Stability Analysis for Linear Systems under State Constraints

Transcription

1 Stabilit Analsis for Linear Sstems under State Constraints Haijun Fang Abstract This paper revisits the problem of stabilit analsis for linear sstems under state constraints New and less conservative sufficient conditions are identified under which such sstems are globall asmptoticall stable Based on these sufficient conditions, iterative LMI algorithms are proposed for testing global asmptotic stabilit of the sstem In addition, these iterative LMI algorithms can be adapted for the design of globall stabilizing state feedback gains I INTRODUCTION AND PROBLEM STATEMENT In this paper, we will investigate stabilit analsis of two classes of linear sstems under state constraints, which were recentl studied in 4, 6, 7, 8, The first class of sstems are defined as follows, ẋ = hax), ) where x D n = {x =x,x 2,,x n ) T R n : x i i,n}, A =a ij R n n, and ) h n j= a jx j ) h 2 n j= hax)= a 2 jx j ) h n n j= a njx j Let r i = n j= a ijx j, then, for each i,n, {, if xi = and r h i r i )= i x i >, r i, otherwise Such sstems are defined on a closed hpercube as all state variables are constrained to the unit hpercube D n For this reason, sstem ) is sometimes referred to as a linear sstem subject to state saturation Clearl, ) saturation occurs in the state x i if x i = and n j= a ijx j x i > The other class of sstems are sstems with partial state constraints and are described as { ẋ = Ax B, 2) ẏ = hcx E), where x R n m with n m, {, 2,, m ) T : i, i,m}, A,B,C and E are real matrices of appropriate dimensions, and ) h n m j= c jx j m k= e k k ) h 2 n m hcx E)= j= c 2 jx j m k= e 2k k 3) ) h m n m j= c mjx j m k= e mk k Department of Electrical and Computer Engineering, Universit of Virginia, POBox 4743, Charlottesville, VA , USA {hf2r,zl5}@virginiaedu Work supported in part b the National Science Foundation under grant CMS Zongli Lin Let s i = n m j= c ijx j m k= e ik k ), then, for each i,m, {, i = and s h i s i )= i i > 4) s i, otherwise We note that the class of sstems 2) reduces to the class ) if m = n These two classes of sstems are encountered in a variet of applications, including signal processing, recurrent neural networks and control sstems, and have been studied extensivel see, eg, 3, 4, 5, 6, 8, 2 and the references therein) In this paper, we revisit the problem of stabilit analsis for these two classes of sstems In particular, we are interested in conditions under which such sstems are globall asmptoticall stable at the origin Here, b global asmptotic stabilit of the origin we mean that the origin is locall asmptoticall stable within D n or R n m D m ), rather than the usual R n, being the domain of attraction Global asmptotic stabilit of these sstems has been studied in 4, 8, For second order sstems in the form of ), necessar and sufficient conditions for global asmptotic stabilit were established in 4, For higher order sstems in the form of either ) or 2), various sufficient conditions for the global asmptotic stabilit were identified Under the sufficient condition of 8, an sstem trajector starting from inside D n will never reach the boundar of D n, ie, the state never saturates This saturation avoidance sufficient condition leads to a degree of conservatism Using a Lapunov function V : D n R that satisfies V x x) V hax) x x) Ax, 5) 4 arrives at a sufficient condition that is less conservative than that of 8 Motivated b the observation that the hpothesis 5) might be a source of conservatism, we will in this paper re-examine global asmptotic stabilit of such sstems b exploring the special propert of the function h The sufficient conditions we thus arrive at are given in terms of matrix inequalities, which are shown to be less conservative than those of 8 and 4 Based on these new sufficient conditions, iterative LMI algorithms are proposed for testing global asmptotic stabilit In addition to the stabilit analsis, the proposed sufficient conditions and the iterative LMI algorithms can be readil adapted for designing globall stabilizing feedback gains for the following sstems: where x R n and u R m ẋ = hax Bu), u = Fx, 6)

2 This paper is organized as follows The main result is presented in Section II Numerical examples are given in Section III A brief concluding remark is made in Section IV II STABILITY ANALYSIS In this section, we will establish new sufficient conditions for global asmptotic stabilit for both the sstem ) and 2) To this end, we first establish some technical lemmas Lemma : Consider a nonlinear sstem ẋ = f x), x Ω R n, with f )= Assume that all trajectories remain inside Ω If there exists a function V : Ω R such that and φ x ) Vx) φ 2 x ), x Ω, Vx) φ 3 x ), x Ω, for some class functions φ,φ 2 and φ 3, then the origin is globall asmptoticall stable in the sense that the origin is locall asmptoticall stable with Ω being the domain of attraction Proof The main idea of the proof comes from 4 Under these conditions, it follows from the standard Lapunov theor that the sstem is locall asmptoticall stable with a neighborhood of the origin S Ω contained in the domain of attraction Let d = min x,x S, where S denotes the boundar of the set S Then, an trajector starting from Ω \ S will remain in Ω and enter S at some finite time t and converge to the origin asmptoticall Otherwise, we must have xt) > d for all t, and Vxt)) = Vx)) Vx)) t t Vxτ))dτ φ 3 xτ) )dτ Vx)) tφ 3 d), t, which is a contradiction to the fact that Vxt)), t Recall that for a group of points, u,u 2,,u, their convex hull is defined as, co { u i : i, } { } := α i u i : α i =,α i i= i= Lemma 2: 9 Let u,u,u 2,,u R m, v,v,v 2,,v R m 2, if u co { u i : i, } and v co { v i : i, }, then { } u u i co v v j : i,, j, Let n be the set of n n diagonal matrices whose diagonal elements are either or n contains 2 n elements Let D i be an element of n, denote D i = I D i Definition : A matrix M =m ij R n n is said to be row) diagonall dominant if m ii > n j=, j i m ij, i,n Lemma 3: Let G =g ij R n n be row) diagonall dominant and the diagonal be composed of negative elements ie, g ii < for all i,n) Then, haxk) co { D i Ax K)D i Gx,i,2n }, x D n, for an matrix K R n independent of x Proof Since G is row) diagonall dominant and g ii <,i,n, for an x D n, G i x < ifx i =, or G i x > ifx i = In the absence of state saturation, ie, ha i xk i )=A i xk i, it is obvious that ha i x K i ) co{a i x K i,g i x} Inthe event of state saturation, ha i x K i )=, either when x i = and A i x K i >, or when x i = and A i x K i < When x i = and A i xk i >, G i x < and hence, ha i xk i )= co{a i xk i,g i x} Similarl, when x i = and A i xk i <, G i x > and hence ha i x K i )= co{a i x K i,g i x} It then follows from Lemma 2 that haxk) co { D i Ax K)D i Gx,i,2n }, x D n We are now read to establish a new sufficient condition under which the sstem ) is globall asmptoticall stable at the origin Theorem : If there exist a smmetric positive definite matrix P R n n and a G R n n such that D i A D i G)T P PD i A D i G) <, i,2n, 7) and G is row) diagonall dominant with negative diagonal elements, then the sstem ) is globall asmptoticall stable at the origin Proof Let Vx)=x T Px Wehave Vx)=hAx) T Px x T PhAx) Since the matrix G is row) diagonall dominant and the diagonal is composed of negative elements, b Lemma 3, hax) co { D i Ax D i Gx,i,2n }, x D n It then follows that Vx) max i,2 n xt D i AD i G)T PPD i AD i G))x, x Dn Condition 7) then implies that V max i,2 n xt D i A D i G)T P PD i A D i G))x δx T x, x D n, for some δ > The results of the theorem then follow from Lemma Remark : In 8, it is established that the sstem ) is globall asmptoticall stable at the origin if A is row) diagonall dominant and the diagonal is composed

3 of negative elements This condition on A implies that A is Hurwitz stable and hence there is a positive definite matrix P such that A T P PA <, 8) which can be written as 7) with G = A Thus, the sufficient condition established in Theorem is less conservative than the sufficient condition of 8 Remark 2: In 4, it is established that the sstem ) is globall asmptoticall stable at the origin if there exists a smmetric positive definite matrix P such that 8) is satisfied and n p ii p ij, i,n 9) j,n, j i Existence of such a P implies the existence of a P that satisfies 8) and 9) with in 9) replaced with > Indeed, if P satisfies 8) and 9), then, for a sufficientl small ε, P εi will satisfies 8) and 9) with > in 9) Let P be such that it satisfies 8) and p ii > p ij, i,n ) j,n, j i Define a = max i,n j,n b = min i,n p ii a ij, ) p ij j,n, j i Obviousl, a > and b > Let c > a b, then A cp is row) diagonall dominant and the diagonal is composed of negative elements Inequalit 8) then implies D i AD i PPD A cp))t i AD i A cp)) <,i,2n, ) which is 7) with G = A cp We thus see that the sufficient condition of Theorem is less conservative than the sufficient condition of 4 In what follows, we will follow the idea of 2 to propose an iterative LMI algorithm for verifing the sufficient condition of Theorem Let be the set of n-dimensional row vectors in which there is onl one nonzero element which is Denote h i,i,n, as an element of in which the ith element is Let i be the set of n- dimensional column vectors in which the ith element is and other elements are either or The elements of i are denoted as ij, j,2 n Then, the condition that G is row) diagonall dominant and the diagonal is composed of negative elements can be expressed as the following LMIs, h i G ij <, i,n, j,2 n 2) Algorithm : Global Asmptotic Stabilit of Sstem ) Step Select a Q >, and solve P from the following Lapunov equation, A T P PA = Q Set k = Step 2 Using P obtained previousl, solve the following LMI optimization problem for G and α, st inf G α D i A D i G)T P PD i A D i G) < αp, i,2 n, h i G ij <, i,n, j,2 n If k = and α, go to Step 4 If k >, α or α α k, go to Step 4 Otherwise, set k = k,α k = α, go to the next step Step 3Using G obtained in the previous step, solve the following LMI optimization problem for P and α, st P> D i A D i P PD G)T i A D i G) < αp, i,2 n If α orα α k, go to Step 4 Otherwise, let k = k, α k = α, go to Step 2 Step 4If α, the sstem ) is globall asmptoticall stable at the origin Otherwise, no conclusion can be drawn A different Q ma be selected and the algorithm ma be repeated from Step Remark 3: We note that solutions to the LMI optimization problems in Algorithm alwas exist and α k is nonincreasing However, the number of constraints increases exponentiall as n, the order of the sstem increases The large number of constraints ma cause numerical difficulties B viewing F as an additional variable, Algorithm can also be readil adapted for the design of a globall stabilizing feedback law u = Fx for the sstem 6) In particular, we have the following algorithm Algorithm 2: Design of Globall Stabilizing Feedback Gain F Step Select a Q >, and solve P from the following Lapunov equation, A BF) T P PA BF)= Q, where F is chosen such that A BF is Hurwitz Set k = Step 2 Using P obtained previousl, solve the following LMI optimization problem for G, F and α, G st D i A BF)D i G)T P PD i A BF)D i G) < αp, i,2n, h i G ij <, i,n, j,2 n If k = and α, go to Step 4 If k >, α or α α k, go to Step 4 Otherwise, set k = k,α k = α, go to the next step

4 Step 3 Using G and F obtained in the previous step Solve the following LMI optimization problem for P and α, P> st D i A BF)D i G)T P PD i A BF)D i G) < αp, i,2n If α orα α k, go to Step 4 Otherwise, set k = k, α k = α, go to Step 2 Step 4If α, the sstem 6) is globall asmptoticall stable at the origin And the current F is the calculated feedback gain Otherwise, no conclusion can be drawn A different Q ma be selected and the algorithm ma be repeated from Step We next consider the second class of sstems 2) Again, our interest here is to establish conditions under which the sstem is globall asmptoticall stable at the origin We have the following result Theorem 2: If there exist a smmetric positive definite matrix P R n n and a G R m m such that T D i C D i ED i G PP D i C D i ED <, i G i,2 m, 3) where D i m and G is row) diagonall dominant and the diagonal is composed of negative elements, then the sstem 2) is globall asmptoticall stable at the origin Proof Let Vx,)= x T x T P Then, Vx,)= x T Ax B T P hcx E) Ax B) T hcx E) x T P Recalling Lemma 3, we have hcx E) co { D i Cx E)D i G}, i,2 m, and hence, V x,) max x T P T i,2 m D i C D i E D i G T x P) D i C D i E D i G B 3), we have Vx,) δ x T x T, for some constant δ > It then follows from Lemma that the sstem 2) is globall asmptoticall stable at the origin Remark 4: In 4, it is concluded in Theorem 3 that the sstem 2) is globall asmptoticall stable at the origin if there exist smmetric positive matrices P R n m) n m), P 2 R m m and Q R n n, with P 2 satisfing 9), such that T P P < Q P 2 C E C E P 2 4) As explained in Remark 2, it is without loss of generalit to assume that P 2 satisfies ) Let a = max i,n j,n b = min i,n e ij, p 2ii ) p 2ij j,n, j i Then, choose c > a b to be large enough such that E cp 2 is row) diagonall dominant and the diagonal is composed of negative elements, and Q c CT C c E T E 5) Because for an x T, T ) T R n \{}, the following inequalit is valid, c T E T E c T P 2 D i P 2 c Cx T c cd i P 2 ) Cx ) cd i P 2 <, i,2 m, 6) which implies c C T T C P c E T E D i C D i cp 2 P 2 P P 2 D i C,i,2 m 7) D i cp 2 Therefore, b 5) and 7), the inequalit 4) implies T P P 2 D i C E D i cp 2 P P 2 D i C E D i cp 2 <,i,2 m,8) P which is 3) with G = E cp 2 and P = P 2 Therefore, we can see that Theorem 3 of 4 is more conservative than Theorem 2 We next propose an iterative LMI algorithm for the verification of the conditions of Theorem 2 Algorithm 3: Global Asmptotic Stabilit of Sstem 2) Step Select a Q >, and solve the following Lapunov equation for P >, T P P = Q C E C E Set k =

5 Step 2 Using P obtained previousl, solve the following optimization problem for G and α, st G T D i C D i E D i G P P D i C D i E D i G < αp, i,2 m 2) If k = and α, go to Step 4 If k >, α or α α k, go to Step 4 Otherwise, set k = k,α k = α, go to the next step Step 3Let G = G k Solve the following optimization problem for P and α, Fig A trajector of the sstem 9) 5 st P> T D i C D i E D i G P P D i C D i E D i G < αp, i,2 m 5 5 If α orα α k, go to Step 4 Otherwise, let k = k, α k = α, go to Step 2 Step 4If α, the sstem 2) is globall asmptoticall stable at the origin Otherwise, No conclusion can be drawn A different Q ma be selected and the algorithm ma be repeated from Step t III NUMERICAL EXAMPLES 2 Example Consider the following sstem 25 ẋ = hax)=h x 9) 2 2 The matrix A is not row) diagonall dominant, hence no conclusion can be drawn based on the condition of 8 It can also be verified that the condition 8)-9) from 4) can t be satisfied either However 7) of Theorem is satisfied with P = G = , Thus, b Theorem, the sstem is globall asmptoticall stable A trajector of 9) is shown in Fig Fig 2 shows that Theorem relaxes the hpothesis 5) from 4) Example 2 Consider the following sstem 2 ẋ = hax Bu)=h x 2 ) Fx 2) Fig 2 line) V x haxt))solid xt)) line) vs V x Axt)dash-dotted xt)) This sstem is open loop unstable as the matrix A is unstable Using Algorithm 2, we obtain a globall stabilizing feedback gain F as with F = , P =, G = t

6 A trajector of the closed-loop sstem 2) is shown in Fig Fig 3 A trajector of the sstem 2) Example 3 Consider the following sstem ẋ = x 5 5, 2 ẏ = h x ) 2) It can be verified that the sstem 2) does not satisf the condition of 4 and hence no conclusion on the global asmptotic stabilit can be drawn based on the results of 4 However, b using Algorithm 3, it can be verified that the condition of Theorem 2 is satisfied with P = G = , 22) We thus conclude that the sstem 2) is globall asmp- linear sstems with partial state saturation are globall asmptoticall stable at the origin These conditions were shown to be less conservative than the existing conditions Based on these conditions, iterative LMI algorithms are proposed for verifing global asmptotic stabilit of these sstems Numerical examples were used to show the effectiveness of the proposed algorithms REFERENCES S Bod, L El Ghaoui, E Feron and V Balakrishnan, Linear Matrix Inequalities in Sstems and Control Theor, SIAM Studies in Appl Mathematics, Philadelphia, Y-Y Cao, J Lam and Y-X Sun, Static output feedback stabilization: an ILMI approach, Automatica Vol 34, No 2, pp , JJ Hopfield, Neural computation of decisions in optimization problems, Biol Cbern, Vol 52, pp 4-52, L Hou and AN Michel, Asmptotic Stabilit of Sstems with Saturation Constraints, Proc of the 35th IEEE Conference on Decision and Control, pp , J-H Li, AN Michel and W Porod, Analsis and snthesis of a class of neural networks: linear sstems operating on a closed hpercube, IEEE Transactions on Circuits and Sstems, Vol 36, pp , D Liu and AN Michel, Asmptotic stabilit of sstems operating on a closed hpercube, Sstems & Control Letters, Vol 9, pp , D Liu and AN Michel, Stabilit analsis of sstems with partial state saturation nonlinearities, Proceedings of the 33rd IEEE Conference on Decision and Control, pp 3-36, D Liu and AN Michel, Dnamical Sstems with Saturation Nonlinearities: Analsis and Design, Springer, London, T Hu and Z Lin, Control Sstems with Actuator Saturation: Analsis and Design, Birkhäuser, Boston, 2 R Mantri, A Saberi and V Venkatasubramanian, Stabilit analsis of continuous time plannar sstems with state saturation nonlinearit, Proc of IEEE ISCAS 96, pp 6-63, Atlanta, GA, 996 RK Miller and AN Michel, Ordinar Differential Equations, Academic Press, New York, DW Tank and JJ Hopfield, Simple neural optimization networks: an A/D converter, signal decision circuit, and a linear programming circuit, IEEE Transactions on Circuits and Sstems, Vol 33, pp , Fig 4 A trajector of the sstem 2) toticall stable at the origin A trajector is shown in Fig 4 IV CONCLUSIONS In this paper, we established simple conditions under which linear sstems defined on a closed hpercube and