The Randomized Ellipsoid Algorithm with Application to Fault-Tolerant Control. Stoyan Kanev and Michel Verhaegen

Transcription

1 The Randomized Ellipsoid Algorithm with Application to Fault-Tolerant Control Stoyan Kanev and Michel Verhaegen Delft University of Technology, Delft Center for Systems and Control, Mekelweg 2, 2628 CD Delft, The Netherlands. Abstract In this paper we present a randomized ellipsoid algorithm with application to robust active fault-tolerant control (FTC) design problems. The majority of FTC design approaches that have been developed usually deal with some restricted class of faults, e.g. faults that enter the state-space matrices in a linear way such as some sensor and actuator faults. Many deterministic approaches to FTC have been proposed that can deal with some specific cases of faults. In practice, however, many faults (for instance parametric faults) do not enter the state-space matrices necessarily in a linear manner. Moreover, as faults are not directly measurable, but are estimated online by dedicated algorithms, their estimates are often only known to lie in some time-varying uncertainty intervals. Addressing the problem of FTC for general nonlinear faults with uncertainty in a deterministic framework becomes an NPhard problem. In this paper this problem is instead addressed in a probabilistic framework. We present a robust active FTC method based on randomized parameter-varying controller (LPV) design that can deal with nonlinear faults with time-varying uncertainties. keywords: randomized ellipsoid algorithm, active FTC, nonlinear faults, fault estimate uncertainty. 1 Introduction Faults in controlled systems represent malfunctioning of the system that results in performance degradation or even instability of the system. When untreated, small faults in some subsystems of the controlled system can easily develop into serious overall system failures. In safe-critical systems such failures can have serious consequences ranging from economical losses to human deaths. It is therefore important that such safe-critical systems possess the properties of increased reliability and safety. These properties can be enhanced by means of increasing the fault-tolerance of the system. In the literature, many algorithms have been proposed to the design of FTC systems [1, 2, 3]. The majority of these approaches, however, are focussed on a specific subclass of faults as, for instance, linear sensor and actuator faults, i.e. faults that affect linearly the B and C matrices of the state-space representation of the system. Treating component faults is much more involved as such faults may affect any/all of the state-space matrices in a nonlinear way. In addition to that, the usual assumption is imposed that the faults are exactly known. This can be a serious limitation in practice since faults can, in general, not be measured but a dedicated fault-detection and diagnosis (FDD) scheme is used to estimate them. These fault estimates, however, are never perfect due to measurement noise, modelling errors etc. In this paper we model the imperfection of the fault estimates as parametric uncertainty. Clearly, this fault estimate uncertainty is usually time-varying: it increases immediately after the occurrence of a fault due to the lack of sufficient measurement data for the FDD scheme to make an accurate estimate of the fault, and decreases as more data becomes available. Dealing with such time-varying, nonlinear system faults with time-varying uncertainties in a deterministic way is, in general, an NP-hard problem. 1

2 On the other hand, the probabilistic framework [4, 5, 6, 7, 8] can provide us with a numerically tractable solution at the expense that we allow (with some possibly very small probability) that there exist some faults and/or fault estimate uncertainties for which our design specifications are not met. To this end we propose a randomized ellipsoid algorithm (EA) [8] to active fault-tolerant control design based on robust LPV controller design. Under the assumption that the estimates of the faults ˆf as well as estimates of their uncertainty sizes γ f are available online, the goal will be to design an LPV controller that is scheduled by both ˆf and γ f and achieves certain guaranteed closed-loop performance. In addition to fault estimates uncertainty, model uncertainty is also considered. The complete uncertainty is assumed to be bounded in the structured uncertainty set, and to be coupled with a probability density function f ( ). It is further assumed that it is possible to generate samples of according to f ( ). See [9] for more details on the available algorithms for uncertainty generation. 2 Problem Formulation Consider the following discrete-time system with faults x k+1 = A δ (f)x k + Bξ δ(f)ξ k + Bu(f)u δ k S F : z k = C δ z(f)x k + Dzξ δ (f)ξ k + Dzu(f)u δ k y k = Cy(f)x δ k + Dyξ δ (f)ξ k + Dyu(f)u δ k, (1) where x k R n is the state of the system, u R m is the control action, y R p is the measured output, z R nz represents the controlled output of the system, and ξ R ξ is the disturbance to the system. The vector f(k) F R n f represents faults in the system. Note that this general representation could be used to model a wide class of sensor, actuator and component faults. δ represents the uncertainty in the system, which is assumed to belong to some bounded set δ with a given probability density function f δ (δ) inside δ. In addition to that it is assumed that random uncertainty samples can be generated with the specified probability density function f δ (δ). Furthermore, δ is assumed to be such that A δ (f) Bξ δ(f) Bδ u(f) Cz(f) δ Dzξ δ (f) Dδ zu(f) Cy(f) δ Dyξ δ (f) <, δ δ, f F. Dδ yu(f) As discussed in the introduction, the estimates ˆf(k) are assumed to be imprecise. Here the i-th entry of f is represented as 2 f i (k) = (1 + γ f,i (k)ˆδ i ) ˆf i (k), i = 1,2,...,n f, (2) where ˆδ i 1 represents the fault estimate uncertainty, and where γ f,i (k) defines the size of the uncertainty in the sense that (2) is equivalent to f i (k) = (1+ δ i (k)) ˆf i (k) with δ i (k) γ f,i (k). We will however make use of the FDD uncertainty representation in equation (2) where the uncertainty ˆδ i is normalized since we will later on design a controller scheduled by both the fault estimates ˆf(k) and the uncertainty sizes γ f,i (k). The uncertainty sizes γ f,i are allowed to be time varying and are assumed provided by some FDD scheme together with the fault estimates. In addition, we will denote the set in which the matrix ˆδ, representing the fault estimate uncertainty, can take values as ˆδ =. {ˆδ = diag(ˆδ1,..., ˆδ } nf ) : ˆδ i 1, i = 1,2,...,n f. Both the fault estimates ˆf(k) and the uncertainty sizes γ f (k) are assumed to belong to some known interval sets, γ f (k) Ω γ = {w R n f : γ f,min w γ f,max } ˆf(k) Ω f = {w R n f : f min w f max }. 2

3 In this paper the control objective is specified as an LPV design problem, where the goal is to design a controller that can be scheduled by the fault estimates ˆf i and the fault estimate uncertainty sizes γ f,i, i.e. an LPV controller of the form K = K( ˆf,γ f ). For some given bounded functions g i ( ˆf,γ f ) : Ω f Ω γ R we consider the following parametrization of the LPV controller: where K( ˆf,γ f ) = n f K( ˆf,γ f ) = K 0 + g i ( ˆf,γ f )K i, (3) [ A c ( ˆf,γ f ) B c ( ˆf,γ f ) F( ˆf,γ f ) 0 i=1 ] [ A c, K i = i Bi c F i 0 ], i = 0,1,...,n f, (4) with A c R n n, B c R n p, and F R m n. Thus, only strictly proper full-order controllers are considered. Let T δ (f) denote the (suitably weighted) closed-loop system from the disturbance signal ξ(k) to z(k). Clearly, in view of (2), this operator depend on ˆf and γ f. The problem considered in this paper is formulated as follows: find an LPV controller (3) by solving the problem min F( ˆf,γ f ) subject to γ sup δ δ, ˆδ ˆδ ˆf Ω f, γ f Ω γ T δ ( ˆf,γ f ) 1 (5) Two main difficulties arise when one tries to solve problem (5). The first difficulty in solving the problem (5) lies in the fact that the function is not convex in the parameters ˆf and γ f since both the system and the controller depend on them. This problem is, in general, an NP-hard problem. In the state-feedback case, however, the optimization problem can be transformed to an equivalent optimization problem that is affine in the optimization variables (but still non-convex in the parameters ˆf and γ f ). This makes it possible to approach the problem in a probabilistic framework by means of randomized algorithms. This circumvents the NP-hardness of the problem at the expense that we can only guarantee optimality of the resulting solution with some probability. In other words, we can guarantee that the obtained controller would result in a closed-loop system with L 2 -gain less than or equal to some γ opt for almost all possible values of ˆf Ω f and γ f Ω γ. The second difficulty in solving the problem (5) appears when output-feedback LPV controller needs to be designed. In the output-feedback case the problem (5) cannot be transformed to an equivalent problem that is affine in the optimization variables which is necessary in order to apply the randomized algorithm. In this paper we will address this problem by making use of a two-step method, proposed in [10]. There the output-feedback case is approached by designing a statefeedback gain F( ˆf,γ f ) at the first step which is subsequently kept fixed at the second step where the remaining controller matrices A( ˆf,γ f ) and B( ˆf,γ f ) are computed. As we discussed above, the state-feedback design in the first step can be made convex in the optimization variables and we can deal with the remaining non-convexity of ˆf and γ f by means of the randomized algorithm. The problem at the second step is approached in the same way by first transforming the problem to an equivalent one that is convex in the new optimization parameters and then making use of the randomized approach proposed in this paper. Clearly, this approach introduces some conservatism due to the decoupling of the state-feedback matrix from the remaining controller matrices. 3 Parameter-dependent LMI Formulation In this section we reformulate the FTC design problem (5) in term of parameter-dependent linear matrix inequalities (LMIs), i.e. LMIs that depend on the fault estimates ˆf and their uncertainty sizes γ f. In the next section we will present the randomized ellipsoid algorithm that can be used for solving them. 3

4 3.1 State-Feedback Case We will begin by considering the state-feedback case, assuming that the state is directly measured. The controller is in this case taken of the form (3) with K( ˆf,γ f ) = F( ˆf,γ f ), K i = F i, i = 0,2,...,n f. Applying the state-feedback control law ( n u(k) = F( ˆf,γ f f )x(k) = F 0 + g i ( ˆf,γ f )F i )x(k) (6) i=1 results in the following closed-loop system { Tst.fb.( δ ˆf,γ x k+1 =(A δ (f) + B δ f ) : u(f)f( ˆf,γ f ))x k + Bξ δ(f)ξ k z k =(Cz(f) δ + Dzu(f)F( δ ˆf,γ f ))x k + Dzξ δ (f)ξ k (7) The following result is a standard generalization of the discrete version of the bounded real lemma (see, e.g., [10]). Lemma 3.1. Suppose that the matrices P = P T and L i, i = 0,1,...,n f, be such that for all δ δ, ˆδ ˆδ, ˆf Ω f, γ f Ω γ it holds that P A δ (f)p + B δ u(f)l( ˆf, γ f ) B δ ξ(f) 0 P 0 (C δ z(f)p + D δ zu(f)l( ˆf, γ f )) T I (D δ zξ(f)) T I N g > 0, (8) where it is denoted L( ˆf,γ f ) = L 0 + g i ( ˆf,γ f )L i, and where the elements of the vector f are i=1 defined in (2). Then the state-feedback gain (6) with F i = L i P 1, i = 0,1,...,n f, (9) achieves sup δ δ, ˆδ ˆδ ˆf Ω f, γ f Ω γ δ Tst.fb.( ˆf,γ f ) 1. (10) This result can be used to compute a feasible solution (if such exists) to the problem (5) in the state-feedback case with fixed γ (note that γ enters the matrix inequality (8) via the matrices C z(f), δ D zu(f), δ and D zξ δ (f). A bisection algorithm on γ can then be used to approach the solution to the original optimization problem (5). Note that the matrix inequality (8) is linear in the unknowns, but not necessarily convex in ˆf, δ, ˆδ, and γ f. For that reason the standard LMI tools cannot be used and we will instead approach the problem by a randomized algorithm. 3.2 Output-Feedback Case We next consider the output-feedback case and we will be searching for controller of the form (3)-(4) needs to be designed. This controller results in the following closed-loop system T δ out.fb.(f) : { xk+1 = A cl x k + B cl ξ k z k = C cl x k + D cl ξ k, (11) 4

5 with [ A cl = [ B cl = A δ (f) Bu(f)F( δ ˆf,γ f ) B c ( ˆf,γ f )Cy(y) δ A c ( ˆf,γ f ) + B c ( ˆf,γ f )Dyu(f)F( δ ˆf,γ f ) B δ ξ (f) B c ( ˆf,γ f )D δ yξ (f) ], [ C cl = Cy(f) δ Dyu(f)F( δ ˆf,γ f ) ], D cl = D δ zξ (f)ξ k. ], Suppose that a state-feedback controller has been found in the form (6) by using the result in Lemma 3.1. The remaining matrices A c ( ˆf,γ f ) and B c ( ˆf,γ f ) of the LPV controller (4) can be found using the following result Lemma 3.2. Suppose that the matrices X = X T, Y = Y T, Z i and G i, i = 0,1,...,n f, are such that for all δ δ, ˆδ ˆδ, ˆf Ω f, γ f Ω γ it holds that P M( ˆf,γ f ) N( ˆf,γ f ) 0 P 0 R( ˆf,γ f ) T I Dzξ T I > 0 (12) where the elements of the vector f are defined in (2), and where the matrices M( ˆf,γ f ), N( ˆf,γ f ), R( ˆf,γ f ), and P are defined as [ M( ˆf,γ M11 ( f ) = ˆf,γ f ) M 12 ( ˆf,γ ] f ) M 21 ( ˆf,γ f ) M 22 ( ˆf,γ, f ) M 11 ( ˆf,γ f ) = X(A δ (f) + Bu(f)F( δ ˆf,γ f )) M 21 ( ˆf,γ f ) = Y(A δ (f) + Bu(f)F( δ ˆf,γ f )) Z( ˆf,γ f ) G( ˆf,γ f )( C y(f) δ + D yu(f)f( δ ˆf,γ f )) M 12 ( ˆf,γ f ) = XBu(f)F( δ ˆf,γ f ) M 22 ( ˆf,γ f ) = Z( ˆf,γ f ) + G( ˆf,γ f ) D yu(f)f( δ ˆf,γ f ) YBu(f)F( δ ˆf,γ f ) [ N( ˆf,γ XBξ δ f ) = (f) ] [ ] X YBξ δ(f) G( ˆf,γ f )Dyξ,i δ (f), P =, Y R( ˆf,γ f ) = [ Cδ z (f) + D zu(f)f( δ ˆf,γ f ) D zu(f)f( δ ˆf,γ f ) ], n G( ˆf,γ f n f ) = G 0 + g i ( ˆf,γ f )G i, Z( ˆf,γ f f ) = Z 0 + g i ( ˆf,γ f )Z i. i=1 Then the LPV controller (3)-(4) with i=1 (13) A c ( ˆf,γ f ) = Y 1 Z( ˆf,γ f ), B c ( ˆf,γ f ) = Y 1 G( ˆf,γ f ), (14) is such that sup δ δ, ˆδ ˆδ ˆf Ω f, γ f Ω γ δ Tout.fb.(f) 1. (15) For proof see [10]. Similarly to the state-feedback case, the standard deterministic LMI solvers are not directly applicable to this problem due to the nonlinear dependence of (12) on the parameters ˆf and γ f. 5

6 4 The Randomized Ellipsoid Algorithm In this section we propose a randomized ellipsoid algorithm for solving the parameter-dependent optimization problems (8) and (12). To this end, note that they both can be rewritten into the following general parameter-dependent LMI representation: where minimize γ over x X R N subject to U γ (x, ) = U γ,0 ( ) + N i=1 U γ,i( )x i 0,,,. = (δ, ˆδ, ˆf,γ f ) {δ ˆδ Ω f Ω γ }. =, is a generalized uncertainty with probability density function f ( ), and U γ,i ( ) = U T γ,i ( ) R q q are given matrices. It is assumed that the set X is convex. In (16) we have collected all free optimization parameters into the new vector x (i.e. x contains the free entries in the matrices (P,L i ) in (8), or the free entries of the matrices (Y,Z i,g i ) in (12)). Having found a solution x to (16), one needs to go back to the original optimization variables and to parametrize the FTC controller via (9) or (14). For a given γ, the set of all feasible solutions to the control problem, called the feasibility set, is denoted as S γ. = {x X : Uγ (x, ) 0, }. (17) The goal is the development of an iterative algorithm capable of finding a solution to the control problem defined (16). To this end the following cost function is defined (16) v γ (x, ). = Π + [U γ (x, )] 2 F 0, (18) which is non-negative for any x X and. The usefulness of the so-defined function v γ (x, ) stems from the fact that v γ (x, ) = 0 holds for all if and only if x S γ. In this way the initial problem is reformulated to the following optimization problem Then the following result holds [5]. x γ = arg min sup v γ (x, ). (19) x X Lemma 4.1. The function v γ (x, ), defined in equation (18), is convex and differentiable in x and its gradient is given by trace ( U γ,1 ( )Π + [U γ (x, )] ) v γ (x, ) = 2. trace ( U γ,n ( )Π + [U γ (x, )] ) (20) The approach proposed in this chapter is based on the Ellipsoid Algorithm (EA) [11]. The starting point in EA is the computation of an initial ellipsoid that contains the solution set S γ. Then at each iteration of the EA two steps are performed. In the first step a random uncertainty sample (i) is generated according to the given probability density function f ( ). With this generated uncertainty the convex function U γ (x, (i) ) is parametrized and used at the second step of the algorithm where an ellipsoid is computed, in which the solution set is guaranteed to lie. In this way the EA produces a sequence of ellipsoids with decreasing volumes, all containing the solution set. Using some existing facts, and provided that the solution set has a non-empty interior, it will be established that this algorithm converges to a feasible solution in a finite number of iterations with probability one. To initialize the algorithm, a method is presented for obtaining an initial ellipsoid that contains the solution set. It is also shown that even if the solution set has a zero volume, the EA converges to the solution set when the iteration number tends to infinity. Define the ellipsoid E( x, P) = {x R N : (x x) T P 1 (x x) 1} S γ (21) 6

7 with center x R N and matrix P = P T > 0 describing its shape and orientation. We begin by constructing an initial ellipsoid E(x (0),P 0 ) that contains the solution set S γ. This is an important step since, as will become clear later on, the larger the initial ellipsoid the more iterations it takes for the proposed randomized EA to converge. 4.1 Construction of E(x (0),P 0 ) In this section we consider the problem of finding initial ellipsoid that contains the solution set S γ. The idea that is exploited here is that for any fixed value ˆ of the uncertainty set it holds that the set S γ ( ˆ ). = {x : U γ (x, ˆ ) 0, ˆ } Sγ. Therefore, a reasonable option would be to search for the minimum volume ellipsoid (known as the outer Löwner-John ellipsoid) E(x (0),P) that contains the set S γ ( ˆ ). This could be achieved by solving the optimization problem min x (0),Z subject to log detz 1 sup Zx Zx (0) 1 x S ˆ γ and then taking P = Z 2. However, as stated in [12], this is in general an NP-hard problem. For that reason we will not be interested here with finding any outer Löwner-John ellipsoid, but will rather propose a fast algorithm capable of finding some outer ellipsoidal approximation of the set S γ ( ˆ ). The following additional assumption needs to be imposed. Assumption 4.1. It is assumed that the set S γ ( ˆ ) is bounded. Assumption 4.1 could be restrictive for some problems. If it does not hold, one can enforce it by including in U γ (x, ) 0 additional hard constraints on the elements of the vector of unknowns x. One way to find an outer approximation of the set S γ ( ˆ ) is as follows [13]. Define the following barrier function for S γ ( ˆ ) φ(x) =. { log det( Uγ (x, ˆ )) 1, if x S γ ( ˆ ), otherwise Denote the analytic center of S γ ( ˆ ) as x = arg min x φ(x). (22) Note that computing the analytic center is a convex optimization problem. It is then shown in [13] that an outer approximating ellipsoid E(x (0),P 0 ) of the set S γ ( ˆ ) is given by S γ ( ˆ ) x (0) = x, P 0 = N(N 1)H 1 (x ), (23) where the symmetric positive definite matrix H(x) = [h ij (x)] is the Hessian of φ(x) with elements h ij (x) = trace[uγ 1 (x, ˆ )U 1 γ,i ( ˆ )U γ (x, ˆ )U γ,j ( ˆ )], i,j = 1,2,...,N. (24) 4.2 The Randomized Ellipsoid Algorithm In the previous subsection we discussed how an initial ellipsoid that contains the feasibility set S γ can be computed. The complete randomized ellipsoid algorithm is summarized in Algorithm 4.1 (see also [8]). Steps 1-3 of the algorithm are used to compute an initial ellipsoid that contains the solution set S γ. Subsequently, it iterates between Steps 4 and 8 until convergence. The convergence of the EA algorithm can be established provided that the following technical assumption holds. 7

8 Algorithm 4.1 (The Randomized Ellipsoid Algorithm for (P F )). Initialization: i = 0, ε > 0 small, integer L > 0. Step 1. Generate a random uncertainty sample ˆ with the selected pdf f ( ). Compute the analytic center of S γ ( ˆ ) (22). Step 2. Compute the Hessian H(x ) using equation (24). Step 3. Compute x (0) and P 0 from equation (23). Step 4. Form the ellipsoid Step 5. Set i i + 1. E (i) = {x : (x x (i) ) T P 1 i (x x (i) ) 1} S γ. Step 6. Generate a random sample (i) with probability distribution f. Step 7. If v γ (x (i), (i) ) 0 then take x (i+1) = x (i) 1 P i v γ (x (i), (i) ) N + 1 ( T v γ (x (i), (i) )P i v γ (x (i), (i) ) P i+1 = N2 N 2 P i 2 P i v γ (x (i), (i) ) T v γ (x (i), (i) )Pi T 1 N + 1 T v γ (x (i), (i) )P i v γ (x (i), (i) ) ) else take x (i+1) = x (i), P i+1 = P i. ( det(p) ) Step 8. If < ε or ( v γ (x (i+j L), (i+j L) ) = 0 for j = 0,1,...,L ) then Stop else Goto Step 4. Assumption 4.2. For any x (i) S γ there is a non-zero probability to generate a sample (i) for which v γ (x (i), (i) ) > 0, i.e. Prob(v γ (x (i), (i) ) > 0) > 0. This assumption is not restrictive in practice. Note that a sufficient for the assumption to hold is that the density function f is nonzero everywhere. The assumption implies that for any x (i) S γ there exists a non-zero probability for the execution of a correction step (i.e. there is a non-zero probability for generation of (i) such that v γ (x (i), (i) ) > 0). Correction step means an iteration with v γ (x (i), (i) ) 0. The convergence of the approach is established immediately, provided that Assumption 4.2 holds. Lemma 4.2 (Convergence of Algorithm 4.1). Consider Algorithm 4.1 without the stopping condition in Step 5 (or with ε = 0 and L ), and suppose that Assumption 4.2 holds. Suppose also that (i) vol(s γ ) > 0. Then a feasible solution will be found in a finite number of iterations with probability one. (ii) vol(s γ ) = 0. Then with probability one. lim i x(i) = x S γ 8

9 Proof. Suppose that at the i-th iteration of Algorithm 4.1 k(i) correction steps have been performed. Algorithm 4.1 generates ellipsoids with geometrically decreasing volumes so that for the i-th iteration we can write [11] vol(e(x (i),p i )) e k(i) 2N vol(e(x (0),P 0 )), Due to Assumption 4.2, for any x (i) S γ there exists a non-zero probability for the execution of a correction step. Therefore, at any infeasible point x k(i) the algorithm will execute a correction step after a finite number of iterations with probability one. This implies that lim i vol(e(x(i),p i )) = 0. (25) (i) If we then suppose that the solution set S γ has a non-empty interior, i.e. vol(s) > 0, then from equation (25) and due to the fact that E(x (i),p i ) S γ for all i = 0,1,..., it follows that in a finite number of iterations with probability one the algorithm will terminate at a feasible solution. (ii) If we now suppose that vol(s) = 0, then due to the convexity of the function, and due to equation (25), the algorithm will converge to a point in S γ with probability one. Remark 4.1. It needs to be noted, however, the Lemma 4.2 considers Algorithm 4.1 with L, which in practice is never the case. For finite L the solution found by the algorithm can only be analyzed in a probabilistic sense. To be more specific, let some scalars ǫ (0,1) and δ (0,1) be given, and let x be the output of Algorithm 4.1 for ε = 0 and L ln 1 β /ln 1 1 ǫ. Then [14, 15] Prob{Prob{v γ (x, ) > 0} ǫ} 1 β. (26) Therefore, if we want with high confidence (e.g. β = 0.01) that the probability that x is an optimal solution is very high (1 ǫ = 0.999) then we need to select L larger than In practice, however, a much smaller value for L suffices. Remark 4.2. In [16] the author proposes to substitute the constant L with one that depends on the number of correction steps k(i) at iteration i. To this end they define L(k(i)) = (ln π2 (k(i) + 1) 2 ) / ln 1 6β 1 ǫ where a denotes the largest integer less or equal to a. It is then shown that if the algorithm terminates due to the second condition in Step 8 of Algorithm 4.1, namely that the no correction step has been made in the last L(k(i)) iterations, then (26) holds. Remark 4.3. A main disadvantage of the probabilistic method is that the computational speed strongly increases with the number of variables in the optimization as a result of the significant increase of the volume of the initial ellipsoid. In the state-feedback case, considered in this paper, the number of optimization variables remains significantly small for systems of reasonably small dimension so that the computational complexity is not a big issue. 5 Conclusions In this paper the randomized ellipsoid algorithm was presented with a discussion on its application to active fault-tolerant control design. It was shown that when estimates of the faults ˆf and their uncertainty intervals γ f are available online can design an LPV controller, scheduled by γ f and ˆf, that achieves internal stability and a certain bound on the L 2 -gain of the closed-loop system. Due to the randomized setting this approach is applicable to a very general class of faults as no restriction is imposed on the way the faults enter the system matrices. A disadvantage is that computational speed strongly increases with the number of variables in the optimization as a result of the significant increase of the volume of the initial ellipsoid. 9

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