Nonlinear Observers: A Circle Criterion Design and Robustness Analysis

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1 Nonlinear Observers: A Circle Criterion Design and Robustness Analysis Murat Arcak 1 arcak@ecse.rpi.edu Petar Kokotović 2 petar@ece.ucsb.edu 1 Department of Electrical, Computer and Systems Engineering Rensselaer Polytechnic Institute Troy, NY Department of Electrical and Computer Engineering University of California Santa Barbara, CA Automatica 00-HKK-059R Abstract Globally convergent observers are designed for a class of systems with monotonic nonlinearities. The approach is to represent the observer error system as the feedback interconnection of a linear system and a time-varying multivariable sector nonlinearity. Using LMI software, observer gain matrices are computed to satisfy the circle criterion and, hence, to drive the observer error to zero. In output-feedback design, the observer is combined with control laws that ensure input-to-state stability with respect to the observer error. Robustness to unmodeled dynamics is achieved with a small-gain assignment design, as illustrated on a jet engine compressor example. Keywords- nonlinear observer design, output-feedback control, axial compressors, robustness, unmodeled dynamics. 1 Introduction Progress in nonlinear observers has been slower than in other areas of nonlinear control theory. Early results were obtained under a global Lipschitz restriction which excludes common nonlinearities such as x 3,expx), etc. The results of Thau 1973), Kou et al. 1975), Banks 1981) and their recent extensions by Raghavan This research was supported in part by the National Science Foundation under grant ECS and the Air Force Office of Scientific Research under grant F All correspondence should be addressed to Murat Arcak, Department of Electrical, Computer and Systems Engineering, Rensselaer Polytechnic University, Troy, NY, arcak@ecse.rpi.edu, tel: 518) , fax:518)

2 and Hedrick 1994), and Rajamani 1998) make use of quadratic Lyapunov functions for the observer error system. A more advanced framework was introduced by Tsinias 1989, 1993), who pursued an observer analog of control Lyapunov functions and used them for output feedback design. Another common restriction is to require that the nonlinearities appear as functions of the measured output. Systems which admit such a representation have been characterized by Krener and Isidori 1983), Bestle and Zeitz 1983), and other authors. The observer design for these systems is linear because the nonlinearity is canceled by an output injection term. A design which dominates state-dependent nonlinearities by high-gain linear terms has been developed by Esfandiari and Khalil 1992). To avoid the destabilizing effect of the peaking phenomenon analyzed by Sussmann and Kokotović 1991), this design employs judicious scaling of the observer gain matrix, and saturation of the peaking observer signals before they are fed to the controller. To achieve global convergence of high-gain observers, Gauthier et al. 1992) resorted to a global Lipschitz restriction. The design presented in this paper removes the global Lipschitz restriction and avoids high-gain. Instead, it makes two restrictions which allow the observer error system to satisfy the well-known multivariable circle criterion. First, a linear matrix inequality LMI) is to be feasible, which implies a strict positive real SPR) property for the linear part of the observer error system. The second restriction is that the nonlinearities be nondecreasing functions of linear combinations of unmeasured states. This restriction ensures that the vector time-varying nonlinearity in the observer error system satisfies the sector condition of the circle criterion. A preliminary idea of this global observer design was presented in Arcak and Kokotović 1999) for systems in strict-feedback form. In the current paper we develop a complete design procedure for more general systems and provide a robustness analysis for observer-based feedback design. In the proposed design, presented in Section 2, observer matrices satisfying the circle criterion are evaluated by efficient LMI computations. The robustness property against inexact modeling of nonlinearities is established, and the feasibility of a reduced-order variant of the observer is shown to be equivalent to that of the full-order observer. In Section 3, the new observer is incorporated in output-feedback design with control laws that ensure input-to-state stability ISS) with respect to the observer error. For unmodeled dynamics such as those studied by Praly and Jiang 1993), a small-gain assignment design achieves robust observerbased stabilization. As illustrated by a jet engine compressor example in Section 4, this design broadens the applicability of the new observer to systems which violate feasibility of the exact observer design. 2 Nonlinear Observers: Circle Criterion Design For our observer design we consider the plant ẋ = Ax + GγHx)+ϱy, u) 1) y = Cx, where x IR n is the state, y IR p is the measured output, u IR m is the control input, the pair A, C) is detectable, and, γ ) andϱ, ) are locally Lipschitz. The state-dependent nonlinearity γhx) isan r-dimensional vector where each entry is a function of a linear combination of the states n γ i = γ i H ij x j ), i =1,,r. 2) j=1 2

3 Our main restriction is that each γ i ) be nondecreasing, that is, for all a, b IR, it satisfies a b)[γ i a) γ i b)] 0. 3) If γ i ) is continuously differentiable, then dγ i v)/dv 0 for all v IR. If, instead, γ i ) satisfies dγ i v)/dv g i, g i 0, we can still represent the system as in 1)-3) by defining a new function γ i v) :=γ i v) g i v which satisfies d γ i v)/dv 0. When the nonlinearity γhx) also depends on y and u, the nondecreasing property 3) is to hold for each y IR p and u IR m. For the plant 1), we construct an observer in the form ˆx = Aˆx + LC ˆx y)+gγhˆx + KC ˆx y)) + ϱy, u). 4) Our task is to determine the observer matrices K IR r p and L IR n p. At this point we assume that the solution xt) of 1) does not escape to infinity in finite time. In Section 3 this assumption will be satisfied by our output feedback control design. From 1) and 4), the dynamics of the observer error e = x ˆx are governed by ė =A + LC)e + G [γv) γw)], 5) where v := Hx, w := H ˆx + KC ˆx y). 6) We begin the observer design by representing the observer error system 5) as the feedback interconnection of a linear system and a multivariable sector nonlinearity. To this end, we view γv) γw) as a function ϕ of v and z := v w =H + KC)e; that is, a time-varying nonlinearity in z: ϕt, z) :=γv) γw), 7) where the time dependence is due to vt). Substituting 7), we rewrite the observer error system 5) as ė = A + LC)e + Gϕt, z) 8) z = H + KC)e, and note from 3) that each component of ϕt, z) satisfies z i ϕ i t, z i ) 0, z i IR. 9) Thanks to this sector property of each ϕ i t, z i ), the product ϕt, z) T Λz is nonnegative for any diagonal Λ > 0. This means that the feedback path in the observer error system depicted in Figure 1 is a multivariable sector nonlinearity. Thus, from the circle criterion, asymptotic stability is guaranteed if the linear system with input ϑ and output Λz is SPR, that is, if a matrix P = P T > 0, a constant ν>0, and a diagonal matrix Λ > 0 exist such that [ ] A + LC) T P + P A + LC)+νI PG +H + KC) T Λ 0. 10) G T P +ΛH + KC) 0 Thus, the observer design for system 1) consists in finding observer matrices K and L to satisfy 10) with some P = P T > 0, Λ > 0, and ν>0. Efficient numerical tools are available for this task because 10) is a 3

4 ϑ ė =A + LC)e Gϑ z =H + KC)e z Λ Λz ϕ 1 t, ) ϕt, z)... ϕ r t, ) z Λ 1 Λz Figure 1: Observer error system. LMI in P, PL,Λ,ΛK and ν. It is important to note that the feasibility of 10) is not known a priori, and is to be determined numerically. An analytical test of feasibility has been derived in Arcak 2000). Robustness issues are critical in every observer design. To analyze the robustness of the above observer against inexact modeling of nonlinearities, we suppose that instead of 1), the plant is ẋ = Ax + G[γHx)+ Hx)µt)] + ϱy, u), 11) where µt) is an unknown bounded disturbance. Then, using the observer 4), we get the observer error system ė =A + LC)e + G [γv) γw)+ v)µt)], 12) where v and w are as in 6). We now characterize uncertain nonlinearities ) for which the observer 4) guarantees an ISS property from the disturbance µt) to the observer error et). Theorem 1 Consider the plant 11) and the observer 4). Suppose xt) exists for all t 0, and that the LMI 10) holds with a matrix P = P T > 0, a constant ν>0, and a diagonal matrix Λ > 0. If, for each i =1,,r, there exists a class-k function σ i ) such that a b)[γ i a) γ i b)+ i a)µ] σ i µ ) a, b, µ IR, 13) then the observer error et) satisfies, for all t 0, ) et) κ e0) exp βt)+ρ sup µτ), 14) where κ = λmaxp ) λ minp ), β = ν 2λ maxp ), and the ISS-gain from µt) to et) is ρ ) =κ 2 ν r λ i σ i ). 15) i=1 4

5 Proof: We use V = e T Pe as an ISS-Lyapunov function, and evaluate its derivative for 12): Substituting 13), we obtain V ν e 2 2 V ν e 2 +2 r λ i v i w i )[γ i v i ) γ i w i )+ i v i )µ]. 16) i=1 r λ i σ i µ ) 2βV +2 from which it follows that r et) κ e0) exp βt)+ 1 βλ min P ) i=1 r λ i σ i µ ), 17) i=1 λ i sup i=1 ) σ i µτ) ), 18) so that 14) and 15) result from 1 βλ minp ) = 2κ2 ν. The ISS property established by Theorem 1 shows that et) decreases with the decrease in the magnitude of the disturbance µt). As µt) vanishes, we recover exponential convergence of the observer error to zero. The dependence of uncertain nonlinearities ) onγ ) is characterized by 13). For example, if γ ) is cubic, then ) is allowed to be linear. In this case, 13) is satisfied because a b)[a 3 b 3 + aµ] 1 3 µ2 19) holds for all a, b, µ IR due to the identity a 3 b 3 =a b)a 2 + ab + b 2 ). In applications it may be more convenient to employ a reduced-order observer, which generates estimates only for the unmeasured states. The design of such an observer starts with a preliminary change of coordinates such that the output y consists of the first p entries of the state vector x =[y T x T o ] T. In the new coordinates, the system 1) is ẏ = A 1 x o + G 1 γh 1 y + H 2 x o )+ϱ 1 y, u) ẋ o = A 2 x o + G 2 γh 1 y + H 2 x o )+ϱ 2 y, u), 20) where the linear terms in y are incorporated in ϱ 1 y, u) andϱ 2 y, u). An estimate of x o will be obtained via χ := x o + Ny, where N IR n p) p is to be designed. From 20), the derivative of χ is: χ =A 2 + NA 1 )χ +G 2 + NG 1 )γh 2 χ +H 1 H 2 N)y)+ ϱy, u), 21) where ϱy, u) :=Nϱ 1 y, u)+ϱ 2 y, u) A 2 + NA 1 )Ny. In this χ-subsystem, the output injection matrix N has altered the A 2 and G 2 matrices of the x o -subsystem 20). To obtain the estimate ˆx o =ˆχ Ny, 22) we employ the observer ˆχ =A 2 + NA 1 )ˆχ +G 2 + NG 1 )γh 2 ˆχ +H 1 H 2 N)y)+ ϱy, u). 23) From 21) and 23), the dynamics of e o := x o ˆx o = χ ˆχ are governed by ė o =A 2 + NA 1 )e o +G 2 + NG 1 )[γv o ) γw o )], 24) 5

6 where v o := H 2 χ +H 1 H 2 N)y and w o := H 2 ˆχ +H 1 H 2 N)y. Weletz := v o w o = H 2 e o, and denote ϕt, z) =γv o ) γw o ). Then, the nondecreasing property 3) implies that z i ϕ i t, z i ) 0 for all i =1,,r, and derivations similar to those in Section 2 yield the following LMI in P o, P o N, ν and Λ: [ ] A2 + NA 1 ) T P o + P o A 2 + NA 1 )+νi P o G 2 + NG 1 )+H2 T Λ 0. 25) G 2 + NG 1 ) T P o +ΛH 2 0 If this LMI is satisfied with a matrix P o = Po T > 0, a constant ν 0, and a diagonal matrix Λ > 0, then it is not difficult to show that, with appropriate modifications, Theorem 1 holds for the observer error e o t). Furthermore, as we prove in Appendix A.1, the reduced-order observer exists whenever the full-order observer exists: Theorem 2 Let the constant ν 0 and the diagonal matrix Λ > 0 be given. statements are equivalent: Then, the following two 1. There exist matrices P = P T > 0, K and L satisfying the full-order observer LMI 10). 2. There exist matrices P o = P T o > 0 and N satisfying the reduced-order observer LMI 25). To emphasize the importance of the concept of injecting an output signal in the observer nonlinearity K 0), we provide an example in which the observer cannot be designed with K =0: Example 1 With y = x 1, and the nondecreasing nonlinearity γx 2 )=x 5 2, the system ẋ 1 = x 1 + x 2 ẋ 2 = x 2 x u 26) is of the form 1) with C =[1 0], G =[0 1] T, H =[0 1]. Because the off-diagonal term in [ the LMI 10) ] 1+l1 1 must be zero, the restriction K =0implies that P be diagonal. However, because A + LC =, l 2 1 it is not difficult to show that A + LC) T P + P A + LC) cannot be negative definite with a positive definite diagonal P. With K 0the design is feasible and a solution of the LMI 10) is L =[ 2 3] T, K = 1.5. The resulting observer is ˆx 1 = 2ˆx 1 y)+ˆx 2 + y ˆx 2 = 3ˆx 1 y)+ˆx 2 ˆx 2 1.5ˆx 1 y)) 5 + u. 27) Likewise, a reduced-order observer for ˆx 2 =ˆχ +1.5y is ˆχ = 0.5ˆχ ˆχ +1.5y) y + u. 28) 3 Output-Feedback Control The design of a nonlinear output-feedback control law using the certainty equivalence implementation of a full state-feedback law is, in general, impossible as we illustrate on the system 26). This system is stabilizable with full state feedback u = k 1 x 1 + k 2 x 2 + x ) 6

7 When only y = x 1 is measured, the certainty equivalence control law 29) with the unmeasured state x 2 replaced with its estimate ˆx 2 results in the closed-loop system ẋ 1 = x 1 + x 2 ẋ 2 = k 1 x 1 +k 2 +1)x 2 +[x 2 e 2 t)) 5 x 5 2] k 2 e 2 t) 30) where e 2 t) =x 2 t) ˆx 2 t) is exponentially decaying. As we prove in Appendix A.2, this system exhibits finite escape time for any choice of k 1 and k 2. Rather than attempting a certainty equivalence design, we view the state observer error e = x ˆx as a disturbance acting on the system 1), and design a control law u = αy, ˆx) =αy, x e) to render the system ISS with respect to the observer error et). Such ISS control laws have already been designed for classes of nonlinear systems in Freeman and Kokotović 1996), Krstić et al. 1995), and Marino and Tomei 1995). Here, we pursue a more general output-feedback design, where we achieve robustness against dynamic modeling errors. We consider the plant ẋ = Ax + G[γHx)+ Hx)µ]+ϱy, u), y = Cx 31) ξ = qξ,hx)) 32) µ = pξ,hx)), where γ ) and ) are as in 13), and the ξ-subsystem 32) represents unmodeled dynamics. This formulation extends the applicability of our observer because, if there is a subsystem to which the observer is not applicable, it can be treated as unmodeled dynamics. The unmodeled dynamics 32) are assumed to possess the following input-to-output stability IOS), and ISS properties: )} µt) max {β µ ξ0),t),ρ µh sup hxτ)) 33) )} ξt) max {β ξ ξ0),t),ρ ξh sup hxτ)), 34) where β µ, ), β ξ, ) are class-kl functions, and ρ µh ), ρ ξh ) are class-k functions. The ISS property of the observer error 14) is now rewritten in the max form of Teel 1996), which is suitable for the small-gain analysis we will employ: )} et) max {β e e0),t),ρ eµ sup µτ). 35) To prepare for a small-gain design of u = αy, ˆx), we represent the closed-loop system with the blockdiagram in Figure 2. The gains of the unmodeled dynamics block and the observer error block are ρ µh ) and ρ eµ ), respectively. Let ρ he ) andρ hµ ) be the plant gains from e and µ to hx), respectively. The task for the control law u = αy, ˆx) is to render ρ he ) andρ hµ ) small enough for the inner loop gain ρ hµ ρ µh ), and the outer loop gain ρ he ρ eµ ρ µh ) to satisfy, for all s>0, ρ hµ ρ µh s) <s 36) ρ he ρ eµ ρ µh s) <s. 37) Then, global asymptotic stability GAS) of the closed-loop system will be guaranteed as in the nonlinear small-gain theorem of Jiang et al. 1994), and Teel 1996). 7

8 µ Observer error system e µ Plant hx) Unmodeled dynamics Figure 2: Closed-loop system with observer feedback. Theorem 3 Consider the system 31)-32), in which γ ) and ) satisfy 13), and the ξ-subsystem satisfies 33) and 34). Suppose that the observer ˆx = Aˆx + LC ˆx y)+gγh ˆx + KC ˆx y)) + ϱy, u) 38) is such that the LMI 10) holds with a matrix P = P T > 0, a constant ν>0, and a diagonal matrix Λ > 0. If the control law u = αy, ˆx) guarantees ) )} hxt)) max {β h x0),t), ρ hµ sup µτ), ρ he sup eτ) 39) ) )} xt) max {β x x0),t), ρ xµ sup µτ), ρ xe sup eτ), 40) where ρ hµ ) and ρ he ) satisfy 36) and 37), respectively, then the origin of the closed-loop system 31), 32), 38) is globally asymptotically stable. 4 Design Example An axial compressor model, which has been the starting point for jet engine control studies, is the following single-mode approximation of a PDE model due to Moore and Greitzer 1986), φ = ψ φ φ +1)3 3φ +1)R 41) ψ = 1 φ +1 u) 42) β2 Ṙ = σr 2φ φ 2 R), R0) 0, 43) where φ and ψ are the deviations of the mass flow and the pressure rise from their set points, the control input u is the flow through the throttle, and, σ and β are positive constants. This model captures the main surge instability between the mass flow and the pressure rise. It also incorporates the nonnegative magnitude R of the first stall mode. A state feedback GAS control law in Krstić et al., 1995, Section 2.4) was replaced in Krstić et al. 1998) by a design using φ and ψ. With a high-gain observer, Isidori, 1999, Section 12.7), and Maggiore and Passino 2000) obtained a semiglobal result using the measurement of ψ alone. With y = ψ, we will now achieve GAS for 41)-43). While the exact observer cannot be designed because of the nonlinearities φr 8

9 and φ 2 R, the φ, ψ)-subsystem 41)-42) contains the nondecreasing nonlinearity φ +1) 3, and is of the form 31) with disturbance µ = R. This suggests that we treat the R-subsystem 43) as unmodeled dynamics and apply the small-gain design of Section 3. First, we prove that µ = R satisfies the IOS property 33) with hx) =φ as the input. With V = R 2 as an ISS-Lyapunov function, R 2.1 φ implies V 0.09σR 3, because Rt) 0 for all t 0. This means that 33) holds with the linear gain ρ µh ) =2.1 ), 44) and, since µ = ξ = R, the ISS property 34) is also satisfied. To design the reduced-order observer of Section 2 for the φ, ψ)-subsystem 41),42), we let χ = φ + Nψ, and obtain 3 χ = 2 + N ) β 2 χ 1 2 χ Nψ +1)3 3χ Nψ +1)R + ϱψ, u), 45) where 3 ϱψ, u) := 2 + N ) β 2 Nψ ψ N 1 u). 46) β2 The resulting observer for ˆφ =ˆχ Nψ is the scalar equation 3 ˆχ = 2 + N ) β 2 ˆχ 1 2 ˆχ Nψ +1)3 + ϱψ, u). 47) For its implementation, the LMI 25) is satisfied by selecting N such that 3 k := 2 + N ) β 2 > 0. 48) To prove the ISS property 35) for the observer error e φ = φ ˆφ, we employ the ISS-Lyapunov function V e = e 2 φ, and evaluate its derivative for ė φ = ke φ 1 2 a3 b 3 +6aR), 49) where a := χ Nψ +1andb =ˆχ Nψ + 1. Employing the inequality 19), and substituting a b = e φ, we get V e 2ke 2 φ +12R 2, 50) from which e φ 6.1 k R implies V e 0.03ke 2 φ, and, hence, the ISS property 35) holds with the linear gain 6.1 ρ eµ ) = ). 51) k We are now ready to design a control law as in Theorem 3. Noting that 41)-42) is in strict feedback form, we apply one step of observer backstepping. For ψ, we design the virtual control law α 0 = c 1 ˆφ. Denoting ω := ψ c 1 ˆφ = ψ c1 φ + c 1 e φ, 52) we rewrite 41) as φ = c 1 φ 3 2 φ2 1 2 φ3 3φR ω 3R + c 1 e φ. 53) The substitution of 47) in 52) yields ω =1+Nc 1 )ψ c 1 ˆχ, and, from 42) and 47), ω = 1+Nc 1 β 2 φ + 1 β 2 1 u)+γˆφ, ψ), 54) 9

10 where Γ ˆφ, ψ) :=c 1 ψ + c 1 k ˆφ + c1 2 ˆφ +1) 3 c1 2. Then, the control law is implementable using the signals ψ and ˆφ, and results in u =1+1+Nc 1 ) ˆφ + β 2 c 2 ω +Γˆφ, ψ)) 55) ω = c 2 ω + 1+Nc 1 β 2 e φ. 56) The remaining task is to select the design parameters c 1 and c 2 such that 39) and 40) are satisfied. For the ISS-Lyapunov function W φ, ω) := 1 2 φ ω2, the inequalities 3 2 φ3 9 8 φ φ4, φω 1 2 φ ω2, 3φR 9 4 φ2 +R 2, 3φ 2 R 0 because Rt) 0), c 1 φe φ c1 2 φ2 + c1 2 e2 1+Nc1 φ,and β ωe 2 φ 1+Nc1)2 2β 4 c 1 ω 2 + c1 2 e2 φ, yield c1 Ẇ 2 31 ) φ 2 c Nc 1) 2 ) 2β 4 ω 2 + R 2 + c 1 e 2 φ. 57) c 1 We let c>0, and select c 1 and c 2 to satisfy c ) >c, c Nc 1) 2 ) 2β 4 >c, 58) c 1 so that Ẇ cφ 2 + ω 2 )+R 2 +2c )e2 φ, 59) from which 40) follows for x =φ, ψ). For hx) =φ and µ = R, we now compute the gains ρ hµ ) andρ he ) in 39). Using the fact that for each constant θ>0, a + b max { 1 + θ 1 )a, 1 + θ)b } for all a, b 0, we obtain Ẇ cφ 2 + ω 2 )+R 2 +2c + 31 { 4 )e2 φ 2cW + max 1 + θ 1 )R 2, 1 + θ)2c + 31 } 4 )e2 φ, 60) from which it follows that { 1 + θ 1 ) W max R 2, 1.9c 1 + θ) 1.9c 2c + 31 } 4 )e2 φ Ẇ 0.1cW. 61) Then, 39) follows because φ 2W, and the gains are 1 + θ 1 ) 2 ρ hµ ) = ), ρ he ) = 1 + θ) 0.95c ) ). 62) 3.8c Using 44), 51) and 62), the inner and outer loop small-gain conditions, 36) and 37) are, respectively, 1 + θ 1 ) c θ) k ) 3.8c < 1, 63) < 1. 64) Selecting c>0andk>0sufficiently large ensures that 63) and 64) hold. Additional freedom for the selection of c and k is obtained from θ>0, which allocates the inner and outer loop gains. 10

11 5 Conclusion Output-feedback control of nonlinear systems continues to be a challenging research area in which every new effort attempts to remove previous restrictions and improve robustness. The advance made in this paper is to allow the presence of nondecreasing nonlinearities to depend on linear combinations of unmeasured states. Global convergence of the nonlinear observer is achieved without the usual global Lipschitz restriction or high-gain feedback. Instead, admissible nonlinearities and system structures are determined via verifiable feasibility conditions, while the design is carried out with computationally efficient LMI software. The ubiquitous question of observer and closed-loop system robustness is also answered for a significant class of modeling errors. It is hoped that these results will reinvigorate the search for new structure-specific nonlinear observer designs. Appendix A.1 Proof of Theorem 2 1 2) Suppose the full-order observer LMI 10) is feasible for the system 20). Partitioning P and L as [ ] [ ] P1 P 2 L1 P =, L =, P2 T P 3 L 2 65) and substituting [ 0 A1 ] [ G1 ] A = 0 A 2, G = G 2, H =[H 1 H 2 ], and C =[I 0 ] 66) in 10), we obtain A + LC) T P + P A + LC)+νI = 67) [ ] 0 P 3 A 2 + P3 1 P2 T A 1 )+A 2 + P3 1 P2 T A 1 ) T P 3 + νi [ ] PG+H + KC) T Λ= =0. 68) P 3 G 2 + P3 1 P2 T G 1 )+H2 T Λ Defining P o := P 3,andN := P 1 3 P T 2, we note that 67) and 68) imply which is the reduced-order observer LMI 25). P o A 2 + NA 1 )+A 2 + NA 1 ) T P o + νi 0 69) P o G 2 + NG 1 )+H T 2 Λ=0, 70) 2 1) Suppose 69) and 70) hold for the system 20). To prove the existence of P = P T > 0, K and L satisfying 10), we rewrite system 20) in y and χ = x o + Ny coordinates, so that [ ] [ ] L1 A 1 G 1 A + LC =, G =, 71) L 2 A 2 + NA 1 G 2 + NG 1 11

12 H + KC =[H 1 H 2 N + K H 2 ], 72) where L 1 := L 1 A 1 N and L 2 = L 2 A 2 + NA 1 )N. Welet [ ] I 0 P =, 73) 0 P o and obtain [ L A + LC) T 1 + P + P A + LC) = L T 1 A 1 + L ] T 2 P o P o L2 + A T 1 P o A 2 + NA 1 )+A 2 + NA 1 ) T P o [ ] PG+H + KC) T G1 +H 1 H 2 N + K) T Λ Λ=. 74) P o G 2 + NG 1 )+H2 T Λ In view of 69) and 70), the choice L 1 = ν 2 I, L2 = P 1 0 A T 1, K = H 1 + H 2 N Λ 1 G T 1 75) results in the full-order observer LMI 10). A.2 Proof of finite escape time in system 30) To prove that system 30) exhibits finite escape time for any choice of k 1 and k 2, we use a variant of Lemma 3 from Mazenc et al. 1994): Lemma 1 Consider the system ζ 1 = ζ 2 ζ 2 = f 2 ζ 1,ζ 2,t), 76) where ζ 1,ζ 2 IR, andf 2 C 1. If there exists a nonempty open set U IR, and positive constants a, b, d, s and n>2 such that, for all t [0,s], ζ 1 U and ζ 2 >d, bζ n 2 <f 2 ζ 1,ζ 2,t) <aζ n 2, 77) then, for ζ 1 0) U and sufficiently large ζ 2 0) >d, ζ 2 t) escapes to infinity in finite time. With the new variables ζ 1 := exp t)x 1 and ζ 2 := exp t)x 2, system 30) is transformed into the form 76) with f 2 ζ 1,ζ 2,t)=k 1 ζ 1 + k 2 ζ 2 +exp t){[x 2 e 2 t)) 5 x 5 2] k 2 e 2 t)}. 78) Noting that the leading term of [x 2 e 2 t)) 5 x 5 2]is 5e 2 t)x 4 2, it is not difficult to show that the hypotheses of Lemma 1 hold with n = 4 on any compact time interval t [0,s]inwhiche 2 t) is negative. Thus, with e 2 0) < 0, and x 2 0) > 0 sufficiently large, the closed-loop system 30) exhibits finite escape time. References Arcak, M. 2000). Unmodeled Dynamics in Robust Nonlinear Control. PhD thesis. University of California, Santa Barbara. 12

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