Non-quadratic Lyapunov functions for performance analysis of saturated systems

Transcription

1 Non-quadratic Lyapunov functions for performance analysis of saturated systems Tingshu Hu, Andrew R. Teel and Luca Zaccarian Abstract In a companion paper [4, we developed a systematic Lyapunov approach to the regional stability and performance analysis of saturated systems via quadratic Lyapunov functions. The corresponding conditions are expressed in terms of LMIs but can be too conservative in some cases. To obtain less conservative conditions, we use in this paper two types of conjugate Lyapunov functions: the convex hull quadratic function and the max quadratic function. These functions yield bilinear matrix inequalities (BMIs) as conditions for stability and guaranteed performance level. The BMI conditions cover the LMI conditions for quadratic stability as special cases and hence the reduction of conservativeness is guaranteed. A numerical example demonstrates the effectiveness of this paper s methods and the great potential of the non-quadratic Lyapunov functions. keywords: saturation, deadone, nonlinear L gain, reachable set, domain of attraction, Lyapunov functions. I. INTRODUCTION Saturation nonlinearities are very common in control systems. The development of analysis and synthesis tools for stability and performance within this context has captured increasing attention from the control community in the past ten years (see the companion paper [4 for some key references in this field). One important approach to characteriing stability and performances for systems with saturation is the Lyapunov approach. Generally the Lyapunov approach consists of two main steps. In the first step the saturation or the deadone functions are bounded locally or globally with sectors. As a result, the system is described with a linear differential inclusion (LDI). In the second step, the LDI or the system satisfying a sector condition is analyed with tools based on (or extended from) absolute stability theory which uses Lyapunov functions to characterie stability or performances. Many existing results on systems with saturation adopt the Lyapunov approach and most of them use quadratic Lyapunov functions. While significant results have been developed for the characteriation of global stability and L gain, more recent efforts have been devoted to regional Work supported in part by AFOSR grant number F , NSF under Grants ECS and ECS , by ENEA-Euratom, ASI and MIUR under PRIN and FIRB projects. A.R. Teel is with the Department of Electrical and Computer Engineering, University of California, Santa Barbara, CA 9306, USA teel@ece.ucsb.edu T. Hu is with the Dept. of ECE, Umass Lowell, Lowell, MA 0854, USA tingshu hu@uml.edu L. Zaccarian is with the Dipartimento di Informatica, Sistemi e Produione, University of Rome, Tor Vergata, 0033 Rome, Italy ack@disp.uniroma.it stability and performance analysis (see the extensive references in [4). For example, in our recent work [3, we characteried nonlinear L gains for saturated systems, and used an example to illustrate that the global characteriation of the L gain can be misleading when the system operates within a bounded region. The regional performance analysis in [3 was made possible through an effective tool of describing a saturated system with a parameteried LDI. In our companion paper [4, we enhanced the LDI description by proposing two forms of parameteried LDIs. One of them is polytopic LDI (PLDI) and the other is norm-bounded LDI (NLDI). The NLDI is derived from the PLDI and is generally more conservative but could be easier to handle in some cases. It turns out that the analysis results based on the NLDI is equivalent to those in [3. With these LDI descriptions, there is yet another great potential to be explored in the second step about the analysis of LDIs. It is now generally accepted that quadratic Lyapunov functions could be very conservative even for stability analysis of LDIs (see, e.g., [3, [5, [5). For this reason, considerable attention has been paid to the construction and development of non-quadratic Lyapunov functions (e.g., see [, [, [3, [5, [6, [7, [0). In [8, piecewise quadratic Lyapunov functions were used for the study of saturated systems. Recently, two types of conjugate Lyapunov functions have both demonstrated a great potential in the analysis of LDIs and saturated linear systems [7, [6, [9, [. One is called the convex hull quadratic function since its level set is the convex hull of a family of ellipsoids. The other is called max quadratic function since it is obtained by taking the pointwise maximum over a family of quadratic functions and its level set is the intersection of a family of ellipsoids. Some conjugate relationships about these two functions were established in [7, [6. Since these functions are natural extensions to quadratic functions, they can also be used to perform quantitative performance analysis beyond stability, such as to estimate the L gain, the reachable set, and the dissipativity, for LDIs. A handful of dual bilinear matrix inequalities (BMIs) have been derived for these purposes in [6. As compared to the corresponding LMIs resulting from quadratic Lyapunov functions, these BMIs contain extra degrees of freedom in the bilinear terms, which are injected through the non-quadratic functions. Experience with low order systems shows that these BMIs can be effectively solved with the path-following method in [8. Although it is possible that numerical difficulties may arise for higher order systems, the great potential of these non-quadratic Lyapunov functions has been demonstrated in [7, [6, [ through

2 a set of numerical examples. In this paper, we will use these two conjugate Lyapunov functions to enhance regional performance analysis of saturated systems. This paper is organied as follows. In Section I-A we describe the problems to be studied and briefly summarie the key tools developed in [4 which are crucial to the description of a saturated system with Polytopic Differential Inclusions (PDI). Section II contains the main results on the characteriation of stability and performances via the two non-quadratic Lyapunov functions. Section III uses the same example from [4 to show that tighter estimations of the nonlinear L gain can be achieved by using the non-quadratic functions. The paper is concluded with proofs of key and technical results. Notation - I[k, k : For two integers k, k, k < k, I[k, k = k, k +,, k }. - sat( ): The standard saturation function. For u R m, [sat(u) i = sign(u i ) min, u i }. - d(u): The deadone function, d(u) = u sat(u). - co S: The convex hull of a set S. - HeX: For a square matrix X, HeX := X + X T. - E(P ): For P R n n, P = P T > 0, - L(H): For H R m n, E(P ) := x R n : x T P x }. L(H) := x R n : Hx }. About the relationship between E(P ) and L(H), for a given s > 0, we have (see, e.g., [), [ /s H se(p ) L(H) l 0 () P H T l for all l I[, m, where H l is the lth row of H. A. Problem statement The type of closed-loop system that we address in this paper corresponds to the following general representation of a linear system subject to saturation: ẋ = Ax + B q q + B w w y = C y x + D yq q + D yw w = C x + D q q + D w w q = d(y). where x R n, q, y R m, w R r, R p and d is the standard vector-valued deadone function. This system can be graphically depicted as in Fig., where w is the exogenous input or disturbance and is the performance output. Many linear systems with saturation/deadone components Fig.. w q H d Compact representation of a system with saturation/deadone. y c () can be transformed into the above general form through loop transformation. Due to this fact, this type of closedloop system has received great attention from the control community over the past decade. In most of the literature, various restrictive assumptions are made on the general configuration (), such as the absence of algebraic loops (namely, D yq = 0), or the exponential stability of certain subsystem (typically, an open-loop plant driven by saturated signals), and so on. More detailed discussions on the background can be found in our companion paper [4. What distinguishes our current effort from most of the existing results is that the only assumptions we make on system () are that A is Hurwit and that the nonlinear algebraic loop is well posed (these are clearly basic requirements for the system to be functional). In [4, great attention has been devoted to the characteriation of the well-posedness of () and the development of two forms of parameteried differential inclusions for (): the polytopic differential inclusion (PDI) and the norm-bounded differential inclusion (NDI). In this paper, we will continue to use the PDI description, which is summaried in the following proposition. Proposition : (Polytopic differential inclusion (PDI)) Let K i : i I[, m } be the set of diagonal m m matrices with 0 or at the diagonal elements. For i I[, m, denote T i = (I K i D yq ) K i, A i = A + B q T i C y, B i = B w + B q T i D yw, C i = C + D q T i C y, D i = D w + D q T i D yw. Let h : R n R m be a given map and let h l be the lth component of h. For system (), if h l (x) for all l I[, m, then [ ẋ co i I[, m [ Ai x + B i w B q T i h(x) C i x + D i w D q T i h(x) }. (3) The NDI description in [4 is more conservative than (3) but may simplify the computation under certain situations. However, we will not consider the NDI description in this paper since detailed investigation reveals that the two nonquadratic Lyapunov functions will yield the same results as those by quadratic functions when NDIs are concerned. We will consider the same stability and performance analysis problems as in [4. Instead of using quadratic Lyapunov functions as in [4, we will apply non-quadratic Lyapunov functions to address the following problems:. Estimation of the domain of attraction: (in the absence of w) by using invariant level sets of the non-quadratic Lyapunov functions.. Estimation of the nonlinear L gain from w to : With w s for a given s, we would like to determine a number γ > 0 as small as possible, so that under the condition x(0) = 0, we have γ w. Performing this analysis for each s (0, ), we obtain an estimate of the nonlinear L gain. 3. Estimation of the reachable set (from L bounded inputs): With a given bound on the L norm of w, i.e, w s for a given s, we would like to determine a set S as small as possible so that under the condition

3 x(0) = 0, we have x(t) S for all t. This set S will be considered as an estimate of the reachable set. II. ANALYSIS WITH NON-QUADRATIC LYAPUNOV FUNCTIONS In this section, we will use a pair of conjugate functions, the convex hull quadratic function and the max quadratic function to perform stability and performance analysis of system (). We first review some results about this pair of conjugate functions. A. The max quadratic function and the convex hull quadratic function Given a family of positive definite matrices P j R n n, P j = Pj T > 0, j I[, J, the pointwise maximum quadratic function is defined as V max (x) := maxx T P j x : j I[, J}. (4) Given Q j R n n, Q j = Q T j > 0, j I[, J. Let } Γ = γ R J : γ + γ + + γ J =, γ j 0. The convex hull quadratic function is defined as V c (x) := min γ Γ xt γ j Q j x. (5) For simplicity, we say that V c is composed from Q j s. It was shown in [7 that V max is conjugate to V c if Q j = P j for each j I[, J. It is evident that V c and V max are homogeneous of degree, i.e., V c (αx) = α V c (x), V max (αx) = α V max (x). Also established in [7, [9 are that V c is convex and continuously differentiable and that V max is strictly convex. The -level set of V max and that of V c are respectively } L Vmax := x R n : V max (x), } L Vc := x R n : V c (x). Since V max and V c are homogeneous of degree, we have sl Vmax = x R n : V max (x) s }, sl Vc = x R n : V c (x) s }. It is easy to see that L Vmax is the intersection of the ellipsoids E(P j ) s. In [9, It was established that L Vc is the convex hull of the ellipsoids E(Q j ) s, i.e., L Vc = γ j x j : x j E(Q j ), γ Γ. For a compact convex set S, a point x on the boundary of S (denoted as S) is called an extreme point if it cannot be represented as the convex combination of any other points in S. For a strictly convex set, such as L Vmax, every boundary point is an extreme point. In what follows, we characterie the set of extreme points of L Vc. Since L Vc is the convex hull of E(Q j ), j I[, J, an extreme point must be on the boundaries of both L Vc and E(Q j ) for some j I[, J. Define E j := L Vc E(Q j ), namely E j = x R n : V c (x) = x T Q j x = }. (6) Then J E j contains all the extreme points of L Vc. The exact description of E j is given by the following lemma (which is proved in the Appendix). Lemma : For each j I[, J, E j =x L Vc : x T Q j (Q k Q j )Q j x 0, k I[, J}. It is clear from Lemma that δ [0, δe j =x L Vc : x T Q j (Q k Q j )Q j x 0, k I[, J}. The following lemma combines results from [9, [0. Lemma : For a given x 0 ( R n, let γ Γ J be an optimal γ such that x T 0 γ j j) Q x0 = ( J ) min γ Γ x T 0 γ jq j x0 = V c (x 0 ). Assume that γj > 0 for j I[, J 0 and γj = 0 for j I[J 0 +, J. Denote J 0 Q 0 = γj Q j, x j = Q j Q 0 x 0, j I[, J 0. Then V c (x j ) = V c (x 0 ) and x j V c (x 0 ) E j, j I[, J 0. Moreover, x 0 = J 0 γ j x j, and V c (x 0 ) = V c (x j ) = Q j x j = Q 0 x 0, j I[, J 0, where V c (x) denotes the gradient of V c at x. The following lemma is adapted from a result of [ to the slightly different definition of V c and V max (the two functions in [ have the coefficient ). Lemma 3: [ Let H R m n and denote the l-th row of H as H l. We have, ) L Vc L(H) if and only if Hl T L Vmax for all l I[, m; ) L Vmax L(H) if and only if Hl T L V c for all l I[, m. B. Analysis with convex hull quadratic functions In this section, we apply the convex hull quadratic function to the analysis of system () through the polytopic differential inclusion (3), which is repeated below for easy reference: [ ẋ co i I[, m [ Ai x + B i w B q T i h(x) C i x + D i w D q T i h(x) }. (7) This PDI is a valid description for () as long as h(x). We will restrict our attention to the level set sl Vc, where h(x) for all x sl Vc. As with the case of using quadratic functions, the crucial point is to guarantee that x(t) sl Vc under the class of norm-bounded w and the set of initial states under consideration. It may appear that choosing h(x) as a linear function Hx within sl Vc should lead to simpler results than choosing it as a nonlinear function. However, it turns out that a nonlinear h(x) not only reduces conservatism but also leads to cleaner and numerically more tractable results. Theorem : (Reachable set by bounded inputs) Given Q j = Q T j > 0, j I[, J and let V c be composed from Q j s. Also given s > 0. For system (), with x(0) = 0, we have x(t) sl Vc for all t > 0 and for all w such

4 that w s if there exist Y j R m n and λ ijk 0, 0, j I[, J and let V c be composed from Q j s. Consider i I[, m, j, k I[, J such that system (). Given s, γ > 0. If there exist Y j R m n and [ λ ijk 0, i I[, m, j, k I[, J such that s Y j,l 0, l I[, m, j I[, J, (8) [ Yj,l T Q j s Y j,l 0, l I[, m, j I[, J. (0) where Y j,l is the lth row of Y j, and such that for all i Yj,l T Q j I[, m, j I[, J, and such that He iq j B q T i Y j + λ ijk (Q j Q k ) B i A i Q j B q T i Y j + λ ijk (Q j Q k ) B i 0 0. (9) k= He k= 0 I 0 I 0 0, Remark : (Optimiation issues) With conditions (9) and C i Q j D q T i Y j D i γ I () (8), we may formulate an optimiation problem to minimie for all i I[, the estimation of the reachable set as with the quadratic function case. We observe that the first block in (9) contains some s and x(0) = 0, we have γ w. m, j I[, J. Then for all w such that w bilinear terms as the product of a full matrix and a scalar. From our computational experience, such kind of bilinear matrix inequalities (BMIs) can be effectively addressed with The proof of the following theorem can be adapted from the proof of Theorem by assuming that B i = 0. Theorem 3: (Estimation of the domain of attraction) the path-following method in [8. We also see that if we take Given Q j = Q T j > 0, j I[, J. Let V c be composed Q j = Q and Y j = Y for all j, then the bilinear terms vanish from Q j s. Consider system (). With w 0, we have and the conditions reduce to LMIs (these LMIs coincide with V c (x) < 0 for all x L Vc \ 0} if there exist λ ijk 0, the ones given in [4, Theorem ). In our computation, Y j R m n, i I[, m, j, k I[, J such that [ we first solve the resulting optimiation problem with LMI Yj,l constraints and then use the optimal Q and Y to start the Y T 0, l I[, m, j I[, J. () new algorithm with BMI constraints, with Q j = Q j,l Q j and Y j = Y and that for all i I[, for all j and λ ijk 0 randomly chosen. This, j I[, J, approach also proves effective for the problems of estimating He(A the L gain and the domain of attraction, which will be i Q j B q T i Y j + λ ijk (Q j Q k )) < 0. (3) k= addressed in Theorems and 3. Although there is no guarantee that the global optimal C. Analysis with max quadratic functions solution can be located, the convergence of the algorithms is satisfactory. Furthermore, since the initial value of the optimiing parameters can be inherited from the optimal solution obtained with quadratic Lyapunov functions, the algorithms always improve on the results from using the The max quadratic function is not differentiable everywhere. We use V max (x) to denote its generalied Jacobian at x (see, e.g., [4). If x T P j x > x T P k x for all k j, then V max is differentiable at x and V max (x) is single valued and equals P j x. If x T P x = x T P x = = x T P J0 x quadratic functions proposed in [4. To avoid redundancy, and x T P x > x T P j x for j > J 0, then V max (x) = we will not discuss the computational issues after we present cop j x : j I[, J 0 }. For simplicity and with some abuse Theorems or 3. of notation, along the trajectory of (), we denote Remark : (About the nonlinear function h(x)) From the V max := maxξ T ẋ : ξ V max (x)}. proof of Theorem, we see that a nonlinear function h(x 0 ) = H 0 (x 0 )x 0 is constructed from Q j s and Y j s so Theorem 4: (Reachable set by bounded inputs) Given that H 0 (x 0 )x 0 for all x 0 sl Vc (see (3) where P j = Pj T > 0, j I[, J and let V max be the max quadratic H 0 is constructed and subsequent discussion up to (6)). function formed by P j s. Also given s > 0. For the system This makes the proof more complicated than with a linear (), with x(0) = 0, we have x(t) sl Vmax for all t > 0 function Hx but the result turns out to be cleaner and more and for all w such that w s if there exist H R m n, easily tractable numerically. If we attempt to use a linear λ ijk 0, α lj 0, j, k I[, J, i I[, m, l I[, m, function h(x) = Hx such that Hx for all x sl Vc, such that J α lj =, we would have Y j in (9) replaced with HQ j and Y j,l in (8) replaced with H l Q j,l. When we formulate an optimiation s H l problem to estimate the reachable set by taking H and Q j s H T 0, l I[, m. (4) J l α ljp j as optimiing parameters, this would result in more complex and for all i I[, m, j I[, J, BMI terms including HQ j which may cause difficulties in [ the algorithms, such as slow convergence or getting easily Pj A He i P j B q T i H + J k= λ ijk(p j P k ) P j B i stuck at a local solution. 0 I 0. We next address the problems of estimating the L gain and the domain of attraction. Theorem : (Nonlinear L gain) Given Q j = Q T j > (5) Theorem 5: (Nonlinear L gain) Consider system (). Given P j = Pj T > 0, j I[, J and s, γ > 0, if there exist

5 H R m n, λ ijk 0, α lj 0, j, k I[, J, i I[, m, l I[, m, such that J α lj =, s H l 0, l I[, m, (6) J α ljp j H T l and for all i I[, m, j I[, J, P j A i P j B q T i H+ λ ijk (P j P k ) P j B i 0 He k= 0 I 0 0, C i D q T i H D i γ I (7) then for all w such that w s and x(0) = 0, we have γ w. The following result can be derived by adapting the proof of Theorem 4. Theorem 6: (Estimation of the domain of attraction) Consider system () and any P j = Pj T > 0, j I[, J. With w 0, then V max (x) < 0 for all x L Vmax \ 0} if there exist H R m n, λ ijk 0, α lj 0, j, k I[, J, i I[, m, l I[, m, such that J α lj =, [ Hl J α 0, l I[, m. (8) ljp j H T l and for all i I[, m, j I[, J, [ He P j A i P j B q T i H + λ ijk (P j P k ) < 0. (9) k= As compared to the counterpart results from using convex hull quadratic functions, the conditions (5), (7) and (9) in Theorems 4 to 6 appear to be less tractable because of the bilinear term P j B q T i H in the first blocks of the matrices. Also, the same H for all P j s seem to offer fewer degrees of freedom as compared to the different Y j for different Q j in Theorems to 3. However, numerical examples show that Theorems 4 to 6 may produce better results in some cases. III. AN EXAMPLE Consider system () with the following parameters: A B q B w 0 3 C y D yq D yw = 0 3 C D q D w This system is the same as the one considered in [4, where quadratic Lyapunov functions are applied to the PDI description and the NDI description, respectively, to obtain two estimates of the nonlinear L gain. Using this paper s methods, we obtain two improved estimates of the nonlinear L gain, one by applying the convex hull function to the PDI description and the other by applying the max quadratic function to the PDI description. In what follows, we compare the two estimates from [4 and the two improved estimates by this paper s methods. Fig. presents these four estimates of the nonlinear L gain. The dotted curve is from applying quadratics via NDI, the dash-dotted one is from applying quadratics via PDI, the dashed one is from applying max quadratics (with J = ) via PDI (Theorem 5) and the solid one is from applying convex hull quadratics (J = ) via PDI (Theorem ). Each of the quadratics via NDI quadratics via PDI V max via PDI V c via PDI Fig.. Different estimates of the nonlinear L gain: Case. four curves tends to a constant value as w goes to infinity. This constant value will be an estimate of the global L gain. As expected, applying non-quadratic Lyapunov functions always leads to better results than applying quadratic ones. However, the relationship between the two non-quadratic functions is not definite. The situation exhibited in Fig. can be reversed if we change the parameters of the system. In what follows, we present several scenarios through some adjustment of the parameters. [ 3.3 Case : If we change D yq to D yq =,.3 4 then the global L gain by using quadratics via NDI is unbounded (or, global stability is not confirmed), while that by using quadratics via PDI is By using max quadratics and convex hull quadratics, the global L gains are respectively and [ 3 Case 3: If we change D yq to D yq =, then 4 the global L gain by using quadratics via either NDI or PDI is unbounded. By using max quadratics and convex hull quadratics, the global L gains are respectively and The two situations above also show how the stability and performance results by the same method can be affected by the parameter D yq which describes the algebraic loop. As discussed in [9, this parameter can be adjusted through anti-windup compensation. Case 4: Next we replace the matrix A with its transpose [ 3 and take D yq =. The four different bounds on 4 the L gain are plotted in Fig. 3, where the dashed curve is from using max quadratics via PDI and the solid curve is from using convex hull quadratics via PDI. We see that for some range of w around, the dashed curve is higher than the solid one but for w > 0, the dashed curve is lower than the solid one.

6 quadratics via NDI quadratics via PDI V max via PDI V c via PDI Fig. 3. Different estimates of the nonlinear L gain: Case 4. Due to space limitation, we will not present computational results about the estimation of the domain of attraction or the estimation of the reachable set. From the different situations exhibited through the L gain, it is not hard to infer that the difference among the estimations by using quadratics/nonquadratics via NDI/PDI can be made arbitrarily large through adjusting the four elements of D yq. For instance, Case suggests that the estimate of the domain of attraction by using quadratics via NDI is bounded while that by using quadratics via PDI is the whole state space. Case 3 suggests that the domain of attraction estimated by non-quadratic functions is the whole state space while that by quadratics (via PDI or NDI) is bounded. On the other hand, the estimate of the reachable set by non-quadratics can be bounded while that by quadratics is not. We should remark that for this particular example, the algorithm for applying convex hull quadratics converges very well for all w and under different parameter changes. The algorithm for applying max quadratics generally converges well but for some w, it may have some difficulties and we need to stop the algorithm and restart it from different initial values of λ ijk which are randomly generated. In any case, improvement is expected from the non-quadratic functions. IV. PROOF OF THE MAIN RESULTS Proof of Theorem. We will prove the theorem by showing that for all x sl Vc and w R r, we have V c (x, w) w T w, where V c (x, w) is the derivative of V c along the trajectory of (), which depends on x and w. Let P j = Q j, H j = Y j Q j. Left- and right-multiplying (9) by diagp j, I}, we have He P ja i P j B q T i H j + λ ijk P j (Q j Q k )P j k= P j B i 0. 0 I This implies that for all i I[, m, j I[, J, x T P j (A i x + B i w B q T i H j x) w T w J k= λ ijkx T (0) P j (Q k Q j )P j x, x R n, w R r. Given j I[, J and any δ > 0. Consider x δe j. By Lemma we have λ ijk x T P j (Q k Q j )P j x 0. () k= It follows from (0) that for all x δe j, w R r, δ > 0, x T P j (A i x + B i w B q T i H j x) w T w 0. () (In view of (7) and condition (8), this actually shows that V c (x, w) w T w for all x s(l Vc E j ). We proceed to show that this inequality holds for all x sl Vc by exploring the properties of V c.) Now consider x 0 sl Vc. Then V c (x 0 ) = δ for some δ (0, s. By Lemma, there exist x j δe j, γ j > 0, j I[, J 0 with J 0 J such that J 0 γ j = and x 0 = J0 γ jx j. Let J 0 J 0 Q 0 = γ j Q j, Y 0 = γ j Y j, H 0 = Y 0 Q 0. (3) Then we also have x T 0 Q 0 x 0 = V c (x 0 ) = δ and V c (x 0 ) = Q 0 x 0 = Q j x j, j I[, J 0. (4) Applying convex combination to the inequalities in (8), we have for l I[, m, [ [ /s Y 0,l /s H 0,l Y0,l T 0 0. (5) Q 0 H T 0,l Q 0 By (), this implies that se(q 0 ) L(H 0). Since x T 0 Q 0 x 0 = δ s, we have H 0 x 0. Applying (7) at x 0 with h(x 0 ) = H 0 x 0, we have ẋ coa i x 0 + B i w B q T i H 0 x 0 : i I[, m } (6) and V c (x 0, w) w T w (7) co i I[, m T V c (x 0 )(A i x 0 + B i w B q T i H 0 x 0 ) w T w}. Recall that, for all x j δe j, x 0 = J 0 γ jx j, V c (x 0 ) = Q 0 x 0 = Q j x j = P j x j. (8) Applying () to x j and replacing x T j P j by T V c (x 0 ), we obtain for all w R r, T V c (x 0 )(A i x j + B i w B q T i H j x j ) w T w 0. (9) By the definition of Q 0, H 0 and Y 0 in (3), H 0 x 0 = Y 0 Q 0 x J 0 0 = γ j Y j Q 0 x 0 (30) and from (4) we have H j x j = Y j Q j x j = Y j Q 0 x 0, j I[, J 0. (3) Combining (8), (30) and (3) we have A i x 0 + B i w B q T i H 0 x 0 = J0 γ j(a i x j + B i w B q T i H j x j ) w R r. (3) Note that this is satisfied for all i I[, m. It follows from (9) that for each i I[, m and w R r, T V c (x 0 )(A i x 0 + B i w B q T i H 0 x 0 ) w T w J 0 = γ j [ T V c (x 0 )(A i x j + B i w B q T i H j x j ) w T w 0

7 By (7), we have V c (x 0, w) w T w 0 w R r. (33) Note that x 0 is an arbitrary point in sl Vc. Hence we have that V c (x, w) w T w for all x sl Vc and w R r. Now suppose x(0) = 0 and w s. Then for any t 0 > 0, as long as x(t) sl Vc for all t (0, t 0 ), we have V c (x(t 0 )) t 0 0 wt (τ)w(τ)dτ s, i.e., x(t 0 ) sl Vc. On the other hand, if there exists t 0 > 0 such that V c (x(t)) s for all t (0, t 0 ) and V c (x(t 0 )) = s then we must have t 0 w T (τ)w(τ)dτ = 0 and V c (x(t), w(t)) 0 for almost all t > t 0. Hence V c (x(t)) s for all t > t 0. Therefore, we conclude that x(t) sl Vc for all t > 0. Proof of Theorem. We will prove the theorem by showing that for all x sl Vc and w R r, V c (x, w) + γ T w T w. Since () implies (9), by Theorem, we have x(t) sl Vc for all t and for all w s, x(0) = 0. Also, all the relationships established in the proof of Theorem are true under the conditions of the current theorem. Let P j = Q j and H j = Y j Q j. Left- and rightmultiplying () by diagp j, I, I}, and applying Schur complement, similar to the proof of Theorem, we obtain that for all x δe j, w R r, δ > 0 x T P j f ij (x, w) + γ gt ij(x, w)g ij (x, w) w T w 0, (34) where f ij (x, w) = A i x + B i w B q T i H j x, g ij (x, w) = C i x + B i w D q T i H j x. Relation (34) holds for all i I[, m and j I[, J. Now consider x 0 sl Vc. Then V c (x 0 ) = δ for some δ (0, s. Similarly to the proof of Theorem, there exist x j δe j, γ j > 0, j I[, J 0 such that J 0 γ j = and x 0 = J 0 γ jx j. Let H 0, Q 0, Y 0 be defined as in (3). Then we also have H 0 x 0. Applying Proposition at x 0, we have [ ẋ Let [ Ai x co 0 + B i w B q T i H 0 x 0 i I[, m C i x 0 + D i w D q T i H 0 x 0 f i0 (x 0, w) = A i x 0 + B i w B q T i H 0 x 0, g i0 (x 0, w) = C i x 0 + D i w D q T i H 0 x 0. }. Then V c (x 0, w) + γ T w T w (35) max i I[, m T V c (x 0 )f i0 (x 0, w) + γ g i0(x 0, w) w T w}. Since x T j P j = x T 0 Q 0 = T V c (x 0 ) (see (4)), applying (34) to x j, we obtain for all w R r, i I[, m, T V c (x 0 )f ij (x j, w) + γ g ij(x j, w) w T w 0. (36) Similar to (3), we have f i0 (x 0, w) = J 0 γ jf ij (x j, w), g i0 (x 0, w) = J 0 γ (37) jg ij (x j, w). It follows that T V c (x 0 )f i0 (x 0, w) + γ g i0(x 0, w) w T w 0, and from (35) V c (x 0, w) + γ T w T w 0, (38) which is satisfied for all x 0 sl Vc and w R r. Since x(0) = 0, x(t) sl Vc for all t and for all w s, integrating both sides of (38), we have γ w. This completes the proof. Proof of Theorem 4. By the definition of V c, condition (4) implies that V c (sh l ) for all l I[, m. By Lemma 3, this implies that L Vmax L(sH) = (/s)l(h), i.e., sl Vmax L(H). Hence Hx for all x sl Vmax. By Proposition, we have for all x sl Vmax, ẋ co A i x + B i w B q T i Hx : i I[, m }. (39) On the other hand, it can be verified that (5) implies that x T P j (A i x + B i w B q T i Hx) w T w (40) λ ijk x T (P k P j )x, j I[, J, i I[, m. k= The state space of x can be partitioned in the following subsets (for all j I[, J): S j = x R n : x T (P k P j )x 0, k I[, J}. (4) If x S j \ k j S k, then V max (x) = x T P j x and V max (x) = P j x. If x J0 S j \ J j=j S 0+ j, then V max (x) = x T P j x, j I[, J 0 and V max (x) = cop j x : j I[, J 0 }. We first consider x S j \ k j S k. Then λ ijk x T (P k P j )x 0, (4) k= and Vmax (x, w) w T w max i I[, m (xt P j (A i x + B i w B q T i Hx) w T w). (43) If x J0 S j \ J j=j 0+ S j, then (4) is satisfied for all j I[, J 0 and we have V max (x, w) w T w (44) max j I[,J 0 max i I[, m (xt P j (A i x + B i w B q T i Hx) w T w). It follows from (40) and (4) that Vmax (x, w) w T w 0. The remaining part of the proof is similar to the proof of Theorem. Proof of Theorem 5. Similarly to the proof of Theorem 4, we have x(t) sl Vmax for all t > 0 under the condition w s and x(0) = 0. Also, we have Hx for all

8 x(t) sl Vmax. By Proposition, [ [ } ẋ fi (x, w) co : i I[, m, (45) g i (x, w) where f i (x, w) = A i x + B i w B q T i Hx, g i (x, w) = C i x + D i w D yq T i Hx. By Schur complement it can be verified that (7) implies x T P j f i (x, w) + γ g i (x, w) w T w J k= λ ijkx T (46) (P k P j )x for all j I[, J, i I[, m. With similar arguments as in the proof of Theorem 4, it can be shown that for all x sl Vmax and w R r, V max (x, w) + γ T w T w 0. (47) The remaining part of the proof is similar to the proof of Theorem. V. CONCLUSIONS In this paper we addressed the stability and performance analysis of linear systems with saturation elements. Two conjugate Lyapunov functions are used to improved the performance analysis results from our companion paper [4. The arising BMI conditions reduce the conservatism of the LMI conditions of [4. Numerical experience with low order systems shows that these BMI conditions can be effectively solved with the path following method. Although there is no guarantee that the global optimal solutions will be obtained, the great potential of these non-quadratic Lyapunov functions has been revealed by a numerical example. The effectiveness demonstrated through this example motivates further investigation on these non-quadratic Lyapunov functions and the development of more efficient algorithms to handle them for more complicated situations. APPENDIX Proof of Lemma. Without loss of generality, consider j =. Note that V c (x) = N min Q γ k 0: N xt + γ k (Q k Q ) x. k= γ k j= It is implied here that γ = N k= γ k. For a fixed x, define φ(γ, γ 3,, γ N ) := N (Q xt + γ k (Q k Q )) x. k= Then by Schur complement, for any c > 0, the set (γ, γ 3,, γ N ) : φ(γ, γ 3,, γ N ) c, N k= γ (48) k, γ k 0} is convex. Hence the optimal (γ, γ 3,, γ N ) s that minimie φ form a convex set. If x δ [0, δe, then V c (x) = Q xt x, implying that the minimal value of φ is reached at (γ, γ 3,, γ N ) = (0, 0,, 0). 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