem, Haddad, Gahinet and Feron etc in the mid1990s began to try to use parameters dependent Lyapunov function to solve uncertain system robust stability analysis. On this foundation, many scholars have made a series of creative research achievements, developed a new research direction for robust control, such as slack variable method, polynomial method with positive coefficient, D-Scale method, sum of squares method and so on.Now, the application of the parameters dependent Lyapunov function has been quite common, such as affined parameters dependent Lyapunov function and polynomial Lyapunov function. These methods applied to convex polytopic systems.However, some problems of not convex can be solved by linear matrix inequality (LMI) form, and some nonlinear problems can also be solved through the polynomial approximation. So the application of polynomial parameters-dependent Lyapunov functions in the uncertain systems not only has important theoretical significance, but also can further improve the possibility that the robust control theory is applied to industry. But research achievements of this field are very limited and the research results are not formed work detailed summary. Therefore, Our study will further promote the development of parameters dependent Lyapunov stability theory in this field, and it will inject new vitality for the development of robust control theory.In this paper, the robust control problem has been studied for parameters polynomial uncertain systems. The main contents:1. The robust stability analysis research of the single parameter polynomial type uncertain systems,and get the corresponding stability theorem.(1)study the time-invariant single parameter polynomial dependent linear continuous time uncertain systems robust stabi lity problem.Consider the time-invariant parameter polynomial dependent linear system that can be described by state-space equations of the form x(t)=A(δ)x(t),which A(δ):=A0+δA1+δ2A2+…+δLaALa is a polynomial parameter of order La and depended on the parameter δ.Aα={δ∈R:|δ|≤α} is an uncertain parameter set which is limited by α>O.Taken V(x,δ)=xTP(δ)x for the system candidate Lyapunov function and P(δ):=P0+δP1+δ2P2+…+δLpPLp is polynomial parameter depended on the parameter δ. Used by S-Procedure method to process parameter constraints.As any alternative positive definite matrix group S(x,δ):=∑i=1L(δix)TSi(δix) and any alternative negative definite matrix group T(x,δ):=∑i=1L(δix)TTix,we present a robust stability sufficient conditions of the system if there exist {Pi∈Sn,i=0,1,...,L},{Si∈Sn+,i=1,2...,L} and {Ti∈Tn,i=1,2,...,L} which the robust stability of the system is transformed into feasibility test of a pair of linear matrix inequalities (LMIs).(2)Study the robust stability of the time-varying single parameter polynomial dependent linear continuous uncertain system.Consider the time-varying parameter polynomial-dependent linear system that can be described by state-space equations of the form x(t)=A(δ(t))x(t)which A(δ(t)):=A0+δ(t)A1+δ(t)2A2+…+δ(t)LaALa is a polynomial parameter of order Lα and depended on the parameter δ(t).A,β={δ(·):|δ|≤α,|δ|≤β}.The theorem of a robust uniform-asymptotically stability and robust uniform-asymptotically stable with exponential rate γ are given.When the system is robust uniform-asymptotically stability under the condition of|δ|(t)|≤α*|δ|≤β,we present a method to seek the parameter vatiational α(3)Study the robust stability of the time-invariant single parameter polynomial linear discrete uncertain system.Consider the discrete uncertain system that can be described by state-space equations of the form xk+1=A(δ)xk which A(δ)=A0+δA1+δ2A2+…+δLaALa and Aa={δ∈R:|δ|≤α}. We use the same analytical method as the time-invariant single parameter polynomial dependent continuous time systems, the roubst stable problem is transformed into a set of linear matrix inequalities feasibility problem, the sufficient criteria is given.2. Study the single parameter polynomial dependent system L2gain analysisConsider the time-varying single parameter polynomial dependent systemThe matrices are polynomial parameter and depended on the parameter δ(t). In the case of Lb,Lc,Ld≤La, given γ>0, we propose a theorem for the system with given L2gain γ which is a LMI condition involving parameter-dependent Lyapunov. The theorem may be extended to be applied to the time-invariant system and discrete uncertain system.3. Design robust gain-acheduling controller of time-varying parameter polynomial dependent uncertain systemA time-varying parameter polynomial dependent system state-space model is considered We design a controller’s control law for the given scalar γ>0to guarantee the closed loop system asymptotically stability and let the system has the L2gain γ4. Study the robust stability of multi-parameters polynomial dependent uncertain system Consider the linear time-invariant two parameters polynomial uncertain system where x(·):[0,+∞)â†'Rn is the state, the systems matrix A(δ) satisfied while Aij∈R nxn. and α>0in system, as Aα={δ=[δ1δ2]T∈R2:|δi|≤α,i=1,2) Known A00∈H,taken V(x,δ)=xTP(δ)x for the system candidate Lyapunov function and which has two parameters. We propose a robust stability sufficient conditions of the system which the robust stability of the system is transformed into feasibility test of a pair of linear matrix inequalities (LMIs).The theorem may be applied to single parameter polynomial dependent system and discrete uncertain system.