Stability of uncertain systems using Lyapunov functions with non-monotonic terms

Automatica 82 (2017) 187–193

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Brief paper

Stability of uncertain systems using Lyapunov functions with non-monotonic terms✩ Márcio J. Lacerda a , Peter Seiler b a

Department of Electrical Engineering, Federal University of São João del-Rei - UFSJ, São João del-Rei, MG, Brazil

b

Aerospace Engineering and Mechanics Department, University of Minnesota, Minneapolis, MN, USA

article

info

Article history: Received 18 February 2016 Received in revised form 22 November 2016 Accepted 22 March 2017

Keywords: Robust stability Non-monotonic Lyapunov functions Continuous and discrete-time uncertain systems Time-invariant uncertainty

abstract This paper is concerned with the problem of robust stability of uncertain linear time-invariant systems in polytopic domains. The main contribution is to present a systematic procedure to check the stability of the uncertain systems by using an arbitrary number of quadratic functions within higher order derivatives of the vector field in the continuous-time case and higher order differences of the vector field in the discrete-time case. The matrices of the Lyapunov function appear decoupled from the dynamic matrix of the system in the conditions. This fact leads to sufficient conditions that are given in terms of Linear Matrix Inequalities defined at the vertices of the polytope. The proposed method does not impose sign condition constraints in the quadratic functions that compose the Lyapunov function individually. Moreover, some of the quadratic functions do not decrease monotonically along trajectories. However, if the sufficient conditions are satisfied, then a monotonic standard Lyapunov function that depends on the dynamics of the uncertain system can be constructed a posteriori. Numerical examples from the literature are provided to illustrate the proposed approach. © 2017 Elsevier Ltd. All rights reserved.

1. Introduction Lyapunov theory has proven to be a powerful tool to guarantee the stability of dynamical systems (Khalil, 2002). Most of the results for stability analysis presented in the literature search for a standard Lyapunov function, i.e., one that must be positive definite and must decrease monotonically along trajectories. Some few works have raised the question of why should we require the Lyapunov function to decrease monotonically. In Butz (1969), the problem of inferring asymptotic stability for continuous-time systems has been addressed without requiring the first derivative of the Lyapunov function to be negative definite. Instead of that, a condition based on the existence of a three times continuously differentiable Lyapunov function has been proposed. The work was extended in Meigoli and Nikravesh (2009) to consider higher order derivatives of the Lyapunov function. However, in both cases the

✩ This work was supported by the São Paulo Research Foundation (FAPESP) grant 2015/00269-5. The material in this paper was not presented at any conference. This paper was recommended for publication in revised form by Associate Editor Denis Arzelier under the direction of Editor Richard Middleton. E-mail addresses: [email protected] (M.J. Lacerda), [email protected] (P. Seiler).

http://dx.doi.org/10.1016/j.automatica.2017.04.042 0005-1098/© 2017 Elsevier Ltd. All rights reserved.

use of higher order derivative of the Lyapunov function has led to non-convex conditions, that rely on the search of scalar parameters and the Lyapunov function at the same time. In Ahmadi and Parrilo (2011) it has been shown that once the conditions in Butz (1969) and Meigoli and Nikravesh (2009) are satisfied, then a standard Lyapunov function can be constructed. The Lyapunov function is parameterized by higher order derivatives of the vector field. It was demonstrated that convex conditions based on the existence of this structured Lyapunov function can be obtained to solve the problem. The counterpart of this result for discrete-time, using higher order differences of the vector field, has been presented in Ahmadi and Parrilo (2008). It is also worth mentioning the work in Sassano and Astolfi (2013) that proposes the use of dynamic Lyapunov functions to characterize the stability of linear and nonlinear systems and the recent approach (Chesi, 2015) that does not rely on the use of Lyapunov functions and can provide a certificate of instability for uncertain systems by means of semidefinite programming and determinants of matrices. The well known quadratic stability condition has been used as the first method to certify the stability of linear time invariant (LTI) uncertain systems in polytopic domains (Barmish, 1985). However, the use of a common Lyapunov matrix to assure the stability for all the uncertain domain can be conservative in some cases. To reduce the conservatism of the conditions, the main

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developments were in the sense of improving the structure of the Lyapunov matrix, considering affine Lyapunov functions (Geromel, de Oliveira, & Hsu, 1998; Leite & Peres, 2003; Peaucelle, Arzelier, Bachelier, & Bernussou, 2000) and polynomially parameterdependent Lyapunov functions (Chesi, 2008; Chesi, Garulli, Tesi, & Vicino, 2005; Oliveira & Peres, 2006, 2007; Scherer, 2006) to assure the stability of uncertain systems. In another direction, the authors in Lee, Park, and Joo (2011) have developed an approach to compute stability of LTI uncertain systems using higher order derivatives of the Lyapunov function. As in Meigoli and Nikravesh (2009), the conditions proposed in Lee et al. (2011) depend on scalar parameters, which is the main drawback of the method accordingly to the authors. An algorithm to minimize the effects of the scalar search has been employed, but the results can still be conservative. In Ebihara, Peaucelle, Arzelier, and Hagiwara (2005) higher order time-derivatives of the states (limited to third order derivatives) have been used to construct a Lyapunov function. The work was extended in Peaucelle, Arzelier, Henrion, and Gouaisbaut (2007), considering a Lyapunov function composed by a generic number of higher order time-derivatives of the states, to deal with the problem of topological separation. Another related work is (Ebihara, Peaucelle, & Arzelier, 2015, Chapter 2), that addresses the robust performance analysis of LTI uncertain systems using conditions with the presence of slack variables. In the main results comments will be provided to clarify the relation between the proposed approach and the use of time-derivatives of the states to construct Lyapunov functions. At this point, it is important to remember that the development of efficient stability conditions is the first step towards achieving effective synthesis conditions. This paper provides a systematic procedure to check the stability of LTI uncertain systems in polytopic domains. The conditions are based on the existence of a Lyapunov function composed by a generic number of quadratic functions. Higher order derivatives (differences) of the vector field in the continuoustime (discrete-time) case are employed. The quadratic functions do not have sign condition constraints individually, and some of them do not decrease monotonically. The proposed method decouples the matrices of the Lyapunov function from the dynamic matrix of the uncertain system, preventing the computation of power of uncertain matrices that show up in the higher order derivatives of the vector field. This fact leads to sufficient conditions that are given in terms of Linear Matrix Inequalities (LMIs) defined at the vertices of the polytope. If the conditions are fulfilled, then a monotonic Lyapunov function that depends on the dynamic matrix of the system can be constructed a posteriori. The proposed approach also contains as particular cases some well known conditions from the literature. Numerical experiments show the potential of the technique of requiring a smaller number of scalar decision variables and LMI rows to certify the stability of uncertain systems. Notation. For two symmetric matrices of same dimensions A and B, A > B means that A − B is positive definite. For matrices or vectors (T ) indicates transpose. Matrix He(Z ) = Z + Z T is used to simplify the developments. In continuous-time case, V p (x) represents the derivative of order p of the function V (x), while in the discrete-time case V p (x) represents the function V (x)1 evaluated at the instant p. A ⊗ B represents the Kronecker product between A and B. The factorial of d is denoted by d!. Identity (null) matrices of dimension n × n (n × m) are denoted by In (0n×m ).

1 For simplicity of notation, the dependence of V (x) on x is omitted in some of the formulations.

2. Background Consider the dynamical system

δ [x] = f (x)

(1) n

n

n

where x ∈ R is the state vector and f : R → R . The operator δ[x] denotes the time-derivative for continuous-time systems and the shift operator for discrete-time systems. We are interested in verifying if f (0) = 0 is the unique stable equilibrium point of the system, i.e., if the system is globally asymptotically stable (GAS). Before introducing the main problem of this paper, let us state some results presented in the literature that make use of higher order derivatives (differences) of the vector field in the continuoustime (discrete-time) case to infer GAS of (1). The first result can be seen as a generalization of the conditions proposed in Butz (1969) and Meigoli and Nikravesh (2009) and has been presented in Ahmadi and Parrilo (2011). Instead of search for scalar parameters and just one single function V (x), convex conditions were obtained by using different functions Vi (x), as stated in the next lemma. Lemma 1 (Ahmadi & Parrilo, 2011). If there exists a radially unbounded function W (x) such that (N −1)

W (x) = VNN+1 (x) + VN

(x) + · · · + V˙ 2 (x) + V1 (x) ˙ (x) < 0, ∀x ̸= 0 W (x) > 0, W

W (0) = 0,

then the origin is a GAS equilibrium point of (1) and W (x) is a standard Lyapunov function. Note that Lemma 1 does not require sign condition for any individual function Vi (x), i = 1, . . . , N + 1. Moreover, W (x) is a standard Lyapunov function that has been parameterized in a very special way using derivatives of the vector field, which show up in the derivatives of the functions Vi (x). Concerning the discrete-time case, the following condition is of interest to our work Lemma 2 (Ahmadi & Parrilo, 2008). If there exist continuous  N +1 functions V1 , . . . , VN +1 : Rn → R such that i=1 iVi (0) = 0, N +1 

Vi radially unbounded for j = 1, . . . , N + 1

i=j N +1 

Vi > 0 ∀x ̸= 0 for j = 1, . . . , N + 1

i=j N +1 (VNk+ − VNk +1 ) + · · · + (V1k+1 − V1k ) < 0 +1

then the origin is a GAS equilibrium point of (1) and W k (x) =  N +1 N +1 k+j−1 j=1

Vi

i =j

is a standard Lyapunov function.

3. Problem formulation Consider the following linear time invariant uncertain system

δ[x] = A(α)x

(2)

where x ∈ R is the state vector. The uncertain matrix A(α) belongs to a polytopic domain parameterized in terms of a time-invariant vector α , being given by n

A(α) =

Z  z =1

αz Az ,

α ∈ ΛZ

(3)

M.J. Lacerda, P. Seiler / Automatica 82 (2017) 187–193

where Az , z = 1, . . . , Z are the vertices of the polytope and ΛZ is the unit simplex Z



ΛZ = α ∈ RZ :





αz = 1, αz ≥ 0, z = 1, . . . , Z .

189

and after further simplifications one has (N )

V1 (x) + V˙ 2 (x) + V¨3 (x) + · · · + VN +1 (x) > 0 with Vi (x) = xT Pi x. Suppose (7) holds. Multiplying it by αz ∈ ΛZ , z = 1, . . . , Z and summing up one has

z =1

In continuous-time, the dynamic matrix A(α) is said to be Hurwitz stable if the eigenvalues lie in the open left-half plane for all α ∈ ΛZ , likewise, in discrete-time the dynamic matrix A(α) is said to be Schur stable if the eigenvalues lie inside the unit disk for all α ∈ ΛZ . The problem to be addressed in this paper is to certify if the matrix A(α) in (2) is Hurwitz stable (continuous-time) or Schur stable (discrete-time). The approach relies on the specialization of Lemmas 1 and 2 to deal with uncertain systems making use of a generic number of quadratic functions within its higher order derivatives (continuous-time) or differences (discrete-time). Differently from Lemmas 1 and 2 the main results presented in the next section do not exhibit products between the Lyapunov function and the dynamic matrix of the system A(α).



Z 

   αz Q N +2 + He X2 IN +1 ⊗ A(α)



0

z =1

 + X2 0

IN + 1 ⊗ − In



< 0.

(9)

Then, multiplying (9) by S on the left and by S T on the right with S = xT In



A(α)T

···

(A(α)T )N

(A(α)T )N +1



and after further simplifications one has ... (N +1) V˙ 1 (x) + V¨2 (x) + V 3 (x) + · · · + VN +1 (x) < 0 with Vi (x) = xT Pi x. The proof is concluded by taking (N )

W (x) = V1 (x) + V˙ 2 (x) + V¨3 (x) + · · · + VN +1 (x)

4. Main results In order to simplify the presentation of the results let us define the matrix BKz BKz





= IK ⊗ Az



IK ⊗ − In

0Kn×n + 0Kn×n

and the upper left triangular matrix Q are computed as

N +a



(4)

, whose blocks matrices

(i + j − 2)! Pj+i−a , (i − 1)! (j − 1)! i = 1, . . . , N + a, j = 1, . . . , N + a + 1 − i.

Q N +a (i, j) =

(5)

4.1. Continuous-time Theorem 1. If there exist symmetric matrices Pi ∈ Rn×n , i = 1, . . . , N + 1, P0 = 0n×n , matrices BNz , BNz +1 computed as in (4), matrices Q N +1 , Q N +2 computed as in (5), matrices X1 ∈ Rn(N +1)×Nn and X2 ∈ Rn(N +2)×n(N +1) , such that the following inequalities hold T

Q N +1 + X1 BNz + BNz X1T > 0, Q N +2 + X2 BNz +1 +

T BNz +1 X2T

(6)

< 0,

(7)

for z = 1, . . . , Z , then system (2) is asymptotically stable and W (x) = V1 (x) + V˙ 2 (x) + V¨3 (x) + · · · +

(N ) VN +1 ( x )

   αz    z =1 Z

P1 P2

.. .

PN PN +1

··· ···

P2 2P3

.. .

..

NPN +1 0

··· ···

  + He X1 IN ⊗ A(α)



PN NPN +1

.

0 0 0



0 + X1 0

(8)

P N +1 0 

  

0   0 0 IN ⊗ − In



> 0.

Then, multiplying it by R on the left and by RT on the right with R = x T In



A(α)T

(A(α)T )2

···

(A(α)T )N

Theorem 1 presents a sufficient condition for stability analysis of continuous LTI uncertain systems described as a polytope. Theorem 1 makes use of Lemma 1 with a specific structure for the function Vi (x), i.e., all the functions Vi (x) are quadratic in the state. Moreover, in Theorem 1 the matrices of the Lyapunov function are decoupled from the dynamic matrix of the system A(α), avoiding the computation of powers of the uncertain matrix A(α). Less conservative results can be obtained by considering affine parameter-dependent functions and parameter-dependent matrices X1 and X2 . To this end, it suffices to impose a polytopic structure as in (3) to the matrices Pi (α), i = 1, . . . , N + 1, X1 (α) and X2 (α). The next lemma states the conditions for this case. Lemma 3. If there exist symmetric matrices Pi,z ∈ Rn×n , i = 1, . . . , N + 1, z = 1, . . . , Z , P0 = 0n×n , matrices BNz , BzN +1 computed as in (4), matrices QzN +1 , QzN +2 computed as in (5) and matrices X1z ∈ Rn(N +1)×Nn , X2z ∈ Rn(N +2)×n(N +1) z = 1, . . . , Z , such that the following inequalities hold



Proof. Suppose (6) holds. Multiplying it by αz ∈ ΛZ , z = 1, . . . , Z and summing up one has



Remark 1. Theorem 1 with N = 0 can recover the classical stability condition with slack variables as in de Oliveira and Skelton (2001) and Peaucelle et al. (2000).

QzN +1 + He X1z BNz > 0,

with Vi (x) = xT Pi x, i = 1, . . . , N + 1 is the Lyapunov function that assures the stability of the uncertain system (2).



˙ (x) < 0, for all then, one has W (0) = 0, W (x) > 0 and W x ̸= 0. 

QzN +2 QzN +1 QzN +2

+ He



X2z BzN +1



+

QrN +1

+

QrN +2



< 0,

X1z BNr

+ He



+ He



z = 1, . . . , Z z = 1, . . . , Z

+

X2z BNr +1

X1r BNz

+



> 0,  < 0,

(11) (12)

X2r BNz +1

z = 1, . . . , Z − 1, r = z + 1, . . . , Z

(13)

then system (2) is asymptotically stable and W (x) as in (8) with Vi (x) = xT Pi (α)x is the Lyapunov function that assures the stability of the uncertain system (2). Proof. By multiplying (10) by αz2 , z = 1, . . . , Z , multiplying (12) by αz αr ≥ 0, z = 1, . . . , Z − 1, r = z + 1, . . . , Z , and summing up one has T

Q N +1 (α) + X1 (α)BN (α) + BN (α) X1 (α)T > 0. The same procedure applied to (11) and (13) gives T



(10)

Q N +2 (α) + X2 (α)BN +1 (α) + BN +1 (α) X2 (α)T < 0.

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M.J. Lacerda, P. Seiler / Automatica 82 (2017) 187–193

The rest of the proof follows the same steps as in the proof of Theorem 1.  Note that Q N +a in (5) needs to be slightly modified to be used with Lemma 3. The term Pj+i−a must be changed by Pj+i−a,z . More complex structures for the matrices P and X could be exploited, such as polynomially parameter-dependent functions proposed in Oliveira and Peres (2006), at the price of an increase in the computational burden.

4.2. Discrete-time Theorem 2. If there exist symmetric matrices Pi ∈ Rn×n , i = 1, . .  . , N + 1, matrices BNz , BzN +1 computed as in (4), M =

diag −

 N +1 i=1

and matrix Y ∈ Rn(N +2)×n(N +1) ,

Pi , P1 , . . . , PN +1

such that the following inequalities hold N +1 

Pi > 0,

j = 1, . . . , N + 1

(17)

i=j

Remark 2. Lemma 3 with N = 0 and X2 as a constant matrix is equivalent to the result presented in Ebihara et al. (2015, Theorem 2.4). The main difference between the proposed approach and the methods in Ebihara et al. (2015) and Ebihara et al. (2005) is the background on how the Lyapunov functions are constructed. In order to establish a connection between the proposed work and the method in Ebihara et al. (2005), let us consider Lemma 3 with N = 1 (remember that Lemma 3 is an extension of Theorem 1 with parameter dependent matrices Pi (α)) and Theorem 2 in Ebihara et al. (2005). If the conditions proposed in Lemma 3 hold for N = 1 one can construct a Lyapunov function with the following form W (x) = xT P1 (α)x + x˙ T P2 (α)x + xT P2 (α)˙x > 0.

(14)

Defining V1 (x) = x P1 (α)x and V2 (x) = x P2 (α)x one can write T

T

W (x) =V1 (x) + V˙ 2 (x) > 0

(15)

where W (x) is a standard Lyapunov function. Now, let us take a look at the method studied in Ebihara et al. (2005). Theorem 2 in Ebihara et al. (2005) makes use of a Lyapunov function given by V (x) = xT

x˙ T Π (α)





P1 (α)

x x˙

P2 (α)

P2 (α)

we have P3 (α)

T

V (x) = xT P1 (α)x + x˙ T P2 (α)T x + xT P2 (α)˙x + x˙ T P3 (α)˙x.

x

x˙

T

···

x

NT

(18)

then system (2) is asymptotically stable and W k ( x) =

N +1  N +1 

k+j−1

Vi

(19)

j =1 i =j

with Vik = xTk Pi xk is the Lyapunov function that assures the stability of the uncertain system (2). Proof. Multiplying (18) by αz ≥ 0, z = 1, . . . , Z and summing up one has



Z 

   αz M + He Y IN +1 ⊗ A(α)

0



z =1

 +Y 0

IN + 1 ⊗ − In



< 0.

(20)

Then, multiplying it by S on the left and by S T on the right with A(α)T

S = xTk In



···

(A(α)T )N

(A(α)T )N +1



and after further simplifications one has N +1 (VNk+ − VNk +1 ) + (VNk+N − VNk ) + · · · + (V1k+1 − V1k ) < 0 +1

N +1 VNk+ + VNk+N + · · · + V1k+1 < VNk +1 + VNk + · · · + V1k . +1

By adding one term on both sides of the inequality yields N +1 VNk+ + VNk+N + · · · + V1k+1 + +1

N  N +1 

k+j

Vi

j =1 i =j +1 N +1  N

(16)

In this case, the presence of P3 (α) in (16) prevents the conditions of being written as a sum of higher order derivatives of quadratic functions. Moreover, the method in Ebihara et al. (2005) requires Π (α) > 0, so it is not possible consider P3 = 0. The term x˙ T P3 (α)˙x in (16) produces xT A(α)T P3 (α)A(α)x and it implies on the existence of a Lyapunov function of degree 3 in α . More complex structures can be constructed considering higher order derivatives ... ... of the states providing terms as x¨ T P (α)¨x, x T P (α) x , indicating the existence of Lyapunov functions of degrees 5, 7 and so forth in the parameter α . In contrast with that, the conditions proposed by Lemma 3 with N = 1 do not have signal constraint for P1 (α) and P2 (α) and the constructed Lyapunov function has degree 2 in α (see W (x) in (14)). More complex structures with N = 2, 3 for example, will construct Lyapunov functions of degrees 3, 4 function W (x) = x˜ T Π (α)˜x with x˜ T = respectively. The Lyapunov  T

z = 1, . . . , Z

as in Lemma 2 with Vik = xTk Pi xk . The inequality above can be reorganized as

 

V (x) must be positive definite and its derivative must be negative definite. The higher order derivatives of the states are included in the Lyapunov function, however, there is no relation with higher order derivatives of Lyapunov functions. Partitioning the matrix  

Π (α) =

T

M + YBNz +1 + BNz +1 Y T < 0,

has degree 2N + 1 in α . On the other hand,

the Lyapunov function resulted from Lemma 3, i.e., W (x) = V1 (x)+ (N ) V˙ 2 (x) + V¨3 (x) + · · · + VN +1 (x) with Vi (x) = xT Pi (α)x has degree N + 1 in α .

< VNk +1 + VNk + · · · + V1k +

k+j

Vi

.

(21)

j=1 i=j+1

Define W k (x) =

N +1  N +1 

k+j−1

Vi

(22)

j =1 i =j

then, the inequality (21) implies that W k+1 < W k . If (17) holds, then W k as in (22) is positive definite. Then one has W k > 0, W k+1 − W k < 0 and W (0) = 0, concluding the proof.  Remark 3. Applying Theorem 2 with N = 0 yields one of the conditions presented in Peaucelle et al. (2000). By imposing N = 0

T

and Y = 0 GT one has the stability condition proposed in de Oliveira, Bernussou, and Geromel (1999).



Theorem 2 presents a sufficient condition to certify the stability of discrete LTI uncertain systems. As in Theorem 1, the matrices of the Lyapunov function are decoupled from the matrix A(α). Less conservative conditions could be obtained for the discrete-time case by using parameter-dependent matrices, as in Lemma 3 for continuous-time systems.

M.J. Lacerda, P. Seiler / Automatica 82 (2017) 187–193

191

Table 1 Number of LMI rows with n = 4 in function of N and Z for Theorem 2 and in function of g and Z for Oliveira and Peres (2006). Method

Z 2

Theorem 2, N = 1 Theorem 2, N = 2 Theorem 2, N = 3 Oliveira and Peres (2006), T 4g =1 Oliveira and Peres (2006), T 4g =2

5

6

32 44 56 32

3 44 60 76 60

4 56 76 96 96

68 92 116 140

80 108 136 192

44

104

200

340

532

Table 2 Bounds for γ obtained with different methods.

Fig. 1. Number of LMI rows LR in terms of the order n of a system with Z = 5 vertices, for Theorem 2 (with different values of N) and for Oliveira and Peres (2006, Theorem 4) (with different degrees g).

Remark 4. All conditions proposed in this paper can be combined with the existing relaxations for positive polynomials (Henrion, 2005). Moreover, this work could be extended to deal with the problem of exact robust stability analysis (Chesi, 2013) following the lines of Zhang, Tsiotras, and Iwasaki (2010). Before presenting the numerical experiments let us provide some comments about the parameter N used in Theorems 1 and 2 and Lemma 3. – If there exists a Lyapunov function solution of the problem for N = N0 , then there is a Lyapunov function of higher order degree on α that also certifies the stability of the system for N ≥ N0 . For instance, suppose that N = 0 provides a solution for continuous time-case, then we have V1 (x) > 0 and V˙ 1 (x) < 0. Then, there exists small enough ϵ such that V1 (x) + ϵ V˙ 2 (x) > 0 and V˙ 1 (x) + ϵ V¨2 (x) < 0. This argument can be extended for greater values of N and also for the discrete-time case. Although we can guarantee the existence of the Lyapunov function we cannot guarantee the feasibility of the LMIs because the condition is only sufficient. – If a solution for N = N0 is not found, there is no guarantee that N > N0 will find a feasible solution. However, as will be shown in the numerical examples increasing N can provide extra degrees of freedom that can help to find better solutions. One important point is that increasing N can be easier than considering more complex structures for the matrices Pi . 5. Numerical complexity and experiments The numerical complexity is associated with the number of scalar decision variables (Sv ) and with the number of LMI rows (LR ). The number of scalar variables in Theorem 1 is given by Sv T 1 = (n/2)(N + 1)(1 + n(4N + 5)) while the number of LMI rows is LRT 1 = Zn(2N + 3). For Theorem 2 the number of scalar variables is Sv T 2 = (n/2)(N + 1)(1 + n(2N + 5)) with the number of LMI rows equal LRT 2 = n(N + 1 + Z (N + 2)). Fig. 1 presents the increase in the number of LMI rows for a system with Z = 5 vertices and different values of n, when applying Theorem 2 (with different values of N) and (Oliveira & Peres, 2006, Theorem 4) (with different degrees g). It can be seen that the number of LMIs increases faster in the polynomial approach (Oliveira & Peres, 2006). The method in Oliveira and Peres (2006) can assure the stability for a large enough degree g of the parameter-dependent Lyapunov function, however, when the order of the system, the degree g

Method

γ

Sv

LR

Lee et al. (2011, Theorem 7), κ = 3 Oliveira et al. (2008, Theorem 8), κ = 1 Oliveira et al. (2008, Theorem 8), κ = 9 Theorem 1, N = 1

12.4631 3.8284 >100 000 >100 000

99 19 1467 38

26 14 114 20

and the number of vertices increase, the problem becomes bigger and difficult to be solved. The strategy proposed in this paper presents a smooth increment in the number of LMI rows with the augmentation of the order of the system and with the augment in the number of quadratic functions used to certify the stability. The presence of a greater number of uncertain parameters implies in a greater number of vertices. It is important to emphasize that the number of variables of Theorems 1 and 2 does not change with the number of vertices of the system. The increase in the number of LMI rows for Theorem 2 with n = 4 in function of N and Z is presented in Table 1. It can be seen that the proposed method presents a slower increment in the number of LMI rows when compared to the approach in Oliveira and Peres (2006). That is one of the reasons why the proposed method should be considered as an alternative to certify the stability of uncertain systems. The main goal of the experiments is to illustrate the benefits of the proposed approach in terms of its efficiency. The routines were implemented in Matlab, version 8.2.0.701 (R2013b) using YALMIP (Löfberg, 2004) and SeDuMi (Sturm, 1999) in an Intel(R) Core(TM) i7-4770, 3.4 GHz, 8 GB RAM, Windows 8.1. Example 1. Consider the uncertain system borrowed from Lee et al. (2011) with

 A1 =

0 −2



1 , −1

 A2 =



0

−(2 + γ )

1 . −1

The objective is to find the maximum value of γ > 0 such that the conditions can assure that the system is robustly asymptotically stable. Table 2 presents the bounds obtained by using different methods within the number of scalar decision variables (Sv ) and number of LMI rows (LR ). As can be seen, Theorem 1 and (Oliveira, de Oliveira, & Peres, 2008, Theorem 8) can certify the stability for the uncertain system even for γ > 100 000. However, the proposed approach requires a smaller number of scalar decision variables and LMI rows to certify the stability. All the methods presented in Table 2 are based on Lyapunov functions. The method in Chesi (2013) that is not a Lyapunov based approach requires no more than 4 scalar decision variables to solve this problem. For γ = 12.4631 Theorem 1 with N = 1 provides a Lyapunov function W (x) = V1 (x) + V˙ 2 (x) with Vi = xT Pi x, i = 1, 2 and 0.6803 P1 = −0.6639



 −0.6639 , 3.0529

0.0359 P2 = −0.7499



 −0.7499 . −0.0617

192

M.J. Lacerda, P. Seiler / Automatica 82 (2017) 187–193 Table 3 Bounds for ρ obtained with different methods with number of LMI rows LR and scalar variables Sv . Method

ρ

Sv

LR

Ebihara et al. (2005, Theorem 2) Ebihara et al. (2005, Condition (17)) Ebihara et al. (2015, Theorem 2.4) Chesi et al. (2005, Theorem 1), m = 2 Theorem 1, N = 1 Theorem 1, N = 2 Theorem 1, N = 3 Theorem 1, N = 4 Lemma 3, N = 1

1.930 1.930 1.4972 2.2237 1.1514 1.5211 1.5954 1.7801 2.2237

204 426 62 750 148 318 552 850 444

60 84 36 64 60 84 108 132 120

Table 4 Comparison of the results of Theorem 2 with the methods proposed in Oliveira and Peres (2006) in terms of number of scalar variables (K ), number of LMI rows (LR ) and solver computational time.

Fig. 2. Evolution of the Lyapunov function computed using Theorem 1 and the



Lyapunov function presented in Lee et al. (2011), x0 = −3 α2 = 0.7.

T

5 , α1 = 0.3 and

In this way, W (x) can be rewritten in function of α as W (x, α) = xT (P1 + A(α)T P2 + P2 A(α))x or simply W (x, α) = xT



α1

3.6799 0.2453

0.2453 22.3721 + α2 1.6765 1.0143







1.0143 1.6765



x.

Fig. 2 presents the evolution of the Lyapunov function W (x, α) and the Lyapunov function presented in Lee et al. (2011) for the initial

T

condition x0 = −3 5 , α1 = 0.3 and α2 = 0.7. As mentioned before, the method proposed in Lee et al. (2011) does not require V˙ (x) < 0 as shown in Fig. 2. In contrast, the function W (x) provided by Theorem 1 has its derivative negative definite for all x ̸= 0.



Example 2. Consider the continuous-time system from Chesi et al. (2005, Example 3), with order n = 4, where specific data can be found. The three vertices of the polytopic representation are constructed as Ai = (A¯ 0 + ρ A¯ i ),

i = 1, 2, 3..

The problem is to find the maximum value of ρ such that the system is Hurwitz stable, i.e., find the value of ρ that provides the stability margin. Table 3 presents the bounds of ρ obtained with different methods with the number of scalar variables Sv and the number of LMI rows LR . Note that, Theorem 1 cannot reach the value of ρ that provides the stability margin even with N = 4. One can see that the conditions from Ebihara et al. (2015, Theorem 2.4) can guarantee ρ = 1.4972 with a small number of LMI rows and scalar decision variables. The method proposed in Ebihara et al. (2005) is also not capable of achieving the bound for the stability margin ρ = 2.2237. It is important to remember that both Theorem 1 and the method in Ebihara et al. (2005) produce parameter-dependent Lyapunov functions. The difference is that Theorem 1 makes use of constant matrices Pi while the method in Ebihara et al. (2005) is based on the use of parameter-dependent matrices Π (α). For this example, Lemma 3 can certify the stability for the system with N = 1 requiring a smaller number of scalar variables than the method in Chesi et al. (2005) with a greater number of LMI rows. Example 3. Consider the discrete-time uncertain system borrowed from Oliveira and Peres (2006, Example 5) with order n = 3 and Z = 3 vertices. The objective is to investigate the robust stability of the polytope. Table 4 provides a comparison of the results of Theorem 2 with the method proposed in Oliveira and Peres

Method

Sv

LR

Time (s)

Oliveira and Peres (2006, Theorem 3), g = 2 Oliveira and Peres (2006, Theorem 4), g = 1 Theorem 2, N = 1

36

78

0.0578

72

45

0.0411

66

33

0.0355

(2006) in terms of number of scalar variables (Sv ), number of LMI rows (LR ) and the solver computational time. As can be seen, the proposed approach requires a smaller number of LMI rows than the method in Oliveira and Peres (2006) to assure the robust stability of the polytope. Moreover, besides the fact of having more scalar variables than (Oliveira & Peres, 2006, Theorem 3) the technique presented in this paper requires the lowest computational time to be solved. In this example, a monotonic Lyapunov function can be constructed as in (22) with W (x) = V1k + V2k + V2k+1 or x(k)T P1 + P2 + A(α)T P2 A(α) x(k)





−0.5358

with P1 = −0.0309 0.2386

1.3834 0.4421 −0.4682

 P2 =

−0.0309 0.0221 −0.1833

0.4421 0.5680 0.3075

(23)

 0.2386

−0.1833 , and −0.4592

 −0.4682 0.3075 . 1.1880

Example 4. Consider the discrete-time uncertain system borrowed from Henrion, Arzelier, Peaucelle, and Lasserre (2004) described by the matrix

  −0.7 0.7 0 A¯ = −0.1 −0.3 −0.3 −0.1 0.3 0.3    −0.7 −0.3 0.4 −1 0.7 −0.5 + q2 0.4 + q1 0.7 −1.5 0.1 0.7 −2.7

−1 0.9 −1.2

0.6 0.1 −0.6



with |q1 | ≤ ρ and |q2 | ≤ ρ . The affine representation can be converted to a polytope of four vertices and then the condition proposed in Theorem 2 can be applied. The objective here is to find the maximum value of ρ such that the system is asymptotically stable. Table 5 presents the computational effort in terms of number of scalar variables (Sv ) and number of LMI rows (LR ) required by different approaches to achieve the stability margin value ρ = 0.38409. It can be noted that Theorem 2 reaches the stability margin ρ = 0.38409 with N = 2. Moreover, Theorem 2 requires the smaller number of LMI rows to assure the stability of the system for ρ = 0.38409. However, for smaller values of ρ , Ebihara et al. (2015, Theorem 2.8) can provide better results in terms of number of LMI rows and scalar decision variables.

M.J. Lacerda, P. Seiler / Automatica 82 (2017) 187–193 Table 5 Comparison of the results of Theorem 2 with the methods proposed in Ebihara et al. (2015), Henrion et al. (2004) and Oliveira and Peres (2006) in terms of number of scalar variables (Sv ) and number of LMI rows (LR ). Method

Sv

LR

ρ

Henrion et al. (2004), k = 10 Oliveira and Peres (2006, Theorem 3), g = 3 Oliveira and Peres (2006, Theorem 4), g = 2 Ebihara et al. (2015, Theorem 2.8) Theorem 2, N = 2

230 120

286 270

0.38409 0.38409

240

150

0.38409

42 126

36 57

0.33747 0.38409

6. Conclusion This paper has provided sufficient conditions to certify the stability of LTI uncertain systems. The Lyapunov function used to certify the stability is composed by an arbitrary number of quadratic functions within higher order derivatives (continuoustime) or differences (discrete-time) of the vector field. The advantages of the proposed approach have been shown by numerical examples. The number of LMI rows, the number of scalar decision variables and the time to certify the stability of uncertain systems have been improved, when compared with techniques available in the literature. It is important to highlight that the conditions presented here can be easily implemented and can be combined with existing LMI relaxations at the price of increasing the computational burden. As future research the authors are investigating the extensions to compute some performance criterion such as the H∞ performance and to cope with the control design problem. Acknowledgments The authors thank the editors and the anonymous reviewers for their comments that helped to improve the paper. References Ahmadi, A.A., & Parrilo, P.A. (2008). Non-monotonic Lyapunov functions for stability of discrete time nonlinear and switched systems. In Proc. 47th IEEE conf. decision control, Cancun, Mexico, December (pp. 614–621). Ahmadi, A.A., & Parrilo, P.A. (2011). On higher order derivatives of Lyapunov functions. In Proc. 2011 Amer. control conf., San Francisco, CA, USA, June–July (pp. 1313–1314). Barmish, B. R. (1985). Necessary and sufficient conditions for quadratic stabilizability of an uncertain system. Journal of Optimization Theory and Applications, 46(4), 399–408. Butz, A. (1969). Higher order derivatives of Liapunov functions. IEEE Transactions on Automatic Control, 14(1), 111–112. Chesi, G. (2008). On the non-conservatism of a novel LMI relaxation for robust analysis of polytopic systems. Automatica, 44(11), 2973–2976. Chesi, G. (2013). Exact robust stability analysis of uncertain systems with a scalar parameter via LMIs. Automatica, 49(4), 1083–1086. Chesi, G. (2015). Instability analysis of uncertain systems via determinants and LMIs. IEEE Transactions on Automatic Control, 60(9), 2458–2463. Chesi, G., Garulli, A., Tesi, A., & Vicino, A. (2005). Polynomially parameter-dependent Lyapunov functions for robust stability of polytopic systems: An LMI approach. IEEE Transactions on Automatic Control, 50(3), 365–370. de Oliveira, M. C., Bernussou, J., & Geromel, J. C. (1999). A new discrete-time robust stability condition. Systems & Control Letters, 37(4), 261–265. de Oliveira, M. C., & Skelton, R. E. (2001). Stability tests for constrained linear systems. In S. O. Reza Moheimani (Ed.), Lecture notes in control and information science: Vol. 268. Perspectives in robust control (pp. 241–257). New York, NY: Springer-Verlag. Ebihara, Y., Peaucelle, D., & Arzelier, D. (2015). S-variable approach to LMI-based robust control. London, UK: Springer-Verlag. Ebihara, Y., Peaucelle, D., Arzelier, D., & Hagiwara, T. (2005). Robust performance analysis of linear time-invariant uncertain systems by taking higher-order timederivatives of the state. In Proc. 44th IEEE conf. decision control — Eur. control conf. ECC 2005, Seville, Spain, December (pp. 5030–5035).

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Márcio J. Lacerda received his D.Sc. degree in electrical engineering from the University of Campinas, UNICAMP, Brazil, in 2014. He has held post-doctoral research positions at UNICAMP from 2014 to 2015 and at the Aerospace Engineering and Mechanics Department, University of Minnesota, Minneapolis, USA, from 2015 to 2016. From October 2012 to April 2013 he was a visitor in the Laboratoire d’Analyse et d’Architecture des Systèmes, Toulouse, France. Currently he is a professor in the Department of Electrical Engineering, Federal University of São João delRei, UFSJ, Brazil, which he joined in 2016. His main research interests include linear systems, robust control, filtering theory and LPV systems.

Peter Seiler received his Ph.D. from the University of California, Berkeley in 2001. His graduate research focused on coordinated control of unmanned aerial vehicles and control over wireless networks. From 2004 to 2008, he worked at the Honeywell Research Labs on various aerospace and automotive applications including the redundancy management system for the Boeing 787, sensor fusion algorithms for automotive active safety systems and re-entry flight control laws for NASA’s Orion vehicle. Since joining the University of Minnesota in 2008, he has been working on fault-detection methods for safety-critical systems and advanced control techniques for wind turbines and unmanned aircraft.