Parameter-Dependent Lyapunov Functions For Linear Systems With Constant Uncertainties

Transcription

1 1 Parameter-Depenent Lyapunov Functions For Linear Systems With Constant Uncertainties Peter Seiler, Ufuk Topcu, Any Packar, an Gary Balas Abstract Robust stability of linear time-invariant systems with respect to structure uncertainties is consiere. The small gain conition is sufficient to prove robust stability an scalings are typically use to reuce the conservatism of this conition. It is known that if the small gain conition is satisfie with constant scalings then there is a single quaratic Lyapunov function which proves robust stability with respect to all allowable timevarying perturbations. In this paper we show that if the small gain conition is satisfie with frequency-varying scalings then an explicit parameter epenent Lyapunov function can be constructe to prove robust stability with respect to constant uncertainties. This Lyapunov function has a rational quaratic epenence on the uncertainties. I. INTRODUCTION Moeling uncertainties can be represente as parametric an/or ynamic perturbations to a nominal moel. The entire collection of moeling uncertainties in a system can be collecte into a single structure uncertainty see, for example, 1, 2. The structure singular value µ 2, 3, 4 provies a necessary an sufficient conition for robust stability with respect to structure linear-time invariant perturbations. However it is known that computing µ is NP Har 5, 6. Thus there has been extensive research into computational algorithms which are fast an provie goo lower/upper bouns for most problems of engineering interest. The small gain conition 7, 8 provies an easily computable sufficient conition for robust stability but is, in general, not necessary. Thus scalings are typically introuce to reuce the conservatism. For example, if there exists constant or frequency-epenent D-scales from an allowable set such that the small gain conition hols, then the system is robustly stable. These tests have their roots in the multiplier approaches use for passivity analysis 7. In 2, these two conitions are referre to as the frequency omain constant D test an the frequency omain upper boun. In this paper we will refer to these as the constant D an varying D tests. The constant D test is necessary an sufficient for robust stability with respect to arbitrarily fast linear time-varying perturbations 9. This conition is connecte to the notion of quaratic stability, i.e. the existence of a single quaratic Lyapunov function which proves stability of all possible trajectories of the uncertain system for both fixe an time-varying perturbations 1, 11, 12, 13, 14, 15. In particular, if the small gain conition hols with constant D-scales then the uncertain system is quaratically stable. The varying D test has a ifferent interpretation. The use of frequency-varying D-scales reners the small gain conition necessary an sufficient for robust stability with respect to arbitrarily slowly-varying linear perturbations 16. This is the conition that is typically use when computing µ versus frequency. Clearly this test is also sufficient for robust stability with respect to all constant perturbations from the allowable set. Thus for each fixe perturbation there exists a Lyapunov function proving stability of the system. In contrast to the constant D test, this Lyapunov function may be a function of the perturbation. This paper provies an explicit expression for a parameter-epenent Lyapunov function PDLF which can be erive from the varying D test. This Lyapunov function has a rational quaratic epenence on the uncertainties. There is a significant amount of relate research on PDLFs. The classical Popov criterion, when applie to a linear uncertainty, can be interprete as using a PDLF to prove robust stability 14, 17. This Lyapunov function is quaratic in the state an has an affine epenence on the uncertainty. There are many approaches to evelop robust stability conitions using more general PDLFs. Lyapunov functions having an affine 14, 18, 19, 2, 17, 21, 22, 23, 24, 25, multi-affine 26, bi-quaratic 27, generic polynomial 28, 29, 3, 31, an linear fractional epenence 32, 33 on the uncertainty have been consiere. Hermite matrices 34 an power forms 35 have also been consiere. The Kalman- Yakubovich-Popov KYP lemma 36 connects the PDLF in the Popov Criterion to a frequency omain conition but there are few aitional connections for these more general PDLF conitions. One connection is mae in 2, 21. In particular, 2, 21 consier linear systems with affine epenence on real parameter uncertainties. They erive a sufficient conition for robust stability using a PDLF having affine epenence on the uncertainties. They emonstrate that this sufficient conition is equivalent to the stanar real µ upper boun 37 but restricte to have constant D scales an G scales having a specific affine epenence on frequency. Thus the conition in 2, 21 is more conservative than the real µ upper boun with generic frequency-varying D G scales which is known to be equal to µ for certain block structures 38. The work on quaratic separators provies another relevant connection to PDLFs 39, 4, 41, 42. The authors erive necessary an sufficient robust stability conitions base on fining a Hermitian matrix-value function, terme a quaratic separator, which topologically separates the graph of the nominal system from the inverse graph of each uncertainty in the allowable set. One version of this conition can be interprete as simultaneously searching for a PDLF an a parameter-epenent quaratic separator which satisfy a linear matrix inequality see Theorem 3 of 4. These necessary an sufficient conitions are computationally ifficult to solve they are equivalent to computing µ an hence various sufficient conitions are erive. One of these sufficient conitions, terme the vertex-separator conition, can be use to construct a PDLF with polytopic epenence on the uncertainties. Another sufficient conition is obtaine for linear parameter varying systems by applying a constant quaratic separator. This sufficient conition is shown to be equivalent to the existence of a PDLF which has a linear fractional epenence on the uncertainty see Theorem 4 of 42. This particular PDLF will be iscusse further in Section IV. II. NOTATION C n m an R n m are complex an real n m matrices, respectively. For M C n m, M is the complex conjugate

2 2 transpose of M. The maximum singular value of M is enote by σ M. For M C n n, the spectral raius of M is enote by ρ M. If M = M then all the eigenvalues of M are real an λ max M enotes the largest eigenvalue. If M = M then M > an M < enote the matrix is positive an negative efinite, respectively. Given A C n m an B C r s, iaga, B C n+r m+s enotes the block iagonal concatenation of the two matrices. A B C rn sm enotes the Kronecker prouct. Let n an m be positive integers an partition M C n+m n+m as M = C A D B where A C n n, B C n m, C C m n, an D C m m. Let C m m be a matrix such that I D is invertible. In this case we efine the linear fractional transformation F l M, := A + B I D 1 C. The subscript l refers to the closure of the lower block of M with the matrix an we can use this transformation to efine an uncertain, autonomous iscrete-time system: x k+1 = F l M, x k. Similarly, for Ω C n n such that I AΩ is invertible, we efine F u M, Ω := D + CΩ I AΩ 1 B. The subscript u refers to the upper block of M being close with the matrix Ω. This transformation can be use to efine a transfer function matrix, e.g. Gz := D + C zi n A 1 B = F u M, 1 z I n. We efine G := θ 2π σ max Ge jθ. III. PRELIMINARY RESULTS This section presents lemmas which are use in Section IV to construct a PDLF from the varying D test. The first lemma is the Schur complement lemma. The next lemma relates a block 2 2 Lyapunov inequality to Lyapunov inequalities for the iagonal blocks. The last lemma relates an algebraic Riccati inequality to a robust stability conition. Lemma 1 Schur Complements 14, 43: Let P11 P P := 12 P C n+m n+m an P = P. The 12 P22 following conitions are equivalent: A P > B P 11 > an P 22 P12 P 11 1 P 12 > C P 22 > an P 11 P 12 P22 1 P 12 > Lemma 2 Block Lyapunov Inequality: Let A := C n+m n+m P11 P an P := 12 A11 A 12 A 22 P 12 P22 C n+m n+m be partitione conformably. If P = P > an A PA P < then: A ˆP := P 11 satisfies ˆP > an A ˆPA ˆP < B ˆQ := P22 P12 P 11 1 P 12 satisfies ˆQ > an A ˆQA ˆQ < Proof: A Any iagonal block of a positive negative efinite matrix must itself be positive negative efinite. Thus P > implies ˆP >. Also, the 1,1 block of A PA P is A ˆPA ˆP an hence this quantity is negative efinite. B By the Schur complement lemma, P > implies ˆQ >. Also by the Schur complement lemma, A PA P < implies AP 1 A P 1 <. The 2,2 block of P 1 is ˆQ 1 see Equation A.1.7 of 43. Thus the 2,2 block of AP 1 A P 1 < implies A ˆQ 1 22 A 22 ˆQ 1 <. One more application of the Schur complement lemma brings the esire result, A ˆQA ˆQ <. Lemma 3 Robust Stability Conition: Let M := A B C D C n+m n+m. If there exists P = P C n n such that P > an A PA P + Z Y 1 Z + C C < 1 where Y := I D D B PB an Z := B PA+D C then P satisfies: max λ max F l M, PF l M, P < 2 C m m, σ 1 Proof: The S-proceure can be use to prove this lemma. For example, Chapter 5 of 14 applies the S-proceure to erive a robust stability conition for continuous-time LTI systems with norm-boune time-varying uncertainties. For completeness, we provie a proof for iscrete-time systems which is base on completion of a square. The algebraic Riccati inequality Equation 1 can be use to show σ D < 1. Thus I D is invertible an F l M, is well-efine for all with σ 1. To simplify notation, efine W := I D 1 so that F l M, = A+BWC. Completing a square an using σ 1 yiels: F l M, PF l M, P = A PA P + Z Y 1 Z Y WC Z Y 1 Y WC Z + C W W W D DW W D DW C = A PA P + Z Y 1 Z Y WC Z Y 1 Y WC Z + C C + C I D II D 1 C A PA P + Z Y 1 Z + C C The esire result follows by applying Equation 1. IV. MAIN RESULT In this section we consier the robust stability of a iscrete time system with respect to structure uncertainties. We consier block structures C m m consisting of s repeate complex scalar blocks an f square full complex blocks. The restriction to square full blocks is for notational simplicity. Given positive integers m 1, m 2,..., m s+f satisfying s+f i=1 m i = m, we can efine the following sets of block structure m m matrices: := { = iagδ 1 I m1,...,δ s I ms, 1,..., f : 3 δ i C, i C ms+i ms+i} B := { : σ 1} 4 Associate with these block structures we can efine sets of constant an varying D-scales: D c := { D c C m m : D c = D c, etd c } { D v := D v z := D + C zi A 1 B : etd, A B } C D C k+m k+m, D v = D v By efinition, any scaling D s from either D c or D v satisfies = Ds 1 D s. Thus we can insert D s at the input to an Ds 1 at the output of. If D s F u M, 1 z ID 1 s < 1 then x k+1 = F l M, Ds 1 D s x k is robustly stable with respect to B. By the equivalence of the scale an unscale

3 3 systems, we can then conclue that x k+1 = F l M, x k is robustly stable with respect to B. This is an inirect proof of robust stability. The two theorems in this section irectly prove robust stability by constructing Lyapunov functions for x k+1 = F l M, x k. The theorems are state in a form which highlights this construction. Theorem 1 Constant D Test: Let Gz := F u M, 1 z I n where M := C A D B Cn+m n+m an ρa < 1. If there exists D c D c such that D c GDc 1 < 1 then: A There exists P = P C n n such that P > an: à Pà P + C C + B Pà + D C 5 I D D B 1 B P B Pà + D C < where à := A, B := BD 1 c, C := Dc C, an D := D c DDc 1. B The solution P > to Equation 5 satisfies max B λ max F l M, PF l M, P < 6 C Let { k } k= B be given. Then x = is a globally exponentially stable equilibrium point of x k+1 = F l M, k x k. Proof: A The Ã, B, C, an D given in the theorem are the state matrices for the system D c GDc 1. The existence of P > satisfying Equation 5 follows from D c GDc 1 < 1 an the iscrete-time Boune Real Lemma 2. B Define M = à B C D. Since P > satisfies Equation 5, we can apply Lemma 3 to conclue max λ max F l M, PF l M, P < C m m, σ 1 Restricting to B, we can use D c = D c to show F l M, = F l M,. C Define the Lyapunov function V x = x T Px. B implies there exists a β < 1 such that for any k B, V x k+1 < βv x k which guarantees the robust stability with respect to time-varying perturbations. Formally, C follows from B an iscrete-time Lyapunov theory Section 5.9 of 44. As note in the introuction, an uncertain system is quaratically stable if there exists a single quaratic Lyapunov function which proves stability of all possible trajectories of the uncertain system. Theorem 1 emonstrates that satisfying the small-gain theorem with constant D-scales implies quaratic stability. The next theorem emonstrates that using frequency varying D-scales implies the existence of a PDLF. One ifficulty is that the state matrices of D v z D v o not commute with the. However, we can erive how is altere as it moves through the state matrices of D v z. Let D v z := D + C zi A 1 B. Since D v z commutes with all, it must be block-iagonal: D v z =iagd v,1 z,..., D v,s z, v,s+1 zi ms+1,..., v,s+f zi ms+f A natural state space realization for D v z is given by: A :=iag A,1,..., A,s, I ms+1 A,s+1,..., 7 I ms+f A,s+f B :=iag B,1,...,B,s, I ms+1 B,s+1,..., 8 I ms+f B,s+f C :=iag C,1,...,C,s, I ms+1 C,s+1,..., 9 I ms+f C,s+f D :=iag D,1,..., D,s, I ms+1 D,s+1,..., 1 I ms+f D,s+f A,i B where,i C,i D,i C ki+mi ki+mi are the state space matrices of D v,i z i = 1,...,s an A,i B,i C,i D,i C ki+1 ki+1 are the state space matrices of v,i z i = s + 1,...,s + f. Next, efine X : C k k by: X =iag δ 1 I k1,..., δ s I ks, s+1 I ks+1,..., s+f I ks+f The imension of the i th block of X epens on the state imension, k i, of the transfer function in i th block of the D v z. For any, the state space realization of D v z given in Equations 7-1 satisfies the commutation relations: A X = X A, B = X B, C X = C, an D = D. Thus commutes with D but is altere when passing through B an C. These relations will be use in the proof of the following theorem. Theorem 2 Varying D Test: Let M := C A D B C n+m n+m with ρa < 1 an efine Gz := F u M, 1 z I n. If there exists D v z D v such that D v GDv 1 < 1 then: A There exists P = P C n+2k n+2k such that P > an: à Pà P + B Pà + D C I D D B P B 1 B Pà + D C + C C < 11 where A B C D C k+m k+m are the state matrices of D v z an: à := A B C B DD 1 C A BD 1 C A B D 1 C, B := B DD 1 BD 1 B D 1 C := C D C D DD 1 C, D := D DD 1 B Given the solution P > to Equation 11, efine: ˆP := P 22 + P 21 + P 23 X P 11 + P 13 X + X P 13 + X X 33 X 1 P 21 + P 23 X 12 where the P ij are a block 3 3 partition of P conformable with the 3 3 blocks of Ã. Then ˆP satisfies: max λ max F l M, ˆP Fl M, ˆP < B 13 C Let B be given. Then x = is a globally exponentially stable equilibrium point of x k+1 = F l M, x k. Proof: A The Ã, B, C, an D given in the theorem statement are the state matrices for the system D v GD 1 v.

4 4 The existence of P > satisfying Equation 11 follows from D v GDv 1 < 1 an the iscrete-time Boune Real Lemma 2. B Define M = Ã B C D. Since P > satisfies Equation 11, we can apply Lemma 3 to conclue max λ max F l M, PF l M, P < C m m, σ 1 Define the coorinate transformation: I T := I X I Define R := T 1 F l M, T an S := T PT. Multiplying F l M, PF l M, P on the left/right by T /T an inserting TT 1 yiels R SR S. This is a congruence transformation an hence R SR S remains strictly negative efinite for all B. Performing block multiplications an applying the commutations relations satisfie by X yiels: R = S = F l M, S 11 P 21+P 23X P 21+P 23X P 22 where S 11 := P 11 +P 13 X +X P 13 +X P 33 X an blocks enote by o not affect the remaining argument in the proof. The proof is conclue by first applying Lemma 2-A to R SR S < an then applying Lemma 2-B. C For each B efine the Lyapunov function V x, = x T ˆP x. C follows from B an iscrete-time Lyapunov theory Section 5.9 of 44. Comments: This paper uses a iscrete-time formulation but the results carry over to the continuous-time case. The lemmas in Section III must be restate in terms of the continuous-time Lyapunov inequality, Boune-Real Lemma, an algebraic Riccati inequality refer to Chapter 5 of 14. The proofs an results in Section IV then require only minor moifications. The continuous-time PDLF in the varying D test has the same structure an epenence on the solution of the continuous-time algebraic Riccati inequality. The algebraic Riccati inequalities Equations 5 an 11 can be converte to linear matrix inequalities by the Schur complement lemma. Thus we can use available software e.g. LMILab 1 an Seumi 45 to solve P >. Both theorems then give an explicit construction for a Lyapunov function which proves robust stability. However, this can be computationally emaning in the case of the varying D test since the variable P has imension n + 2k where k is the state imension of D v z. Fitting the magnitue ata, D v e θ, from a frequency grie mussv 1 calculation with a state-space moel can lea to high state imensions for D v z. This is especially true for the full blocks of D v z associate with repeate scalar uncertainties. If the i th block of D v z is constant then its state imension is k i =. In this case X, an hence P, o not epen on the corresponing block of. Thus we can obtain Lyapunov functions which are partially parameterepenent by fixing some blocks of the D-scale to be constant an others to be frequency varying. The example in the following section will further emonstrate this point. PDLFs with polynomial epenence on the uncertainty are use in 28, 29, 3, 31 for linear robust stability analysis an in 46, 47, 48, 49 for nonlinear region of attraction analysis. While polynomial epenence is without loss of generality for linear robust stability 25, 5, it might require a high egree. It woul be useful to see if algorithms can be evelope base on the particular form of the PDLF given in Equation 12. Theorem 2 emonstrates that satisfying the small gain conition with frequency varying D-scales implies the existence of a PDLF with a rational-quaratic epenence on the uncertainties. The class of PDLFs of this form inclues those which have an affine epenence on the uncertainties. This provies another explanation for why the affine PDLF conition given in 2, 21 is more conservative than using frequency-varying D G scales. We can also compare the PDLF from the varying D test to that obtaine from a special case of the quaratic separator conition for continuous-time, linear parameter varying systems 42. These systems have the form ẋ = A + B I td 1 tc with t having a block iagonal structure of repeate real scalars. A sufficient conition for robust stability with respect to the time-varying real parameters is erive using a constant quaratic separator. This sufficient conition is no more conservative than using constant D G scales but it is, in general, more conservative than using a frequency varying quaratic separator. The sufficient conition with the constant quaratic separator is shown to be equivalent to the existence of a PDLF of the form: ˆP := I I D 1 C T P I I D 1 C 14 This PDLF proves robust stability with respect to time-varying parameter variations an hence it also proves robust stability with respect to constant parameter uncertainties. The class of PDLFs of this form is not irectly comparable to PDLFs of the form given in Equation 12; in general neither form is more general than the other. It is notable that if the nominal system has no irect feethrough D = then the PDLF from the constant quaratic separator conition reuces to a quaratic epenence on the uncertainty. The form of the PDLF from the varying D test can, in principle, have a rational quaratic epenence on the uncertainty for any nominal system. We can briefly summarize four relate cases: 1 Constant D-scales: The small-gain conition with constant D-scales is only a sufficient conition for robust stability. If this sufficient conition is satisfie then the system is quaratically stable an there exists a parameter inepenent Lyapunov function which proves robust stability. 2 Frequency Varying D-scales: The small-gain conition with frequency varying D-scales is only a sufficient conition for robust stability. In this paper we showe that if this sufficient conition is satisfie then one can explicitly construct a PDLF Equation 12 which proves robust stability. 3 Constant Quaratic Separator: The constant quaratic separator conition is only a sufficient conition for robust

5 5 stability. If this sufficient conition is satisfie then there is a PDLF Equation 14 which proves robust stability 42. In general, the form of this PDLF is neither more nor less general than the form erive from the varying D test. 4 Frequency Varying Quaratic Separator: The frequency varying quaratic separator conition is necessary an sufficient for robust stability see Theorem 1 in either 4 or 41. It is not known how to explicitly construct a PDLF when this conition is satisfie. This woul be interesting since it woul provie a form for the PDLF which coul be assume without loss of generality when analyzing the robustness of linear systems with respect to constant uncertainties. In particular, it is well known that quaratic Lyapunov functions are sufficient to prove stability of linear systems. Thus if a linear system is stable then there is a Lyapunov function which is a quaratic function of the state which proves stability. One oes not nee to consier more complicate Lyapunov functions for linear systems. For stability analysis of uncertain linear systems this implies that we only nee to consier Lyapunov functions which are quaratic in the state but with an arbitrary functional epenence on the uncertainty. It woul be useful for algorithm evelopment to know if there is a functional epenence on the uncertainties which can be assume without loss of generality. Since the frequency-varying quaratic separator conition is necessary an sufficient for robust stability, it potentially provies a path to unerstaning this functional epenence. Specifically, if we can construct an explicit PDLF when this conition is satisfie then the form of this PDLF can be assume without loss of generality when analyzing linear uncertain systems. One might then evelop algorithms base on this functional form similar to the current evelopment of algorithms centere aroun affine Lyapunov functions. V. EXAMPLE Consier the two-state system x k+1 = F l M, x k from 12 where := { = iagδ 1, δ 2 : δ i C} an: a 2ba M := a 2ba 1 b 1 b Choose a =.9 an b =.5. This system is not quaratically stable but is robustly stable with respect to constant. Consier the time-varying perturbations k = iag1, for k even an k = iag, 1 for k o. For k o, the two step evolution of the system is x k+1 = xk. This has eigenvalues at.394 an which emonstrates that the system is not stable for all time varying perturbations in B. Hence the system is not quaratically stable. Define Gz := F u M, 1 z I 2. We use LMILab 1 to minimize D c GDc 1 over D c D c. The optimal constant scaling is D c = I 2 an, as expecte, the minimal value of D c GDc 1 = 9.5 which excees 1. Next, consier the scaling D v z = iag1,.9975z.925 z+.9. For this scaling D v GDv 1 =.526 < 1 an by the varying D test we conclue the system is robustly stable with respect to constant B. We again use LMILab to compute P > which satisfies Equation 11. The result is: P = Fig Level Set V==1 for δ 2 = 1 blue, δ 2 =+1 green, 1<δ 2 <+1 re As δ 2 goes from 1 to Level Sets {x : V x, = 1} for δ 2 1,1 The lines enote the 3 3 block partition use in the Lyapunov function construction of Theorem 2. Using this construction, efine the Lyapunov function V x, := x T ˆP x where ˆP is given by: ˆP = δ δ δ 2 5.8δ 2 2 This PDLF proves stability of x k+1 = F l M, x k for each constant B : We verifie this statement on a finite gri of values of δ 1, δ 2, δ i 1. This Lyapunov function oes not epen on δ 1 since the corresponing block of D v z is a constant. Figure 1 shows the unit level sets of this Lyapunov function as δ 2 varies from 1 to +1. The level sets are skewe an rotate with δ 2. VI. CONCLUSIONS This paper consiere robust stability with respect to structure uncertainties. If the small gain conition is satisfie with constant scalings then the uncertain system is robustly stable with respect to norm-boune time-varying perturbations. In this case, there is a single Lyapunov function which proves stability over all possible trajectories, i.e. the system is quaratically stable. If the small gain conition is satisfie with frequency-varying scalings then the uncertain system is robustly stable with respect to norm-boune constant perturbations. In this paper we constructe a PDLF which proves robust stability with respect to constant uncertainties. This Lyapunov function has a rational quaratic epenence on the uncertainties. It might prove fruitful to use this particular form to evelop algorithms for stability an region of attraction analysis for nonlinear, uncertain systems. It woul also be interesting to see if a similar explicit construction can be given for the frequency-varying quaratic separator conition. This woul be interesting since it woul provie a form for the PDLF which coul be assume without loss of generality when analyzing the robustness of linear systems with respect to constant uncertainties. Acknowlegments This research was partially supporte uner a NASA Langley NRA NNH77ZEA1N entitle Analytical Valiation

6 6 Tools for Safety Critical Systems. The technical contract monitor is Dr. Christine Belcastro. This research was also partially supporte by the Air Force Office of Scientific Research, USAF, uner grant number FA entitle Development of Analysis Tools for Certification of Flight Control Laws. The views an conclusions containe herein are those of the authors an shoul not be interprete as necessarily representing the official policies or enorsements, either expresse or implie, of the AFOSR or the U.S. Government. REFERENCES 1 G. Balas, R. Chiang, A. Packar, an M. Safonov, Robust Control Toolbox, MathWorks, A. Packar an J. Doyle, The complex structure singular value, Automatica, vol. 29, no. 1, pp , J. Doyle, Analysis of feeback systems with structure uncertainties, IEE Proc. Part D, vol. 129, no. 6, pp , M. Safonov, Stability margins of iagonally perturbe multivariable feeback systems, IEE Proc., Part D, vol. 129, no. 6, pp , R. Braatz, P. Young, J. Doyle, an M. 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