Nonlinear Control Design for Linear Differential Inclusions via Convex Hull of Quadratics

Transcription

1 Nonlinear Control Design for Linear Differential Inclusions via Convex Hull of Quadratics Tingshu Hu a a Department of Electrical and Computer Engineering, University of Massachusetts, Lowell, MA 854. Abstract This paper presents a nonlinear control design method for robust stabilization and robust performance of linear differential inclusions (LDIs). A recently introduced non-quadratic Lyapunov function, the convex hull of quadratics, will be used for the construction of nonlinear state feedbac laws. Design objectives include stabilization with maximal convergence rate, disturbance rejection with minimal reachable set and least L 2 gain. Conditions for stabilization and performances are derived in terms of bilinear matrix inequalities (BMIs), which cover the existing linear matrix inequality (LMI) conditions as special cases. Numerical examples demonstrate the advantages of using nonlinear feedbac control over linear feedbac control for LDIs. It is also observed through numerical computation that nonlinear control strategies help to reduce control effort substantially. Key words: Linear differential inclusion; nonlinear feedbac; Lyapunov functions; robust stability; robust performance. Introduction A simple and practical approach to describe systems with nonlinearities and time-varying uncertainties is to use linear differential inclusions (LDIs). Such practice can be traced bac to the earlier development of absolute stability theory. The advantages of using LDIs to describe complicated systems are fully demonstrated in Boyd, El Ghaoui, Feron,& Balarishnan (994), where a wide variety of control problems for LDIs are interpreted with linear matrix inequalities (LMIs). The mechanism behind the LMI framewor is a systematic application of Lyapunov theory through quadratic functions. While the LMI technique has been well appreciated and has been widely applied to various control problems, the conservatism introduced by quadratic Lyapunov functions has been revealed in some literature including Boyd et al (994). Considerable efforts have been devoted to the construction and development of nonquadratic Lyapunov functions (see e.g. Blanchini, 995; Chesi, Garulli, Tesi & Vicino, 23; Jarvis-Wlosze & Pacard, 22; Molchanov, 989; Polansi, 997; Xie, Shishin & Fu, 997; Yfoulis & Shorten, 24). In (Molchanov, 989), a necessary and sufficient condition This paper was not presented at any IFAC meeting. Corresponding author Tingshu Hu. Tel Fax This wor was supported by NSF under Grant ECS address: tingshu@gmail.com (Tingshu Hu). for stability of polytopic LDIs was derived as bilinear matrix equations (computational methods for solving these matrix equations are still under development, see Polansi, 997; Polansi, 2; Yfoulis & Shorten, 24). More numerically tractable stability conditions were derived as LMIs in (Chesi et al 23; Jarvis-Wlosze & Pacard, 22; Xie et al, 997) from piecewise quadratic functions and homogeneous polynomial functions. Recently, a pair of conjugate Lyapunov functions have demonstrated great potential in stability and performance analysis of LDIs, saturated systems and uncertain systems with generalized sector condition (Goebel, Hu & Teel, 25; Goebel, Teel, Hu & Lin, 26; Hu, Goebel, Teel & Lin, 25; Hu & Lin, 25; Hu, Teel & Zaccarian, 26). Through these functions, stability and performances of LDIs are characterized in terms of bilinear matrix inequalities (BMIs) which cover the existing LMI conditions in (Boyd et al, 994) as special cases. Since extra degrees of freedom for optimization are injected through the bilinear terms, the analysis results are guaranteed to be at least as good as those obtained by corresponding LMI conditions. Extensive examples have shown that these non-quadratic Lyapunov functions can effectively reduce conservatism in various stability and performance analysis problems. With the effectiveness of non-quadratic Lyapunov functions demonstrated on a number of analysis problems, they can further be applied to the construction of feedbac laws. For linear time-invariant systems, it is well nown that nonlinear controls have no advantage over Preprint submitted to Automatica 7 October 26

2 linear controls when it comes to stabilization or minimization of the L 2 gain (see e.g. Khargonear, Petersen & Rotea, 988). For systems with time-varying uncertainties and LDIs, it is now accepted that nonlinear control can wor better than linear control. In (Blanchini & Megretsi, 999), examples were constructed to demonstrate this aspect and it was suggested that non-quadratic Lyapunov functions would facilitate the construction of nonlinear feedbac laws. In (Blanchini, 995), piecewise linear (or polyhedral) Lyapunov functions was used to guide the construction of variablestructure control laws for robust stability and rejection of bounded persistent disturbances. In this paper, we use one of the pair of conjugate Lyapunov functions considered in (Goebel et al, 25; Hu et al, 25), the convex hull function (i.e., the convex hull of quadratics), for the construction of nonlinear state feedbac laws. This paper is organized as follows. Section 2 describes the problems to be studied and presents some preliminaries on the convex hull function. Section 3 applies the convex hull function to the construction of nonlinear state feedbac laws for robust stabilization. Section 4 constructs nonlinear feedbac laws to achieve a couple of robust performance objectives. Section 5 uses a few examples to demonstrate the effectiveness of nonquadratic Lyapunov functions and nonlinear feedbac design. Section 6 concludes the paper. Notation - :Forx R n, x := max i x i. - 2 :Foru L 2, u 2 := ( u T (t)u(t)dt ) 2. - I[, 2 ]: For two integers, 2, < 2, I[, 2 ]:= {, +,, 2 }. -cos: The convex hull of a set S. - E(P ):={x R n : x T Px }. - L V := {x R n : V (x) }. - L(H) :={x R n : Hx }. About the relationship between E(P )andl(h), we have E(P ) L(H) H l P H T l where H l is the lth row of H. l I[,r], () 2 Problem statement and preliminaries Consider the following polytopic linear differential inclusion (PLDI), [ ] {[ ] } ẋ Ai x + B i u + T i w co : i I[,N], (2) y C i x + D i w where x R n is the state, u R m is the control input, w R p is the disturbance and y R q is the output. A i,b i,t i,c i and D i are given real matrices of compatible dimensions. This type of LDIs can be used to describe a wide variety of nonlinear systems, possibly with time-varying uncertainties (see Boyd et al, 994). Control design problems for LDIs via linear state feedbac of the form u = Fx have been extensively addressed in (Boyd et al, 994), where quadratic Lyapunov functions are used as constructive tools and the control problems are transformed into LMIs. While the LMI technique has gained tremendous popularity and its applications are still expanding to different types of systems, the conservatism resulting from quadratic Lyapunov functions has been recognized and efforts have been devoted to the construction of non-quadratic Lyapunov functions. On the other hand, it has also been recognized (e.g., Blanchini et al, 999) that restriction to linear feedbac laws may also impose unnecessary limitations to the achievable performances. In this paper, we use the convex hull function to construct nonlinear feedbac laws to achieve a few objectives of robust stabilization and performance. In what follows, we give a brief review of the definition and some properties of the convex hull function that will be necessary for the development of the main results. The convex hull function is constructed from a family of positive definite matrices. Let Q j R n n,q j = Q T j >,j I[,J]. Let Γ J := { γ R J : γ + γ γ J =,γ j }. The convex hull function is defined as V c (x) := min γ j Q j x. (3) γ Γ J x T From the definition, V c (x) and the optimal γ can be computed by solving a simple LMI problem obtained via Schur complements. This function was first used in (Hu & Lin, 23) to study constrained control systems, where it was called the composite quadratic function. It was later called convex hull function, or convex hull of quadratics, in (Goebel et al, 26; Hu et al, 25) since it is the convex hull (see Rocafellar, 97) of the family of quadratics x T Q x, or equivalently, the convex hull of g(x) =min{x T Q x : I[,J]}. Asobservedin (Goebel et al, 26) V c (x) =cog(x) { n+ } n+ =min λ g(x ): λ x = x, λ Γ n+. = = By (Rocafellar, 97), g(x )andn + in the above equation can be replaced with x T Q x and J, respectively. It turns out that the level set of V c is the convex hull of a family of ellipsoids. If we define the -level set of V c as L Vc := {x R n : V c (x) }, and denote the -level set of the function x T Px as E(P ):= { x R n : x T Px }, 2

3 then L Vc = γ j x j : x j E(Q j ),γ Γ J. It is established in (Goebel et al, 26; Hu & Lin, 23) that V c is convex and continuously differentiable. From the definition, it can be verified that V c is homogeneous of degree 2, i.e., V c (αx) =α 2 V c (x). For a compact convex set S, apointx 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. A compact convex set is completely determined by its extreme points. In what follows, we characterize the set of extreme points of L Vc.SinceL Vc is the convex hull of E(Q j must be on the boundaries of both L Vc ), j I[,J], an extreme point and E(Q j )for some j I[,J]. Denote E := L Vc E(Q )={ x : V c (x) =x T Q x =}. Then J = E contains all the extreme points of L Vc. The exact description of E is given as follows. Lemma (Hu et al, 26) For each I[,J], E ={x L Vc : x T Q For x R n, define γ (x) :=argmin (Q j Q )Q x,j I[,J]}. γ Γ J x T γ j Q j x. (4) Generally, γ is uniquely determined by x and is a continuous function of x except for some degenerated cases (Hu & Lin, 24). It is evident that γ (αx) =γ (x) for any α. Detailed properties about γ were characterized in (Hu & Lin, 24). The following lemma combines some results from (Hu & Lin, 23, 24). Lemma 2 Let x R n. For simplicity and without loss of generality, assume that γ (x) > for I[,J ] and γ (x) =for I[J +,J]. Denote J Q(γ )= γ Q, x = Q Q(γ ) x, I[,J ]. = Then V c (x )=V c (x) =x T Q x and x (V c (x)) 2 E for I[,J ]. Moreover, x = J = γ x,andforall I[,J ], V c (x) = V c (x )=2Q x =2Q(γ ) x, (5) where V c (x) denotes the gradient of V c at x. Since γ (αx) = γ (x), by (5), we have V c (αx) = α V c (x). Since V c is homogeneous of degree two, to obtain some geometric interpretation of Lemma 2, we may restrict our attention to a point x L Vc. Then by the lemma, x can always be expressed as a convex combination of a family of x s, x E(Q )(notex E ). Furthermore, the gradient of V c at these x s are the same and they all equal to the gradient of V c at x. In other words, x and x s are in the same hyperplane which is tangential to L Vc. In fact, the intersection of the hyperplane with L Vc is a polygon whose vertices include x s (see Hu & Lin, 24). Properties in Lemma and Lemma 2 are essential to system analysis and design via the convex hull function. They have been used in (Hu & Lin, 25; Hu et al, 26) for stability and performance analysis of saturated systems and uncertain systems with generalized sector conditions. 3 Nonlinear feedbac for robust stabilization In the absence of disturbance, the LDI (2) reduces to, ẋ co{a i x + B i u : i I[,N]}. (6) For stability design, we only consider the state inclusion. We would lie to construct a nonlinear state feedbac law to achieve robust stabilization via the convex hull function V c (x). The main result is given as follows. Theorem Consider V c composed from Q R n n, Q = Q T >, I[,J]. Ifthereexistβ>, Y R m n,andλ ij, i I[,N],j, I[,J] such that Q A T i + A iq + Y T BT i + B iy λ ij (Q j Q ) βq i,, (7) then a stabilizing nonlinear feedbac law can be constructed as follows. For each x R n,letγ (x) Γ J be defined as in (4). Let Y (γ )= γ Y, Q(γ )= = γ Q, (8) = F (γ )=Y (γ )Q(γ ). (9) Define f(x) =F (γ (x))x. Then for all x R n, we have max{ V c (x) T (A i x + B i f(x)) : i I[,N]} βv c (x), () which implies that the closed-loop system under u = f(x) is stable. If the vector function γ (x) is continuous in x, then u = f(x) is a continuous feedbac law. Proof. See Appendix A. Since γ (αx) =γ (x), we have f(αx) =αf(x) andthe resulting closed-loop system is homogeneous of degree 3

4 one. Here we give some explanation on the construction of f(x). Let F = Y Q.Ifx E,thenf(x) =F x. For a general x L Vc, we can express it as the convex combination of a family of x E, =, 2,,J, i.e., x =Σ J = γ x by Lemma 2. Then f(x) is the convex combination of f(x ) s with the same coefficients, i.e., f(x) =Σ J = γ f(x ). When the inequality () is satisfied, V c (x(t)) is strictly decreasing and we have V c (x(t)) V c (x())e βt for every solution x( ). Hence β is a measure of convergence rate. To increase the convergence rate, an optimization problem can be formulated to maximize β as follows: sup λ ij,q =Q T >,Y β s.t. (7). () The constraint (7) consists of a family of bilinear matrix inequalities (BMIs) which contain some bilinear terms as the product of a full matrix and a scalar, i.e., λ ij (Q j Q ). Similar bilinear terms are contained in the optimization problems in (Goebel et al, 25, 26; Hu et al, 25, 26). In the aforementioned wors, we adopted the path-following method from (Hassibi, How & Boyd, 999) and our extensive numerical experience shows that the path-following method is very effective. We actually implemented a two-step iterative algorithm which combines the path-following method and the direct iterative method. The first step of each iteration uses the pathfollowing method to update all the parameters at the same time. The second step fixes λ ij s and solves the resulting LMI problem which includes Q j s and Y j s as variables. In (Hu et al, 26), a 2-th order anti-windup system was used to demonstrate nonlinear L 2 gain analysis via convex hull functions. The two-step iterative algorithm converges very well and the results show significant improvement on those obtained via quadratic functions. We note that when J =,V c reduces to a quadratic function and F (γ (x)) reduces to a constant gain. And the optimization problem () reduces to a generalized eigenvalue problem (GEVP) which can be solved under the LMI framewor. In our computation, we first solve the optimization problem for J = andthenusethe optimal Q and Y to start the two-step iterative algorithm for some J>, with Q j = Q and Y j = Y for all j and λ ij randomly chosen. With this approach, the optimization result will be guaranteed to be at least as good as that obtained by solving the corresponding GEVP problem. Similar approaches can be derived for other optimization problems for evaluating the reachable sets and the L 2 gaininsection4. 4 Nonlinear feedbac for robust performance Consider the linear differential inclusion (2) in the presence of disturbances. Lie in (Boyd et al, 994), we consider two types of disturbances, the unit pea disturbances w T (t)w(t) t (2) and the unit energy disturbances ( ) 2 w 2 = w T (t)w(t)dt. (3) Let u = f(x) be a nonlinear state feedbac. The closedloop system is [ ] {[ ] } ẋ Ai x + B i f(x)+t i w co : i I[,N]. y C i x + D i w (4) The control design objective is disturbance rejection, i.e., to eep the state close to the origin or to eep the size of the output small (in terms of certain norm) in the presence of a class of disturbances. The disturbance rejection performance can be characterized by reachable set or the maximal output norm. When the disturbance is of unit pea type, the maximal output norm is associated with the L gain; when the disturbance is of unit energy, the maximal output norm is associated with the L 2 L gain or the L 2 gain. We first consider the reachable set. 4. Suppression of the reachable set The reachable set can be estimated with a level set of a certain Lyapunov function. In (Boyd et al, 994), quadratic Lyapunov functions are considered for LDIs and the reachable set is estimated with ellipsoids. In this section, we use the convex hull of a family of ellipsoids to characterize the reachable set and we attempt to reduce the reachable set by nonlinear feedbac laws. 4.. Reachable set with finite energy disturbances Theorem 2 Consider V c composed from Q R n n, Q = Q T >, I[,J]. Suppose that there exist Y R m n,andλ ij, i I[,N],j, I[,J] such that [ ] Mi T i where T T i I M i = Q A T i +A i Q +Y T B T i +B i Y i,, (5) λ ij (Q j Q ). (6) Let the nonlinear feedbac u = f(x) = F (γ (x))x be constructed from Y s and Q s as in (8) and (9). Then for all w bounded by w 2 and with x =, the state of (4) satisfies x(t) L Vc for all t. With the feedbac law constructed in Theorem 2, the level set L Vc can be considered as an estimate for the reachable set. To eep the state in a small neighborhood of the origin, it is desirable that L Vc satisfying the condition is as small as possible. We may use a reference 4

5 polytope to measure the size of L Vc. The polytope is described in terms of a prescribed matrix H R r n as follows, L(H) :={x R n : Hx }. The outer size of L Vc is defined as α out := min{α : L Vc αl(h)}. (7) The matrix H can be chosen such that H l x is a certain quantity that we would lie to eep small. If we have L Vc αl(h), then H l x(t) α for all t in the presence of the class of disturbances. Since L(H) is a convex set and L Vc is the convex hull of the ellipsoids E(Q ), it is easy to see that L Vc αl(h) =L(H/α) if and only if E(Q ) L(H/α) for all. By(),thisisequivalentto H l Q H T l α2 l I[,r], I[,J]. (8) In view of the above arguments, the problem of reducing the reachable set can be formulated as inf α s.t. (5), (8). (9) λ ij,q =Q T >,Y Proof of Theorem 2. It suffices to show that, under condition (5), we have V c w T w for all x and w satisfying (4). Then by integrating both sides, we have V c (x(t)) w T wdt and hence x(t) L Vc for all t. We need to prove that V c (x) T (A i x + B i f(x)+t i w) w T w x, w, i. (2) Similarly to the proof of Theorem, we can first verify (2) for every x E by using (5). Then extend the results to all other x by expressing it as a convex combination of x E, =, 2,,J with f(x) asthe same convex combination of f(x ) s Reachable set with unit pea disturbances Theorem 3 Consider V c composed from Q R n n, Q = Q T >, I[,J]. Suppose that there exist Y R m n, λ ij, i I[,N],j, I[,J] and β> such that [ ] Mi + βq T i T T i βi i,, (2) where M i is given by (6). Let the nonlinear feedbac law u = f(x) = F (γ (x))x be constructed from Y s and Q s as in (8) and (9). Then L Vc is an invariant set, which means that all trajectories starting from L Vc will stay inside for any possible disturbance satisfying w(t) T w(t), t. Moreover, for all x R n and all possible disturbances, x(t) will converge to L Vc. Proof. With similar arguments as in the proof of Theorem, it can be shown that under the condition (2), we have V c βv c (x)+βw T w, for all x R n,w R p satisfying (4). Since w T w, for V c (x) =,wehave V c andv c is nonincreasing. Hence L Vc is an invariant set. If V c (x) >, then V c is strictly decreasing. Hence any trajectory starting from outside of L Vc will converge to L Vc. Similarly to the unit energy disturbance case, we can formulate the following optimization problem for minimizing the reachable set or the maximal output norm, inf α s.t. (2), (8). (22) λ ij,β,q =Q T >,Y 4.2 Suppression of the L 2 gain For the type of energy bounded disturbances, we have the following result: Theorem 4 Let Q R n n,q = Q T >, I[,J]. Let δ >. Suppose that there exist Y R m n and λ ij, i I[,N],j, I[,J] such that M i T i Q Ci T Ti T I Di T, i,, (23) C i Q D i δ 2 I where M i is given by (6). Let the nonlinear feedbac law u = f(x) =F (γ (x))x be constructed from Y s and Q s as in (8) and (9). Then for system (4) with x =, we have y 2 δ w 2. The proof of Theorem 4 is omitted since the main ideas are similar to those for the previous theorems. We just need to show V c + δ y T y w T w, first for x E 2, then use the properties of V c and the controller to extend the result to other x. By Theorem 4, the quantity δ gives an upper bound for the L 2 gain. The following optimization problem can be formulated for suppression of the L 2 gain: inf δ s.t. (23). (24) λ ij,q =Q T,Y 5 Examples Example Consider a second-order LDI taen from (Blanchini & Megretsi, 999), ẋ co{a x + B u, A 2 x + B 2 u}, where [ ] [ ] [ ] K K A = A 2 =,B =,B 2 =, for some K>. For the LDI to be quadratically stabilizable by linear feedbac, K has to be less than, i.e., 5

6 there exists β> and Q> such that the inequalities Q(A i + B i F ) T +(A i + B i F )Q βq, i =, 2, are satisfied if and only if K<. However, at K =,the LDI can be stabilized with a positive convergence rate via nonlinear feedbac. By solving () with J =2, 3, 4, the convergence rate β can be increased to.5633,.6364 and.7973, respectively. On the other hand, with J = 2, 3, 4, the maximal K for () to have a solution β>is found to be greater than 2.5, 2.9 and 3.4, respectively. As shown in (Blanchini & Megretsi, 999), for any K>, the LDI canbe stabilized bynonlinear feedbac which was explicitly constructed via analytical method based on the geometric structure of the vector field. However, it seems hard to extend the analytical method to other systems. Example 2 Consider an LDI subject to disturbances, ẋ co{a x + B u + Ew,A 2 x + B 2 u + Ew},y = Cx, where [ ] [ ] [ ] 3 A =,B =,E=, 2.5 [ ] [ ] [ ] A 2 =,B 2 =,C=. 2.8 When quadratic Lyapunov function is applied to designing a linear state feedbac law, the problem is a special case of the one studied in Section 4.2 and the optimization problem is a special case of (24) with J =.In this case (24) reduces to an LMI problem. The optimal δ for this case is δ =.767. When δ approaches to δ, the norm of the feedbac gain will approach infinity. If we restrict the norm of the feedbac gain to be less than 5 (via an additional constraint on Q s, i.e., ε I < Q < ε 2 I), the optimal δ is δ = The [ ] 3. feedbac gain is F = Next we apply the convex hull function V c (x)withj =2 to the design of a nonlinear feedbac law. By solving (24) with J = 2, the minimal δ we have obtained is δ 2 =.947. Again, the norm of one of the feedbac gain has to be very large (in the order of ) to produce the value δ 2. If we restrict the norm of F = Y Q to be less than, then the best δ we have computed is δ 2 = Other variables corresponding to this value of δ = δ 2 are [ ] [ ] F = ,F 2 = , [ ] [ ] Q =,Q 2 = Here we compare the output responses for the two designs under the disturbance w(t) = for t [, ] and w(t) = for t >. The switching between ẋ = A x+b f(x)+ew and ẋ = A 2 x+b 2 f(x)+ew is chosen such that V c is maximized at each time instant. The two time responses are compared in Fig., where the dashed curve is produced by the linear state feedbac u = Fx and the solid curve is produced by the nonlinear feedbac constructed from Q,Q 2 and Y = F Q,Y 2 = F 2 Q 2. For the dashed curve, we have ( 5 y 2 (t)dt) 2 =2.6858, output y Linear feedbac Nonlinear feedbac t (sec) Fig.. Two output responses and for the solid curve, we have ( 5 y 2 (t)dt) 2 = Example 3 Consider the same LDI as in Example 2. Assume that the disturbance is of unit pea type, i.e., w T (t)w(t) for all t. We would lie to design a control law such that the pea of the output is suppressed. This is achieved by solving (22). When J =,V c is a quadratic function and the resulting control law is linear. The optimal solution can be obtained by running β from to. If no restriction on the magnitude of the feedbac matrix is imposed, the optimal α is This would require F to go to infinity. If a bound on the norm of F is imposed, say, F 5, we obtain [ ] 3.For α = := α and F = J = 2, we impose a bound F and the best α is := α 2. The other parameters are [ ] [ ] F = ,F 2 = , [ ] [ ] Q =,Q 2 = The two level sets resulting from J =andj =2are plotted in Fig. 2, where the outer dashed boundary is that of the ellipsoid E(Q ) and the inner boundary (in thic curve) is that of L Vc composed from Q and Q 2. A trajectory under the linear control u = Fx is plotted. It starts from near the origin and ends very close to E(Q ). The switching strategy and the value of w 6

7 are chosen such that V c is maximized. Another output response is generated under the nonlinear control. The two responses are plotted in Fig. 3. x x Fig. 2. Two reachable sets and a trajectory by linear control 4 2 satisfied for all extreme points of L Vc,inparticular,for all x E, I[,J]. Next we use Lemma 2 to express an arbitrary x L Vc as a convex combination of a set of extreme points, say, x E, =, 2,,J. Again by Lemma 2 the gradient of V c at x is the same as that at each x. Finally () follows from the fact that f(x) isa convex combination of f(x ) s and that V c (x) =V c (x ) for each. Now consider x E for some I[,J]. Then V c (x) = x T Q x =andγ (x) is a vector whose th element is and the rest are zeros. Hence F (γ (x)) = Y Q and V c (x) =2Q x. By Lemma, λ ij x T Q (Q j Q )Q x, i I[,N]. Let F = Y Q.Thenf(x) =F x. Multiply (7) from left and from right with Q,wehave (A i + B i F ) T Q + Q (A i + B i F ) λ ij Q (Q j Q )Q βq,i I[,N]. y by nonlinear feedbac by linear feedbac t (sec) Fig. 3. Two output responses It follows that x T ((A i + B i F ) T Q + Q (A i + B i F ))x βx T Q x = βv c(x), i I[,N]. Hence for every x E and i I[,N], V c (x) T (A i x + B i f(x)) = 2x T Q (A i + B i F )x βv c (x). (A.) 6 Conclusions We developed LMI-based methods for the construction of nonlinear feedbac laws for linear differential inclusions. The convex hull functions are used to guide the design for achieving a few objectives of robust stabilization and performance. The advantages of nonlinear feedbac over linear feedbac has been demonstrated through some numerical examples. It is expected that the design methods can be extended to deal with other performances, such as the input-to-state, input-tooutput and state-to-output performances studied in (Boyd et al, 994). A Proof of Theorem Since the closed-loop system is homogeneous of degree one, V c is homogeneous of degree two and V c (αx) = α V c (x), we only need to restrict our attention to the boundary of the -level set, L Vc. The rest of the proof is proceeded with two steps. We first prove that () is This implies that () is satisfied for all x E. Next we consider an arbitrary x L Vc. By Lemma 2, x is a convex combination of a set of x s, each of which belongs to a certain E. For simplicity, assume that γ (x) > for =, 2,,J and γ (x) =for>j. Then x = J = γ x. Recalling from Lemma 2 that V c (x) =2Q(γ ) x and we have Q(γ ) x = Q x, I[,J ], (A.2) J F (γ )x = Y (γ )Q(γ ) x = γ F Q Q x = J = γf x. = (A.3) 7

8 Hence J A i x + B i F (γ )x = γ (A ix + B i F x ). It follows that = V c (x) T (A i x + B i F (γ )x) J =2x T Q(γ ) γ (A ix + B i F x ) = J =2 γx T Q(γ ) (A i + B i F )x = J =2 = (A.4) γ x T Q (A i + B i F )x. (A.5) Since x E, by (A.) and noting that V c (x )=V c (x) for each, wehave J V c (x) T (A i x+b i F (γ )x) γβv c (x )= βv c (x), = which shows (). Since Y (γ )andq(γ ) are continuous in γ,andq(γ) > for all γ Γ J, the continuity of f(x) =F (γ (x))x follows from that of γ (x). References Blanchini, F. (995). Non-quadratic Lyapunov functions for robust control. Automatica, 3, Blanchini, F. & Megretsi, A. (999). Robust state feedbac control of LTV systems: nonlinear is better than linear. IEEE Transactions on Automatic Control, 44(4), Boyd, S., El Ghaoui, L., Feron, E. & Balarishnan, V. (994). Linear Matrix Inequalities in Systems and Control Theory. SIAM Studies in Appl. Mathematics, Philadelphia. Chesi, G., Garulli, A., Tesi, A. and Vicino, A. (23). Homogeneous Lyapunov functions for systems with structured uncertainties. Automatica, 39, Goebel,R.,Hu,T.&Teel,A.R.(25).Dual matrix inequalities in stability and performance analysis of linear differential/difference inclusions. In Current Trends in Nonlinear Systems and Control. Birhauser. Goebel,R.,Teel,A.R.,Hu,T.&Lin,Z.(26). Conjugate convex Lyapunov functions for dual linear differential equations. IEEE Transactions on Automatic Control, 5(4), Hassibi, A., How, J. and Boyd, S. (999). A pathfollowing method for solving BMI problems in control. Proc.ofAmericanContr.Conf., Hu, T., Goebel, R., Teel, A. R. & Lin, Z. (25). Conjugate Lyapunov functions for saturated linear systems. Automatica, 4(), Hu, T. & Lin, Z. (23). Composite quadratic Lyapunov functions for constrained control systems. IEEE Trans. Automat. Contr., 48, Hu, T. & Lin, Z. (24). Properties of composite quadratic Lyapunov functions. IEEE Trans. Automat. Contr.,49, Hu, T. & Lin, Z. (25). Absolute stability analysis of discretetime systems with composite quadratic Lyapunov functions. IEEE Trans. Automat. Contr. 5, Hu, T., Teel, A. R. & Zaccarian, L. (26). Stability and performance for saturated systems via quadratic and nonquadratic Lyapunov functions. IEEE Trans. Automatic Control, toappear. Jarvis-Wlosze, Z. & Pacard, A. K. (22). An LMI method to demonstrate simultaneous stability using non-quadratic polynomial Lyapunov functions. IEEE Conf. on Dec. and Contr., , Las Vegas, NV. Khargonear, P. P., Petersen, I. R. & Rotea, M. A. (988). H -optimal control with state-feedbac. IEEE Trans. Automat. Contr., 33(8), Molchanov, A. P. (989). Criteria of asymptotic stability of differential and difference inclusions encountered in control theory. Sys. & Contr. Lett., 3, Polansi, A. (997). Lyapunov function construction by linear programming. IEEE Trans. Automat. Contr., 42, 3-6. Polansi, A. (2). On absolute stability analysis by polyhedral Lyapunov functions. Automatica, 36, Rocafellar, R. T. (97). Convex Analysis. Princeton University Press. Xie, L., Shishin, S. & Fu, M. (997). Piecewise Lyapunov functions for robust stability of linear time-varying systems. Sys. & Contr. Lett., 3, Yfoulis, C. A., & Shorten, R. (24). A numerical technique for the stability analysis of linear switched systems. Int. J. Contr., 77,