witched systems Part 3 doc

Stabilizationofsaturatedswitchingsystems 19 Let each convex set Γ i has µ i vertices ν iκ , κ = 1, . . . , µ i so that for every q i ∈ Γ i , one can write q i = ∑ µ i κ=1 β iκ ν iκ with ∑ µ i κ=1 β iκ = 1,0 ≤ β iκ ≤ 1. The consequence of this, is that each matrix A i (q i (t)), B i (q i (t)) and C i (q i (t)) can be expressed as a convex combination of the corresponding vertices of the compact set Γ i as follows: M (q i ) : = M i + µ i ∑ κ=1 β iκ M(ν iκ ) = µ i ∑ κ=1 β iκ M iκ , M (ν iκ ) = d i ∑ h=1 M ih ν iκh , M iκ = M i + M(ν iκ ), µ i ∑ κ=1 β iκ = 1, 0 ≤ β iκ ≤ 1. where M i represents the nominal matrix. Matrix M can be taken differently as A, B or C. Note that the system without uncertainties can be obtained as a particular case of this representa- tion by letting the vertices ν iκ = 0, ∀i, ∀κ. Besides, equations (69) are directly related to the dimension d i of the convex compact set Γ i . The saturated uncertain switching system given by (68) can be rewritten as: x t+1 = N ∑ i=1 µ i ∑ κ=1 ξ i (t)β iκ (t)[A iκ x t + B iκ sat(K i C iκ x t )] (69) The nominal matrices will be represented by A i , B i and C i . The nominal system in closed-loop is then given by: x t+1 = N ∑ i=1 ξ i (t)[A i x t + B i sat(K i C i x t )] (70) 3.2 Analysis and synthesis of stabilizability This section presents sufficient conditions of asymptotic stability of the saturated uncertain switching system given by (69). The synthesis of the controller follows two different ap- proaches, the first one deals firstly with the nominal system and then uses a test to check the asymptotic stability in presence of uncertainties while the second considers the global representation of the uncertain system (69). Theorem 3.1. If there exist symmetric positive definite matrices P 1 , . . . , P N ∈ R n×n and matrices H 1 , . . . , H N ∈ R m×n such that [ P i [A iκ + B iκ (D is K i C iκ + D − is H i )] T P j ∗ P j ] > 0, (71) ∀κ = 1, . . . , µ i , ∀(i, j) ∈ ℐ 2 , ∀s ∈ [1, η], (72) and ε (P i , 1) ⊂ ℒ(H i ), (73) then the closed-loop uncertain saturated switching system (69) is asymptotically stable ∀ x 0 ∈ Ω := ∪ N i =1 ε(P i , 1) and for all switching sequences α(t). Proof: By using Lemma (2.1), for all H i ∈ R m×n with ∣H ij x t ∣ < 1, j ∈ [1, m], where H ij denotes the jth row of matrix H i , there exist δ iκ1 ≥ 0 , , δ iκη ≥ 0 such that sat(K i C iκ x t ) = ∑ η s =1 δ isκ (t)[D is K i C iκ + D − is H i ]x t , δ iκs (t) ≥ 0, ∑ η s =1 δ iκs (t) = 1. Then the closed-loop system (69) can be rewritten as x t+1 = η ∑ s=1 N ∑ i=1 µ i ∑ κ=1 ξ i (t)β iκ (t)δ iκs (t)Ac iκs x t (74) Ac iκs := A iκ + B iκ (D is K i C iκ + D − is H i ). Consider the Lyapunov function candidate V (x) = x T t ( ∑ N i =1 ξ i (t)P i )x t . Computing its rate of increase along the trajectories of system (69) yields. ∆V (x t ) = x T t +1 ( N ∑ j=1 ξ j (t + 1)P j )x t+1 − x T t ( N ∑ i=1 ξ i (t)P i )x t = x T t ⎧ ⎨ ⎩ Σ T ( N ∑ j=1 ξ j (t + 1)P j )Σ − N ∑ i=1 ξ i (t)P i ⎫ ⎬ ⎭ x t . where, Σ = η ∑ s=1 N ∑ i=1 µ i ∑ κ=1 ξ i (t)β iκ (t)δ sκi (t)Ac iκs Let condition (71) be satisfied. For each i and j multiply successively by ξ i (t), ξ j (t + 1), β iκ (t) and δ iκs (t) and sum. As ∑ N i =1 ξ i (t) = ∑ N j =1 ξ j (t + 1) = ∑ η s =1 δ iκs (t) = ∑ µ i κ=1 β iκ (t) = 1, one gets: [ ∑ N i =1 ξ i (t)P i Π ∗ ∑ N j =1 ξ j (t + 1)P j ] > 0, (75) where Π = Σ T ( N ∑ j=1 ξ j (t + 1)P j ). Inequality (75) is equivalent, by Schur complement, to Σ T ( N ∑ j=1 ξ j (t + 1)P j )Σ − N ∑ i=1 ξ i (t)P i < 0 Letting λ be the largest eigenvalue among all the above matrices, we obtain that ∆V (x t ) ≤ λx T t x t < 0, (76) which ensures the desired result. Besides, following Theorem 2.4, (71)-(73) also allow for a state belonging to a set ε (P i , 1) ⊂ ℒ(H i ), before the switch, if a switch occurs at time t k , the switch will drive the state to the desired set ε (P j , 1) ⊂ ℒ(H j ). That means that the set Ω is a set of asymptotic stability of the uncertain saturated switching system. □ Remark 3.1. It is worth to note that the result of Theorem 2.4 can be obtained as a particular case of Theorem 3.1. SwitchedSystems20 This stability result is now used for control synthesis in two ways: the first consists in com- puting the controllers only with the nominal system and to test their robustness in a second step; while the second consists in computing in a single step the robust controllers. At this end, the result of Theorem 2.6 can be used to compute matrices K i , H i and P i for the nominal switching system (70). At this step, the stabilizing controllers K i and H i of the nominal system are assumed to be known. Then, the following test has to be performed. Corollary 3.1. If there exist symmetric positive definite matrices X i such that [ X i (A iκ X i + B iκ D is K i C iκ X i + B iκ D − is H i X i ) T ∗ X j ] > 0, (77) [ 1 (H i X i ) l ∗ X i ] > 0, (78) ∀(i, j) ∈ ℐ 2 , ∀s ∈ [1, η], ∀l ∈ [1, m], ∀κ ∈ [1, µ i ], with P i = X −1 i , then the closed loop uncertain switching system (69) is asymptotically stable ∀ x 0 ∈ ∪ N i =1 ε(P i , 1) and for all switching sequences α(t). Proof: The proof is similar to that given for Theorem 2.6. □ The second way to deal with robust controller design is to run a global set of LMIs leading, if it is feasible, to the robust controllers directly. However, one can note that this method is computationally more intensive. Theorem 3.2. If there exist symmetric positive definite matrices X i , matrices, Y i , V i and Z i such that [ X i (A iκ X i + B iκ D is Y i C iκ + B iκ D − is Z i ) T ∗ X j ] > 0, (79) [ 1 Z il ∗ X i ] > 0, (80) V i C iκ = C iκ X i , (81) ∀(i, j) ∈ ℐ 2 , ∀s ∈ [1, η], ∀l ∈ [1, m], ∀κ ∈ [1, µ i ] with H i = Z i X −1 i , K i = Y i V −1 i , P i = X −1 i , (82) then, the closed-loop uncertain saturated switching system (69) is asymptotically stable ∀ x 0 ∈ Ω, and for all switching sequences α (t). Proof: The proof is also similar to that given for Theorem 2.6. □ In order to relax the previous LMIs, one can introduce some slack variables as in (Daafouz et al., 2002) and (Benzaouia et al., 2006), as it is now shown: Theorem 3.3. If there exist symmetric positive definite matrices X i , matrices, Y i , V i , G i and Z i such that [ G i + G T i − X i Ψ ∗ X j ] > 0, (83) with Ψ = (A iκ G i + B iκ D is Y i C iκ + B iκ D − is Z i ) T ,  1 Z il ∗ G i + G T i − X i  > 0, (84) V i C iκ = C iκ G i , (85) ∀κ = 1, . . . , µ i , ∀(i, j) ∈ ℐ 2 , ∀s ∈ [1, η], ∀l ∈ [1, m], with H i = Z i G −1 i , K i = Y i V −1 i , P i = X −1 i ; (86) then, the closed-loop uncertain saturated switching system (69) is asymptotically stable ∀ x 0 ∈ Ω and for all switching sequences α (t). Proof: The proof is similar to that given for Corollary 2.1 □ These results can be illustrated with the following example. Example 3.1. Consider a SISO saturated switching discrete system with two modes given by the following matrices: A 1 (q 1 (t)) =  1 1 0 1 + q 11  ; B 1 (q 1 (t)) =  10 5  ; C 1 (q 1 (t)) =  1 + q 12 1  ; A 2 (q 2 (t)) =  0 + q 21 −1 0.0001 1  ; B 2 (q 2 (t)) =  0.5 −2 + q 22  ; C 2 (q 2 (t)) =  2 3  . The vertices of the domain of uncertainties that affect the first mode are: ν 11 = (−0.1, −0.2), ν 12 = (−0.1, 0.2) ν 13 = (0.1, −0.2), ν 14 = (0.1, 0.2). The vertices of the domain of uncertainties that affect the second mode are: ν 21 = (−0.2, 0.5), ν 22 = (−0.2, −0.1) ν 23 = (0.3, 0.5), ν 24 = (0.3, −0.1). Using Theorem 2.6, a stabilizing controller for the nominal system is K 1 = −0.1000, K 2 = 0.1622. To test the robustness, we can use the Corollary 3.1 which leads to the following results: P 1 =  0.0208 −0.0133 −0.0133 0.0257  ; P 2 =  0.0320 0.0023 0.0023 0.0474  On the other hand, the use of Theorem 3.2 leads to the following results: K 1 = −0.0902, K 2 = 0.1858. Stabilizationofsaturatedswitchingsystems 21 This stability result is now used for control synthesis in two ways: the first consists in com- puting the controllers only with the nominal system and to test their robustness in a second step; while the second consists in computing in a single step the robust controllers. At this end, the result of Theorem 2.6 can be used to compute matrices K i , H i and P i for the nominal switching system (70). At this step, the stabilizing controllers K i and H i of the nominal system are assumed to be known. Then, the following test has to be performed. Corollary 3.1. If there exist symmetric positive definite matrices X i such that [ X i (A iκ X i + B iκ D is K i C iκ X i + B iκ D − is H i X i ) T ∗ X j ] > 0, (77) [ 1 (H i X i ) l ∗ X i ] > 0, (78) ∀(i, j) ∈ ℐ 2 , ∀s ∈ [1, η], ∀l ∈ [1, m], ∀κ ∈ [1, µ i ], with P i = X −1 i , then the closed loop uncertain switching system (69) is asymptotically stable ∀ x 0 ∈ ∪ N i =1 ε(P i , 1) and for all switching sequences α(t). Proof: The proof is similar to that given for Theorem 2.6. □ The second way to deal with robust controller design is to run a global set of LMIs leading, if it is feasible, to the robust controllers directly. However, one can note that this method is computationally more intensive. Theorem 3.2. If there exist symmetric positive definite matrices X i , matrices, Y i , V i and Z i such that [ X i (A iκ X i + B iκ D is Y i C iκ + B iκ D − is Z i ) T ∗ X j ] > 0, (79) [ 1 Z il ∗ X i ] > 0, (80) V i C iκ = C iκ X i , (81) ∀(i, j) ∈ ℐ 2 , ∀s ∈ [1, η], ∀l ∈ [1, m], ∀κ ∈ [1, µ i ] with H i = Z i X −1 i , K i = Y i V −1 i , P i = X −1 i , (82) then, the closed-loop uncertain saturated switching system (69) is asymptotically stable ∀ x 0 ∈ Ω, and for all switching sequences α (t). Proof: The proof is also similar to that given for Theorem 2.6. □ In order to relax the previous LMIs, one can introduce some slack variables as in (Daafouz et al., 2002) and (Benzaouia et al., 2006), as it is now shown: Theorem 3.3. If there exist symmetric positive definite matrices X i , matrices, Y i , V i , G i and Z i such that [ G i + G T i − X i Ψ ∗ X j ] > 0, (83) with Ψ = (A iκ G i + B iκ D is Y i C iκ + B iκ D − is Z i ) T ,  1 Z il ∗ G i + G T i − X i  > 0, (84) V i C iκ = C iκ G i , (85) ∀κ = 1, . . . , µ i , ∀(i, j) ∈ ℐ 2 , ∀s ∈ [1, η], ∀l ∈ [1, m], with H i = Z i G −1 i , K i = Y i V −1 i , P i = X −1 i ; (86) then, the closed-loop uncertain saturated switching system (69) is asymptotically stable ∀ x 0 ∈ Ω and for all switching sequences α (t). Proof: The proof is similar to that given for Corollary 2.1 □ These results can be illustrated with the following example. Example 3.1. Consider a SISO saturated switching discrete system with two modes given by the following matrices: A 1 (q 1 (t)) =  1 1 0 1 + q 11  ; B 1 (q 1 (t)) =  10 5  ; C 1 (q 1 (t)) =  1 + q 12 1  ; A 2 (q 2 (t)) =  0 + q 21 −1 0.0001 1  ; B 2 (q 2 (t)) =  0.5 −2 + q 22  ; C 2 (q 2 (t)) =  2 3  . The vertices of the domain of uncertainties that affect the first mode are: ν 11 = (−0.1, −0.2), ν 12 = (−0.1, 0.2) ν 13 = (0.1, −0.2), ν 14 = (0.1, 0.2). The vertices of the domain of uncertainties that affect the second mode are: ν 21 = (−0.2, 0.5), ν 22 = (−0.2, −0.1) ν 23 = (0.3, 0.5), ν 24 = (0.3, −0.1). Using Theorem 2.6, a stabilizing controller for the nominal system is K 1 = −0.1000, K 2 = 0.1622. To test the robustness, we can use the Corollary 3.1 which leads to the following results: P 1 =  0.0208 −0.0133 −0.0133 0.0257  ; P 2 =  0.0320 0.0023 0.0023 0.0474  On the other hand, the use of Theorem 3.2 leads to the following results: K 1 = −0.0902, K 2 = 0.1858. SwitchedSystems22 Figures 5, 6 and 7 concern the first method. In Figure 5, the switching signals α(t) and the evolution of uncertainties used for simulation, are plotted. Figure 6 shows the obtained level set of stability ∪ N i =1 ε(P i , 1) which is well contained inside the sets of saturations, while Figure 7 presents some system motions evolving inside the level set starting from different initial states. 0 5 10 15 0 1 2 3 Switching signal 0 5 10 15 −0.1 0 0.1 0 5 10 15 −0.2 0 0.2 0 2 4 6 8 0.2 0.3 0.4 0 2 4 6 8 0 0.5 uncertainties evolution 0 5 10 15 20 0 1 2 3 0 5 10 15 20 −0.1 0 0.1 0 5 10 15 20 −0.2 0 0.2 0 5 10 15 0.2 0.3 0.4 0 5 10 15 0 0.5 t t t t t t t t t t Fig. 5. Switching signals α(t) and uncertainties evolution −10 −8 −6 −4 −2 0 2 4 6 8 10 −150 −100 −50 0 50 100 150 Fig. 6. Inclusion of the ellipsoids inside the polyhedral sets −10 −8 −6 −4 −2 0 2 4 6 8 10 −10 −8 −6 −4 −2 0 2 4 6 8 10 x2 x1 Fig. 7. Motion of the system with controllers obtained with Theorem2.6 and Corollary 3.1 . Figure 7 shows the level set of stability ∪ N i =1 ε(P i , 1) using the second method of Theorem 3.2 which is well contained inside the sets of saturations. The use of Theorem 3.3 leads to the following results: K 1 = −0.0752, K 2 = 0.1386; Figure 9 shows the level set of stability ∪ N i =1 ε(P i , 1) obtained with Theorem 3.3, which is also well contained inside the sets of saturations. −5 −4 −3 −2 −1 0 1 2 3 4 5 −15 −10 −5 0 5 10 15 Fig. 8. Inclusion of the ellipsoids inside the polyhedral sets obtained with Theorem 3.2 . −10 −8 −6 −4 −2 0 2 4 6 8 10 −150 −100 −50 0 50 100 150 Fig. 9. Inclusion of the ellipsoids inside the polyhedral sets obtained with Theorem 3.3 . 3.3 Synthesis of non saturating controllers The non saturating controllers who works inside a region of linear behavior can be obtained from the previous results by replacing D si = I and D − si = 0. The following result presents the synthesis of such controllers. Theorem 3.4. If there exist symmetric matrices X i and matrices Y i such that [ X i (A iκ X i + B iκ Y i C iκ ) T ∗ X j ] > 0, (87) [ 1 Y il C iκ ∗ X i ] > 0, (88) V i C iκ = C iκ X i , (89) ∀ κ = 1, . . . , µ i , ∀(i, j) ∈ ℐ 2 , ∀l ∈ [1, m], with, K i = Y i V −1 i , P i = X −1 i , then the uncertain closed-loop switching system (69) is asymptotically stable ∀ x 0 ∈ Ω and for all switching sequences α (t). To illustrate this result, the same system of Example 3.1 is used. Theorem 3.4 leads to the following results: P 1 = [ 0.2574 0 0 0.2574 ] ; P 2 = [ 0.2930 0.0535 0.0535 0.3376 ] ; Stabilizationofsaturatedswitchingsystems 23 Figures 5, 6 and 7 concern the first method. In Figure 5, the switching signals α(t) and the evolution of uncertainties used for simulation, are plotted. Figure 6 shows the obtained level set of stability ∪ N i =1 ε(P i , 1) which is well contained inside the sets of saturations, while Figure 7 presents some system motions evolving inside the level set starting from different initial states. 0 5 10 15 0 1 2 3 Switching signal 0 5 10 15 −0.1 0 0.1 0 5 10 15 −0.2 0 0.2 0 2 4 6 8 0.2 0.3 0.4 0 2 4 6 8 0 0.5 uncertainties evolution 0 5 10 15 20 0 1 2 3 0 5 10 15 20 −0.1 0 0.1 0 5 10 15 20 −0.2 0 0.2 0 5 10 15 0.2 0.3 0.4 0 5 10 15 0 0.5 t t t t t t t t t t Fig. 5. Switching signals α(t) and uncertainties evolution −10 −8 −6 −4 −2 0 2 4 6 8 10 −150 −100 −50 0 50 100 150 Fig. 6. Inclusion of the ellipsoids inside the polyhedral sets −10 −8 −6 −4 −2 0 2 4 6 8 10 −10 −8 −6 −4 −2 0 2 4 6 8 10 x2 x1 Fig. 7. Motion of the system with controllers obtained with Theorem2.6 and Corollary 3.1 . Figure 7 shows the level set of stability ∪ N i =1 ε(P i , 1) using the second method of Theorem 3.2 which is well contained inside the sets of saturations. The use of Theorem 3.3 leads to the following results: K 1 = −0.0752, K 2 = 0.1386; Figure 9 shows the level set of stability ∪ N i =1 ε(P i , 1) obtained with Theorem 3.3, which is also well contained inside the sets of saturations. −5 −4 −3 −2 −1 0 1 2 3 4 5 −15 −10 −5 0 5 10 15 Fig. 8. Inclusion of the ellipsoids inside the polyhedral sets obtained with Theorem 3.2 . −10 −8 −6 −4 −2 0 2 4 6 8 10 −150 −100 −50 0 50 100 150 Fig. 9. Inclusion of the ellipsoids inside the polyhedral sets obtained with Theorem 3.3 . 3.3 Synthesis of non saturating controllers The non saturating controllers who works inside a region of linear behavior can be obtained from the previous results by replacing D si = I and D − si = 0. The following result presents the synthesis of such controllers. Theorem 3.4. If there exist symmetric matrices X i and matrices Y i such that [ X i (A iκ X i + B iκ Y i C iκ ) T ∗ X j ] > 0, (87) [ 1 Y il C iκ ∗ X i ] > 0, (88) V i C iκ = C iκ X i , (89) ∀ κ = 1, . . . , µ i , ∀(i, j) ∈ ℐ 2 , ∀l ∈ [1, m], with, K i = Y i V −1 i , P i = X −1 i , then the uncertain closed-loop switching system (69) is asymptotically stable ∀ x 0 ∈ Ω and for all switching sequences α (t). To illustrate this result, the same system of Example 3.1 is used. Theorem 3.4 leads to the following results: P 1 = [ 0.2574 0 0 0.2574 ] ; P 2 = [ 0.2930 0.0535 0.0535 0.3376 ] ; SwitchedSystems24 K 1 = −0.0902; K 2 = 0.1694. 0 2 4 6 8 10 12 14 16 0 0.5 1 1.5 2 2.5 3 0 2 4 6 8 10 12 14 16 18 0 0.5 1 1.5 2 2.5 3 t Switching signals t Fig. 10. Switching supervisor signal α(t) −4 −3 −2 −1 0 1 2 3 4 −20 −15 −10 −5 0 5 10 15 20 x2 x1 Fig. 11. Inclusion of the ellipsoids obtained with Theorem 3.4 −2 −1.5 −1 −0.5 0 0.5 1 1.5 2 −2 −1.5 −1 −0.5 0 0.5 1 1.5 2 x2 x1 Fig. 12. Motion of the system with controllers obtained with Theorem 3.4 In Figure 5, the evolution of uncertainties is plotted, Figure 10 shows the sequence α (t). The level set ∪ N i =1 ε(P i , 1) presented in Figure 11, is also well contained inside the regions of linear behavior. In Figure 12, the trajectories of the system are plotted. Commennt 3.1. The application of all the proposed results to the same example, shows that the result applied in two steps (Theorem 2.6 and Corollary 3.1) is the least conservative. However, it is worth noting that the result introducing slack variables (Theorem 3.3) is also less conservative even applied in one step. One can expect that this same result applied in two steps can be the less conservative one. In this section, two different sufficient conditions of asymptotic stability are obtained for out- put feedback control of uncertain switching discrete-time linear systems subject to actuator saturations. These conditions allow the synthesis of stabilizing controllers inside a large re- gion of saturation under LMIs formulation. Note that the state feedback control case and the unsaturating controller case can be obtained as particular cases of the study presented in this section. An illustrative example is studied by using the direct resolution of the proposed LMIs. A comparison of the obtained solutions is also given. 4. Stabilization of saturated switching systems with structured uncertainties The objective of this section is to extend the results of (Benzaouia et al., 2006) to uncertain switching systems subject to actuator saturations by using output feedback control. This tech- nique allows to design stabilizing controllers by output feedback for switching discrete-time systems despite the presence of actuator saturations and uncertainties on the system param- eters. The case of state feedback control is derived as a particular case. It is also shown that the results obtained in this section with state feedback control are less conservative than those presented in (Yu et al., 2007) where only the state feedback control case is addressed. The main results of this section are published in (Benzaouia et al., 2009c). 4.1 Problem presentation Let us consider the linear uncertain discrete-time switching system described by: { x t+1 = 𝒜 α (t)x t + ℬ α (t)sat(u t ) y t = 𝒞 α (t)x t (90) where x t ∈ R n , u t ∈ R m are the state and the input respectively, sat(.) is the standard satura- tion, y t ∈ R p the output. α is a switching rule taking its values in the finite set I = {1, , N}. The saturation function is assumed here to be normalized, i. e., ∣sat(u i ∣ = min(1, ∣u i ∣), i = 1, . . . m. The system matrices are assumed to be uncertain and satisfy: [ 𝒜 i (t) ℬ i (t) ] = [ A i B i ] + M i Γ i [ N 1i N 2i ] (91) Let the control be obtained by an output feedback control law: u t = K α y k = K α C α x t = F α x t The closed-loop system is given by: x t+1 = 𝒜 α (t)x t + ℬ α (t)sat(K α C α x t ) (92) Defining the indicator function: ξ (t) := [ξ 1 (t), , ξ N (t)] T (93) where ξ i (t) = 1 if the switching system is in mode i and 0 otherwise, yields the following representation for the closed-loop system: x t+1 = N ∑ i=1 ξ i (t)[𝒜 i (t)x t + ℬ i (t)sat(K i C i x t )] (94) Stabilizationofsaturatedswitchingsystems 25 K 1 = −0.0902; K 2 = 0.1694. 0 2 4 6 8 10 12 14 16 0 0.5 1 1.5 2 2.5 3 0 2 4 6 8 10 12 14 16 18 0 0.5 1 1.5 2 2.5 3 t Switching signals t Fig. 10. Switching supervisor signal α(t) −4 −3 −2 −1 0 1 2 3 4 −20 −15 −10 −5 0 5 10 15 20 x2 x1 Fig. 11. Inclusion of the ellipsoids obtained with Theorem 3.4 −2 −1.5 −1 −0.5 0 0.5 1 1.5 2 −2 −1.5 −1 −0.5 0 0.5 1 1.5 2 x2 x1 Fig. 12. Motion of the system with controllers obtained with Theorem 3.4 In Figure 5, the evolution of uncertainties is plotted, Figure 10 shows the sequence α (t). The level set ∪ N i =1 ε(P i , 1) presented in Figure 11, is also well contained inside the regions of linear behavior. In Figure 12, the trajectories of the system are plotted. Commennt 3.1. The application of all the proposed results to the same example, shows that the result applied in two steps (Theorem 2.6 and Corollary 3.1) is the least conservative. However, it is worth noting that the result introducing slack variables (Theorem 3.3) is also less conservative even applied in one step. One can expect that this same result applied in two steps can be the less conservative one. In this section, two different sufficient conditions of asymptotic stability are obtained for out- put feedback control of uncertain switching discrete-time linear systems subject to actuator saturations. These conditions allow the synthesis of stabilizing controllers inside a large re- gion of saturation under LMIs formulation. Note that the state feedback control case and the unsaturating controller case can be obtained as particular cases of the study presented in this section. An illustrative example is studied by using the direct resolution of the proposed LMIs. A comparison of the obtained solutions is also given. 4. Stabilization of saturated switching systems with structured uncertainties The objective of this section is to extend the results of (Benzaouia et al., 2006) to uncertain switching systems subject to actuator saturations by using output feedback control. This tech- nique allows to design stabilizing controllers by output feedback for switching discrete-time systems despite the presence of actuator saturations and uncertainties on the system param- eters. The case of state feedback control is derived as a particular case. It is also shown that the results obtained in this section with state feedback control are less conservative than those presented in (Yu et al., 2007) where only the state feedback control case is addressed. The main results of this section are published in (Benzaouia et al., 2009c). 4.1 Problem presentation Let us consider the linear uncertain discrete-time switching system described by: { x t+1 = 𝒜 α (t)x t + ℬ α (t)sat(u t ) y t = 𝒞 α (t)x t (90) where x t ∈ R n , u t ∈ R m are the state and the input respectively, sat(.) is the standard satura- tion, y t ∈ R p the output. α is a switching rule taking its values in the finite set I = {1, , N}. The saturation function is assumed here to be normalized, i. e., ∣sat(u i ∣ = min(1, ∣u i ∣), i = 1, . . . m. The system matrices are assumed to be uncertain and satisfy: [ 𝒜 i (t) ℬ i (t) ] = [ A i B i ] + M i Γ i [ N 1i N 2i ] (91) Let the control be obtained by an output feedback control law: u t = K α y k = K α C α x t = F α x t The closed-loop system is given by: x t+1 = 𝒜 α (t)x t + ℬ α (t)sat(K α C α x t ) (92) Defining the indicator function: ξ (t) := [ξ 1 (t), , ξ N (t)] T (93) where ξ i (t) = 1 if the switching system is in mode i and 0 otherwise, yields the following representation for the closed-loop system: x t+1 = N ∑ i=1 ξ i (t)[𝒜 i (t)x t + ℬ i (t)sat(K i C i x t )] (94) SwitchedSystems26 Assume that there exist N matrices H 1 , . . . , H N such that x(t) ∈ ℒ(H i ). Using the expression in (10) and rewriting System (94) yields that: x t+1 = η ∑ s=1 N ∑ i=1 ξ i (t)δ is (t)𝒜c is (t)x t ; (95) 𝒜c is (t) := 𝒜 i (t) + ℬ i (t)(D is K i C i + D − is H i ), s ∈ [1, η] 4.2 Analysis and synthesis of stabilizability Consider now the saturated uncertain switching system given by (95). The first result synthe- sizing stabilizing controllers of the uncertain saturated switching system by output feedback is now presented. Theorem 4.1. If there exist symmetric matrices X 1 , . . . , X N , matrices Y 1 , . . . , Y N , Z 1 , . . . , Z N , V 1 , . . . , V N and a set of real positive scalars λ ijs , such that ⎡ ⎣ X i Θ T is Φ T is ∗ X j −λ ijs M i M T i 0 ∗ ∗ λ ijs I ⎤ ⎦ > 0, (96) ∀(i, j) ∈ ℐ × ℐ,∀s ∈ [1, . . . η] C i X i = V i C i (97) [ 1 Z il ∗ X i ] > 0, (98) ∀i ∈ ℐ, ∀l ∈ [1, . . . m] where Θ is = A i X i + B i (D is Y i C i + D − is Z i ) and Φ is = N 1i X i + N 2i (D is Y i C i + D − is Z i ). Then, the uncertain switching system with input saturation in closed-loop (95) with K i = Y i V −1 (99) H i = Z i X −1 (100) is asymptotically stable ∀ x 0 ∈ Ω = ∪ N i =1 ε(X −1 i , 1) and for all switching sequences α(t). Proof: By using Lemma 2.1, for all H i ∈ R m×n with ∣H il x t ∣ < 1, l ∈ [1, m], there exist δ i1 ≥ 0 , , δ iη ≥ 0 such that, sat(K i C i x t ) = ∑ η s =1 δ is (t)[D is K i C i + D − is H i ]x t , δ is (t) ≥ 0, ∑ η s =1 δ is (t) = 1. System (94) is then rewritten as (95). Consider the Lyapunov function candidate V (x) = x T t ( ∑ N i=1 ξ i (t)P i )x t . Computing its rate of increase along the trajectories of system (95) yields: ∆V (x t ) = x T t +1 ( N ∑ j=1 ξ j (t + 1)P j )x t+1 − x T t ( N ∑ i=1 ξ i (t)P i )x t = η ∑ s=1 N ∑ j=1 ξ j (t + 1)δ is x T t [𝒜 i + ℬ i (D is F i + D − is H i )] T P j [𝒜 i + ℬ i (D is F i + D − is H i )]x t − N ∑ i=1 ξ i (t)x T t P i x t Since, ∑ η s =1 δ is (t) = ∑ N j =1 ξ j (t + 1) = ∑ N i =1 ξ i (t) = 1, one should obtain ∆V (x t ) = N ∑ j=1 N ∑ i=1 η ∑ s=1 ξ i (t)ξ j (t + 1)δ is (t)x T t  [𝒜 i + ℬ i (D is F i + D − is H i )] T P j [𝒜 i + ℬ i (D is F i + D − is H i )] − P i  x t A sufficient condition to obtain ∆V(x t ) < 0 is that:  𝒜 i + ℬ i (D is F i + D − is H i )] T P j [𝒜 i + ℬ i (D is F i + D − is H i )  − P i = −Ψ sij < 0 (101) By applying Schur complement to (101), the following equivalent inequality is obtained:  P i [𝒜 i + ℬ i (D is K i C i + D − is H i )] T ∗ P −1 j  > 0, (102) Letting X i = P −1 i , Y i = K i V i , C i X i = V i C i , Z i = H i X i and multiplying the above inequality on both sides by diag (X i , I) we get  X i [𝒜 i X i + ℬ i (D is K i C i + D − is H i )X i ] ⊤ ∗ X j  > 0, (103) Taking account of (91), inequality (103) can be developed as follows: −  X i [A i X i + B i (D is Y i C i + D − is Z i )] ⊤ ∗ X j  +  [N 1i X i + N 2i (D is Y i C i + D − is Z i )] ⊤ 0  Γ ⊤ i  0 −M T i  +  0 −M i  Γ i  [N 1i X i + N 2i (D is Y i C i + D − is Z i )] 0  < 0, by virtue of Lemma2.2, this inequality holds if and only if there exist positive scalars λ ijs such that −  X i Θ T is ∗ X j  + λ ijs  0 −M i   0 −M T i  + 1 λ ijs  Φ is 0   Φ T is 0  < 0, ∀(i, j) ∈ ℐ × ℐ, ∀s ∈ [1, . . . η]. Or in a compact form,  X i − 1 λ ijs Φ is Φ T is Θ T is ∗ X j −λ ijs M i M T i  > 0, (104) ∀(i, j) ∈ ℐ × ℐ, ∀s ∈ [1, η] where Φ is and Θ is are defined before. By Schur complement, inequality (104) is equivalent to (96). One can then bound the rate of increase as follows, ∆V (x t ) ≤ −γ(∥x t ∥); γ (∥x t ∥) = min ijs λ min (Ψ ijs )∥x t ∥ 2 . Stabilizationofsaturatedswitchingsystems 27 Assume that there exist N matrices H 1 , . . . , H N such that x(t) ∈ ℒ(H i ). Using the expression in (10) and rewriting System (94) yields that: x t+1 = η ∑ s=1 N ∑ i=1 ξ i (t)δ is (t)𝒜c is (t)x t ; (95) 𝒜c is (t) := 𝒜 i (t) + ℬ i (t)(D is K i C i + D − is H i ), s ∈ [1, η] 4.2 Analysis and synthesis of stabilizability Consider now the saturated uncertain switching system given by (95). The first result synthe- sizing stabilizing controllers of the uncertain saturated switching system by output feedback is now presented. Theorem 4.1. If there exist symmetric matrices X 1 , . . . , X N , matrices Y 1 , . . . , Y N , Z 1 , . . . , Z N , V 1 , . . . , V N and a set of real positive scalars λ ijs , such that ⎡ ⎣ X i Θ T is Φ T is ∗ X j −λ ijs M i M T i 0 ∗ ∗ λ ijs I ⎤ ⎦ > 0, (96) ∀(i, j) ∈ ℐ × ℐ, ∀s ∈ [1, . . . η] C i X i = V i C i (97) [ 1 Z il ∗ X i ] > 0, (98) ∀i ∈ ℐ, ∀l ∈ [1, . . . m] where Θ is = A i X i + B i (D is Y i C i + D − is Z i ) and Φ is = N 1i X i + N 2i (D is Y i C i + D − is Z i ). Then, the uncertain switching system with input saturation in closed-loop (95) with K i = Y i V −1 (99) H i = Z i X −1 (100) is asymptotically stable ∀ x 0 ∈ Ω = ∪ N i =1 ε(X −1 i , 1) and for all switching sequences α(t). Proof: By using Lemma 2.1, for all H i ∈ R m×n with ∣H il x t ∣ < 1, l ∈ [1, m], there exist δ i1 ≥ 0 , , δ iη ≥ 0 such that, sat(K i C i x t ) = ∑ η s =1 δ is (t)[D is K i C i + D − is H i ]x t , δ is (t) ≥ 0, ∑ η s =1 δ is (t) = 1. System (94) is then rewritten as (95). Consider the Lyapunov function candidate V (x) = x T t ( ∑ N i=1 ξ i (t)P i )x t . Computing its rate of increase along the trajectories of system (95) yields: ∆V (x t ) = x T t +1 ( N ∑ j=1 ξ j (t + 1)P j )x t+1 − x T t ( N ∑ i=1 ξ i (t)P i )x t = η ∑ s=1 N ∑ j=1 ξ j (t + 1)δ is x T t [𝒜 i + ℬ i (D is F i + D − is H i )] T P j [𝒜 i + ℬ i (D is F i + D − is H i )]x t − N ∑ i=1 ξ i (t)x T t P i x t Since, ∑ η s =1 δ is (t) = ∑ N j =1 ξ j (t + 1) = ∑ N i =1 ξ i (t) = 1, one should obtain ∆V (x t ) = N ∑ j=1 N ∑ i=1 η ∑ s=1 ξ i (t)ξ j (t + 1)δ is (t)x T t  [𝒜 i + ℬ i (D is F i + D − is H i )] T P j [𝒜 i + ℬ i (D is F i + D − is H i )] − P i  x t A sufficient condition to obtain ∆V(x t ) < 0 is that:  𝒜 i + ℬ i (D is F i + D − is H i )] T P j [𝒜 i + ℬ i (D is F i + D − is H i )  − P i = −Ψ sij < 0 (101) By applying Schur complement to (101), the following equivalent inequality is obtained:  P i [𝒜 i + ℬ i (D is K i C i + D − is H i )] T ∗ P −1 j  > 0, (102) Letting X i = P −1 i , Y i = K i V i , C i X i = V i C i , Z i = H i X i and multiplying the above inequality on both sides by diag (X i , I) we get  X i [𝒜 i X i + ℬ i (D is K i C i + D − is H i )X i ] ⊤ ∗ X j  > 0, (103) Taking account of (91), inequality (103) can be developed as follows: −  X i [A i X i + B i (D is Y i C i + D − is Z i )] ⊤ ∗ X j  +  [N 1i X i + N 2i (D is Y i C i + D − is Z i )] ⊤ 0  Γ ⊤ i  0 −M T i  +  0 −M i  Γ i  [N 1i X i + N 2i (D is Y i C i + D − is Z i )] 0  < 0, by virtue of Lemma2.2, this inequality holds if and only if there exist positive scalars λ ijs such that −  X i Θ T is ∗ X j  + λ ijs  0 −M i   0 −M T i  + 1 λ ijs  Φ is 0   Φ T is 0  < 0, ∀(i, j) ∈ ℐ × ℐ, ∀s ∈ [1, . . . η]. Or in a compact form,  X i − 1 λ ijs Φ is Φ T is Θ T is ∗ X j −λ ijs M i M T i  > 0, (104) ∀(i, j) ∈ ℐ × ℐ, ∀s ∈ [1, η] where Φ is and Θ is are defined before. By Schur complement, inequality (104) is equivalent to (96). One can then bound the rate of increase as follows, ∆V (x t ) ≤ −γ(∥x t ∥); γ (∥x t ∥) = min ijs λ min (Ψ ijs )∥x t ∥ 2 . SwitchedSystems28 Using (Hu et al., 2002), the inclusion condition (29) can also be transformed to the equivalent LMI (98) by virtue of the results of (Boyd et al., 1994). □ To obtain larger ellipsoid domains ε(P i , 1), we can use a shape reference set 𝒳 R ⊂ R n , in terms of a polyhedron or ellipsoid to measure the size of the domain of attraction. For a set ℒ ⊂ R n which contains the origin, define µ(𝒳 R , ℒ) = sup { µ > 0, µ𝒳 R ⊂ ℒ } . Here, we choose 𝒳 R to be a polyhedral defined as 𝒳 R = co { ω 1 , ω 2 , , ω q } , where ω 1 , ω 2 , , ω q are a prior given points in R n . The problem can be formulated as the following constrained optimization problem (Pb.4) : ⎧   ⎨   ⎩ max X i >0,Y i ,Z i ,λ ijs (µ i ) s.t. µ𝒳 R ⊂ ε(P i , 1) ( 96), (98), i = 1, . . . , N As is explained in (Hu et al., 2001) and ( Hu and Lin, 2002), the constraint µ 𝒳 R ⊂ ε(P i , 1) is satisfied if the following matrix inequalities hold: [ µ −2 i ω T l ω l X i ] ≥ 0, (105) ∀i ∈ ℐ, ∀l ∈ [1, q] The problem of enlarging the domain of attraction can be reduced to an LMI optimization problem defined as follows: (Pb.5) : ⎧ ⎨ ⎩ min X i >0,Y i ,Z i ,λ ijs (γ i ) s.t. (96), (98), (105) i = 1, . . . , N where γ i = µ −2 i . Commennt 4.1. The results of Theorem 4.1 applies directly to switching systems with state feedback control by taking C i = I. In this case, these results can be compared to the one given in (Yu et al.,2007). The fact that the scalars λ ijs are all kept equal in (Yu et al.,2007), makes the result obviously more conservative. An example will show this conservatism. In order to more improve the result of Theorem 4.1 by introducing additional slack variables, the following corollary is presented. Corollary 4.1. If there exist symmetric matrices X i > 0, matrices G i , Y i , V i , Z i and positive scalars λ ijs such that ⎡ ⎢ ⎢ ⎣ G T i + G i − X i Υ T is 0 Λ T is ∗ X j λ ijs M i 0 ∗ ∗ λ ijs I 0 ∗ ∗ ∗ λ ijs I ⎤ ⎥ ⎥ ⎦ > 0, (106) ∀(i, j) ∈ ℐ 2 , ∀s ∈ [1, . . . η] C i G i = V i C i (107) [ 1 Z il ∗ G T i + G i − X i ] > 0, (108) ∀i ∈ ℐ, ∀s ∈ [1, . . . η], ∀l ∈ [1, . . . m] where Υ is = A i G i + B i (D is Y i C i + D − is Z i ) and Λ is = N 1i G i + N 2i (D is Y i C i + D − is Z i ). Then, the uncertain switching system with input saturation in the closed-loop (95) with K i = Y i V −1 (109) H i = Z i G −1 (110) is asymptotically stable ∀ x 0 ∈ Ω = ∪ N i =1 ε(X −1 i , 1) and for all switching sequences α(t). Proof: It was proven in (Benzaouia et al., 2004) and (Benzaouia et al., 2006) that condition (102) is feasible if and only if there exists non singular matrices G i such that the following inequality holds: [ G i + G T i − X i G T i [𝒜 i + ℬ i (D is K i C i + D − is H i )] T ∗ X j ] > 0, (111) ∀(i, j) ∈ ℐ × ℐ, ∀s ∈ [1, η] where X i = P −1 i . The same reasoning is then followed as in the proof of Theorem 4.1 leading to (106). Inequality (108) was also proven in (Benzaouia et al., 2006) by using (Boyd et al., 1994). □ These results can be illustrated with the following example. Example 4.1. Consider a SISO saturated switching discrete-time system with two modes given by the following matrices: A 1 = [ 1 1 0 1 ] , B 1 = [ 10 5 ] , M 1 = 0.1I, N 11 = N 12 = 0.01I, A 2 = [ 0 −1 0 1 ] , B 2 = [ 0.5 −2 ] , M 2 = 0.1I, N 21 = N 22 = 0.01I, By solving the optimization problem (Pb.5) for the above system, we can obtain the following results: P 1 = 10E −03 [ 4.3324 1.2516 1.2516 4.3324 ] ; P 2 = 10E −03 [ 4.3988 2.0934 2.0934 6.1433 ] , H 1 = [ −0.0261536 − 0.0653823]; H 2 = [−0.0000192 0.0717335] K 1 = −0.1000089; K 2 = 0.1256683 The corresponding figures are given by Figure13 and Figure 14. By applying Corollary 4.1, the follow- ing results are obtained: P 1 = 10E −03 [ 1.0047 0.1917 0.1917 2.0626 ] ; P 2 = 10E −04 [ 7.380 1.823 1.823 23.530 ] , [...]... 4 .33 24 1.2516 4 .39 88 2.0 934 P1 = 10E − 03 ; P2 = 10E − 03 , 1.2516 4 .33 24 2.0 934 6.1 433 H1 K1 = = [−0.0261 536 − 0.06 538 23] ; H2 = [−0.0000192 0.071 733 5] −0.1000089; K2 = 0.12566 83 The corresponding figures are given by Figure 13 and Figure 14 By applying Corollary 4.1, the following results are obtained: [ ] [ ] 1.0047 0.1917 7 .38 0 1.8 23 ; P2 = 10E − 04 , P1 = 10E − 03 0.1917 2.0626 1.8 23 23. 530 30 Switched... 0.1917 2.0626 1.8 23 23. 530 30 Switched Systems 40 30 20 10 x2 0 −10 −20 30 −40 −40 30 −20 −10 0 x1 10 20 30 40 Fig 13 Inclusion of the ellipsoids inside the polyhedral sets using Theorem 4.1 Switching signal 3 2.5 2 1.5 1 0.5 0 0 2 4 6 8 t 10 12 14 16 Fig 14 The switching sequences H1 K1 = = [−0.01797 53 − 0.0405 034 ]; H2 = [−0.0000072 0.048 039 7] −0.05971 63; K2 = 0. 036 538 9 The corresponding figures are given... 6750 H1 F1 = = [−0.0002470 − 0.0060812]; H2 = [−0.0000081 0.0081 436 ] [−0.00 032 54 − 0.00998 73] ; F2 = [−0.006 833 9 0.6156659] while the results of (Yu et al., 2007) give: [ ] [ 2 8 4.894 P1 = 10E − 07 ; P2 = 10E − 08 8 7 73 20 H1 F1 = = 20 7 630 ] , [−0.00 032 03 − 0.0066782]; H2 = [−0.0000016 0.0086701] [−0.0050262 − 0 .38 21976]; F2 = [−0.01128 93 0 .36 95061] The corresponding level sets are depicted in Figure... al., 2007) is obvious Stabilization of saturated switching systems 31 40 30 20 10 x2 0 −10 −20 30 −40 −40 30 −20 −10 0 x1 10 20 30 40 Fig 15 Inclusion of the ellipsoids inside the polyhedral sets using Theorem 4.1 Switching signal 3 2.5 2 1.5 1 0.5 0 0 5 10 t 15 20 25 Fig 16 The switching sequences This section studied uncertain switching systems with output feedback control which extends the results... the particular case of the present work, as mentioned by the Comment 4.1, is also presented The obtained improvements with the methods presented in this work are shown in Figure 17 and Figure 18 A numerical example is used to illustrate all these techniques 5 CONCLUSION In this chapter, two main different sufficient conditions of asymptotic stability are obtained for switching discrete-time linear systems. ..Stabilization of saturated switching systems [ 29 Ci Gi = Vi Ci ] > 0, 1 ∗ Zil GiT + Gi − Xi (107) (108) ∀i ∈ ℐ , ∀s ∈ [1, η ], ∀l ∈ [1, m] − − where Υis = Ai Gi + Bi ( Dis Yi Ci + Dis Zi ) and Λis = N1i Gi + N2i ( Dis Yi Ci + Dis Zi... the second applies the idea of Lemma (Hu et al., 2002) which rewrites the saturation function under a combination of 2m elements to obtain stabilizing controllers tolerating saturations to take effect A particular attention is given to the output feedback case which has additive complexity due to the output equation In this sense, three different LMIs are presented for this case The main results of this... formed by the union of all the ellipsoid level sets associated to each subsystem, constitutes a set of asymptotic stability The first time that this important result is established for saturated switching systems is in (Benzaouia et al., 2006) Two illustrative examples are studied by using the solution of the proposed LMIs A comparison of the obtained . 10E − 03 [ 4 .33 24 1.2516 1.2516 4 .33 24 ] ; P 2 = 10E − 03 [ 4 .39 88 2.0 934 2.0 934 6.1 433 ] , H 1 = [ −0.0261 536 − 0.06 538 23] ; H 2 = [−0.0000192 0.071 733 5] K 1 = −0.1000089; K 2 = 0.12566 83 The corresponding. 10E − 03 [ 4 .33 24 1.2516 1.2516 4 .33 24 ] ; P 2 = 10E − 03 [ 4 .39 88 2.0 934 2.0 934 6.1 433 ] , H 1 = [ −0.0261 536 − 0.06 538 23] ; H 2 = [−0.0000192 0.071 733 5] K 1 = −0.1000089; K 2 = 0.12566 83 The corresponding. results: P 1 =  0.0208 −0.0 133 −0.0 133 0.0257  ; P 2 =  0. 032 0 0.00 23 0.00 23 0.0474  On the other hand, the use of Theorem 3. 2 leads to the following results: K 1 = −0.0902, K 2 = 0.1858. Switched Systems2 2 Figures