Lyapunov Stability of Quasiperiodic Systems

Abstract

We present some observations on the stability and reducibility of quasiperiodic systems. In a quasiperiodic system, the periodicity of parametric excitation is incommensurate with the periodicity of certain terms multiplying the state vector. We present a Lyapunov-type approach and the Lyapunov-Floquet (L-F) transformation to derive the stability conditions. This approach can be utilized to investigate the robustness, stability margin, and design controller for the system.

1. Introduction

A large class of engineering systems, such as structures subjected to quasiperiodic excitations, is described by linear ordinary differential equations with time varying coefficients. These linear systems, in general, are described as𝐲̇=𝐀(𝑡)𝐲, (1.1) where 𝐀(𝑡) is an 𝑛×𝑛 quasiperiodic matrix and 𝐲 is an 𝑛 dimensional vector. In general, it is not a trivial problem to determine if (1.1) is asymptotically stable, simply stable, or unstable. The researchers have used perturbation-type techniques or numerical approaches to investigate the stability of this system [1–3].

In this work, we address the stability of a special class of quasiperiodic systems called as periodic quasiperiodic systems where (1.1) can be written as𝐲̇=𝐀₀(𝑡)+𝐀₁(𝑡)𝐲, (1.2) where 𝐀₀(𝑡) has the principal period 𝑇 and 𝐀₁(𝑡) has the period 𝑇₁. It is noted that these periods are incommensurate. These types of equations arise in parametrically excited Micro Electro Mechanical Systems (MEMS) [4]. It is noted that 𝐀(𝑡) has a strong parametric excitation. In this paper, we present the methodology to investigate the stability of system given by (1.2) using the L-Ftransformation and Lyapunov’s method.

This paper is organized as follows. In Section 2, a brief mathematical background on the L-F transformation is provided. Section 3 discusses the stability conditions followed by an example. We present the L-F-transformation-type approach for quasiperiodic systems in Section 4. The discussion and conclusions are presented in Section 5.

2. Mathematical Background

2.1. Floquet Theory and L-F Transformation

Consider (1.1), if 𝐀₁(𝑡)=0, that is, the system is purely time periodic, then the State Transition Matrix (STM) Φ(𝑡) of (1.2) can be factored as [5]𝚽(𝑡)=𝐐(𝑡)𝑒^𝐑𝑡,𝐐(𝑡)=𝐐(𝑡+2𝑇),𝐐(0)=𝐈, (2.1) where the matrix 𝐐(𝑡) is real and periodic with period 2𝑇, 𝐑 is an 𝑛×𝑛 real time invariant matrix, and 𝐈 is the identity matrix. Matrix 𝐐(𝑡) is known as the L-F transformation matrix [5].

The transformation 𝐲(𝑡)=𝐐(𝑡)𝐳(𝑡) produces a real-time invariant representation of purely time periodic system ((1.2) with 𝐀₁(𝑡)=0) given by𝐳̇(𝑡)=𝐀𝐳(𝑡). (2.2) It is to be noted that matrix 𝐀 in (2.2) is time invariant.

2.2. Construction of Lyapunov Functions

Lyapunov’s direct method is widely used in the stability analysis of general dynamical systems. It makes use of a Lyapunov function 𝑉(𝐱,𝑡). This scalar function of the state and time may be considered as some form of time-dependent generalized energy. The basic idea of the method is to utilize the time rate of energy change in 𝑉(𝐱,𝑡) for a given system to judge whether the system is stable or not. The details about Lyapunov’s method and stability theorems can be found in reference [6].

For a linear system with constant coefficients, it is rather simple to find a Lyapunov function. Consider the linear system𝐱_∼̇(𝑡)=𝐀𝐱_∼(𝑡), (2.3) where 𝐀 is a constant matrix. A quadratic form of 𝑉(𝐱) may be assumed as𝑉𝐱_∼=𝐱_∼^𝑇𝐏𝐱_∼, (2.4) where 𝐏 is a real, symmetric, and positive definite matrix. Then𝑉̇𝐱_∼=𝐱̇_∼^𝑇𝐏𝐱_∼+𝐱_∼^𝑇𝐏𝐱_∼̇=𝐀𝐱_∼^𝑇𝐏𝐱_∼+𝐱_∼^𝑇𝐏𝐀𝐱_∼ (2.5) or 𝑉̇𝐱_∼=𝐱_∼^𝑇𝐀^𝑇𝐏+𝐏𝐀𝐱_∼. (2.6)

According to the Lyapunov theorem for autonomous systems, if 𝑉̇(𝐱) is negative definite, then the system is asymptotically stable [6]. Therefore, one can write𝐀^𝑇𝐏+𝐏𝐀=−𝐂, (2.7) where 𝐂 is a positive definite matrix. Equation (2.7) is called the Lyapunov equation. It has been shown by Kalman and Bertram [7] that if there were eigenvalues with negative real parts (asymptotically stable), then for every given positive definite matrix 𝐂, there exists a unique Lyapunov matrix 𝐏. In this study, matrix 𝐂 is always taken as the identity matrix.

3. Stability of Quasiperiodic Systems

Consider the quasiperiodic linear differential equation given by (1.2). In order to determine the stability bounds on 𝐀₁(𝑡), we first use the L-F transformation 𝐲(𝑡)=𝐐(𝑡)𝐳(𝑡) to (1.2). After the L-F transformation, (1.2) can be written as𝐳̇=𝐀+𝐆(𝑡)𝐳, (3.1) where 𝐆(𝑡)=𝐐⁻¹(𝑡)𝐀₁(𝑡)𝐐(𝑡). It is to be noted that 𝐀 is a constant matrix whose eigenvalues have negative real parts. We follow the approach presented by Infante [8] to obtain stability bounds.

Theorem 3.1 (see [8]). If, for some positive definite matrix 𝐁 and some 𝜀>0, 𝐸𝜆_max𝐀^𝑇+𝐆(𝑡)^𝑇+𝐁𝐀+𝐆(𝑡)𝐁⁻¹≤−𝜀, (3.2) then (3.1) is almost surely asymptotically stable in the large, where 𝐸{⋅} is the expectation operator and 𝜆_max is maximum real eigenvalues of a pencil [9].

Proof. Consider the quadratic (Lyapunov) function 𝑉(𝐳)=𝐳^𝑇𝐁𝐳. Then along the trajectories of (3.1), define 𝜆(𝑡)=𝑉̇(𝐳)𝑉(𝐳)=𝐳^𝑇𝐀+𝐆(𝑡)^𝑇𝐁+𝐁𝐀+𝐆(𝑡)𝐳𝐳^𝑇𝐁𝐳. (3.3) It is noted that the numerator and denominator in (3.3) are quadratic forms. The pencil of quadratic forms 𝐀(𝐱,𝐱)=∑^𝑛_𝑖,𝑘=1𝑎_𝑖𝑘𝑥_𝑖𝑥_𝑘 and 𝐁(𝐱,𝐱)=∑^𝑛_𝑖,𝑘=1𝑏_𝑖𝑘𝑥_𝑖𝑥_𝑘 is a matrix-valued function defined over complex numbers 𝜆 given by 𝐀(𝐱,𝐱)+𝜆𝐁(𝐱,𝐱) [9]. From the properties of pencils of quadratic forms [9], we can obtain the following inequality: 𝜆_min𝐀+𝐆(𝑡)^𝑇+𝐁𝐀+𝐆(𝑡)𝐁⁻¹≤𝜆(𝑡)≤𝜆_max𝐀+𝐆(𝑡)^𝑇+𝐁𝐀+𝐆(𝑡)𝐁⁻¹, (3.4) where 𝜆_max is defined before and 𝜆_min is the minimum real eigenvalues of a pencil. Consider ‖𝐳‖_𝑝=(𝐳^𝑇𝐁𝐳)^1/2; it can be shown [10] that ‖𝐳‖_𝑝 satisfies 𝑑‖𝐳‖𝑑𝑡log_𝑝=𝐳^𝑇𝐀+𝐆(𝑡)^𝑇𝐁+𝐁𝐀+𝐆(𝑡)𝐳𝐳^𝑇𝐁𝐳. (3.5) Thus, integrating and dividing (3.5) by 𝑡, 1𝑡log‖𝐳(𝑡)‖_𝑝−log‖𝐳(0)‖_𝑝=1𝑡^𝑡₀𝐳^𝑇𝐀+𝐆(𝑠)^𝑇𝐁+𝐁𝐀+𝐆(𝑠)𝐳𝐳^𝑇𝐁𝐳𝑑𝑠. (3.6) For (1/𝑡)[log‖𝐳(𝑡)‖_𝑝−log‖𝐳(0)‖_𝑝]<0 as 𝑡→∞ follows lim_𝑡→∞‖𝐳‖=0. Thus, algebraic sign of lim_𝑡→∞[(1/𝑡)∫^𝑡₀((𝐳^𝑇[(𝐀+𝐆(𝑠))^𝑇𝐁+𝐁(𝐀+𝐆(𝑠))]𝐳)/𝐳^𝑇𝐁𝐳)𝑑𝑠] provides the condition for stability [10]. The solution of (3.3) can be given as 𝑉[𝐳(𝑡)]=𝑉𝐳𝑡₀𝑒^{∫𝑡𝑡0𝜆(𝜏)𝑑𝜏}≡𝑉𝐳𝑡₀𝑒^{(𝑡−𝑡0)[(1/(𝑡−𝑡0))∫𝑡𝑡0𝜆(𝜏)𝑑𝜏]}. (3.7) It can be observed that if 𝐸{𝜆(𝑡)}≤−𝜀 for some 𝜀>0, 𝑉[𝐳(𝑡)] is bounded and that 𝑉[𝐳(𝑡)]→0 as 𝑡→∞. This is the condition imposed by inequality given by (3.4), which proves the results. Since 𝐲(𝑡)=𝐐(𝑡)𝐳(𝑡), the stability of (3.1) implies the stability of (1.2).
It is remarked that a necessary condition for inequality (3.4) to hold is that the eigenvalues of matrix 𝐀 have negative real parts. It is also possible to obtain a result that is easier to compute but not as sharp.

Corollary 3.2. If, for some positive definite matrix 𝐁 and some ε > 0, 𝐸𝜆_max𝐆^𝑇(𝑡)+𝐁𝐆(𝑡)𝐁⁻¹≤−𝜆_max𝐀^𝑇+𝐁𝐀𝐁⁻¹−𝜀, (3.8) then (3.1) is almost surely asymptotically stable in the large.

Proof. The proof follows immediately from theorem by noting that 𝜆(𝑡)≤𝜆_max𝐀+𝐆(𝑡)^𝑇+𝐁𝐀+𝐆(𝑡)𝐁⁻¹≤𝜆_max𝐀^𝑇+𝐁𝐀𝐁⁻¹+𝜆_max𝐆^𝑇(𝑡)+𝐁𝐆(𝑡)𝐁⁻¹. (3.9) The second inequality is obtained by performing two maximizations separately. Further, using an 𝐸{⋅} operator 𝐸{𝜆(𝑡)}≤𝜆_max𝐀^𝑇+𝐁𝐀𝐁⁻¹+𝐸𝜆_max𝐆^𝑇(𝑡)+𝐁𝐆(𝑡)𝐁⁻¹≤−𝜀, (3.10) yields the desired result. It is obvious that, unless the second inequality in (3.10) is an equality, the stability results obtained will not be as good as those given by the theorem. It is noted that this theorem and corollary can be extended to study stability and robustness of a linear time-periodic system subjected to random perturbations in a straightforward fashion, and for the details, we refer the reader to reference [11].

Example 3.3. Consider the system 𝐲̇=𝐀(𝑡)+𝐀(𝑡)𝐲, (3.11) where 𝐀(𝑡)=𝜔−1+𝛼cos²()()()()()𝜔𝑡1−𝛼sin𝜔𝑡cos𝜔𝑡−1−𝛼sin𝜔𝑡cos𝜔𝑡−1+𝛼sin²(),𝐀(𝑡)=𝜔𝑡𝑓(𝑡)000 (3.12)𝛼 is a system parameter and 𝜔=2𝜋. The state transition matrix (STM), Φ(𝑡), when the quasiperiodic term 𝐀(𝑡)=0, is given as [12] 𝚽(𝑡)=𝑒^{(𝛼−1)𝜔𝑡}cos(𝜔𝑡)𝑒^−𝜔𝑡sin(𝜔𝑡)−𝑒^{(𝛼−1)𝜔𝑡}sin(𝜔𝑡)𝑒^−𝜔𝑡()(𝑡)𝑒cos𝜔𝑡=𝐐^𝐑𝑡. (3.13)
Factoring the state transition matrix as shown above, the L-F transformation matrix 𝐐(𝑡) is found as 𝐐(𝑡)=()()()()cos𝜔𝑡sin𝜔𝑡−sin𝜔𝑡cos𝜔𝑡,𝑒^𝐑𝑡=𝑒^{(𝛼−1)𝜔𝑡}00𝑒^−𝜔𝑡. (3.14)
It is noted that the system is unstable for all α > 1. Using the L-F transformation 𝐳(𝑡)=𝐐(𝑡)𝐲(𝑡) (c.f. (3.14)) (3.11) to yield a time-invariant system given by 𝐳̇(𝑡)=𝐀𝐳(𝑡). (3.15) Let 𝐕=𝐳^𝑇(𝑡)𝐁𝐳(𝑡), where 𝐁 is a constant, symmetric, positive definite matrix.
Then 𝐕̇=𝐳̇^𝑇𝐁𝐳+𝐳^𝑇𝐁𝐳̇=𝐳^𝑇𝐀^𝑇𝐁+𝐁𝐀𝐳≡−𝐳^𝑇𝐂𝐳. (3.16)
Setting 𝐵𝐁=¹¹𝐵₁₂𝐵₁₂𝐵₂₂,𝐂=𝐈₂=1001, (3.17) substituting (3.17) into (3.16), yields 𝐵₁₁=−1/(2𝜔(𝛼−1)) (𝛼<1), 𝐵₁₂=0 and …
Therefore, −1𝐁=()0012𝜔𝛼−12𝜔,𝐁⁻¹=−2𝜔(𝛼−1)002𝜔. (3.18)
Since 𝐵₁₁>0 for 𝛼<1 and (𝐁)=−1Det()0012𝜔𝛼−112𝜔=−2𝜔²(𝛼−1)>0. (3.19) therefore, 𝐁 is a positive definite symmetric matrix and Lyapunov stability conditions are satisfied.
Once the 𝐁 matrix is constructed, the stability theorem and the corollary can be used to determine the stability conditions for the system. Simple computations yield 𝐃=𝐀^𝑇+𝐆^𝑇(𝑡)+𝐁𝐀+𝐆(𝑡)𝐁⁻¹=()(𝑡)()()−)2𝜔𝛼−1+2𝑓sin𝜔𝑡cos𝜔𝑡(𝛼−2cos²(𝜔𝑡)+1𝑓(𝑡))𝛼−1(𝛼−2cos²()𝑓(𝑡)(𝑡)()().𝜔𝑡+1−2𝜔−2𝑓sin𝜔𝑡cos𝜔𝑡 (3.20)
Setting Det[𝐃−𝜆𝐈]=0, the eigenvalues 𝜆 of the 𝐃 matrix are computed as 𝜆_1,2()±=−𝜔2−𝛼𝜔²𝛼²+1()𝑓(𝑡)()+𝛼−12𝜔𝛼𝛼−1sin𝜔𝑡2𝛼cos²(𝜔𝑡)−𝛼²cos⁴(𝜔𝑡)−1𝑓²(𝑡). (3.21)
Application of the theorem yields 𝐸𝜆_max[𝐃()]=−𝜔2−𝛼+𝐸𝜔²𝛼²+1()𝑓(𝑡)()+𝛼−12𝜔𝛼𝛼−1sin𝜔𝑡2𝛼cos²(𝜔𝑡)−𝛼²cos⁴(𝜔𝑡)−1𝑓²(𝑡)≤0 (3.22) or 𝐸𝜔²𝛼²+1()𝑓(𝑡)()+𝛼−12𝜔𝛼𝛼−1sin𝜔𝑡2𝛼cos²(𝜔𝑡)−𝛼²cos⁴(𝜔𝑡)−1𝑓²(𝑡)≤𝜔(2−𝛼). (3.23)
Using Schwarz’s Inequality [13], (𝐸{𝑓(𝑡)})²≤𝐸{𝑓²(𝑡)}, and simplification yields 𝐸𝑓²(𝑡)≤32𝜔²(1−𝛼)²8+3𝛼²−8𝛼. (3.24)
The results obtained from condition (3.24) for 𝛼 from 0 to 1 are shown in Figure 1.
In order to get the conditions for almost sure asymptotic stability from the corollary, matrices [𝐆^𝑇(𝑡)+𝐁𝐆(𝑡)𝐁⁻¹] and [𝐀+𝐁𝐀𝐁⁻¹] are calculated as 𝐆^𝑇(𝑡)+𝐁𝐆(𝑡)𝐁⁻¹(𝑡)()()−()=𝑓2sin𝜔𝑡cos𝜔𝑡𝛼−2cos²(𝜔𝑡)+1()𝛼−1𝛼−2cos²()()(),𝜔𝑡+1−2sin𝜔𝑡cos𝜔𝑡𝐀+𝐁𝐀𝐁⁻¹=2().𝛼−1𝜔00−2𝜔 (3.25)
The maximum eigenvalues of matrices given by (3.25) are computed as 𝜆_max𝐆^𝑇(𝑡)+𝐁𝐆(𝑡)𝐁⁻¹=𝑓(𝑡)1−2𝛼cos²(𝜔𝑡)+𝛼²cos⁴(𝜔𝑡),𝜆1−𝛼_max𝐀^𝐓+𝐁𝐀𝐁⁻¹=−2(1−𝛼)𝜔. (3.26)
Applying the corollary 𝐸𝜆_max𝐆^𝑇(𝑡)+𝐁𝐆(𝑡)𝐁⁻¹≤−𝜆_max𝐀^𝑇+𝐁𝐀𝐁⁻¹−𝜀 (3.27) yields𝐸𝑓(𝑡)1−2𝛼cos²(𝜔𝑡)+𝛼²cos⁴(𝜔𝑡)()1−𝛼≤21−𝛼𝜔. (3.28)
Then using Schwarz’s Inequality in (3.28), one obtains 𝐸𝑓²(𝑡)1−2𝛼cos²(𝜔𝑡)+𝛼²cos⁴(𝜔𝑡)()1−𝛼≤41−𝛼²𝜔² (3.29) or 𝐸𝑓²(𝑡)1−2𝛼𝐸cos²𝜔𝑡+𝛼²𝐸cos⁴𝜔𝑡≤4(1−𝛼)³𝜔². (3.30)
Since 𝐸{cos²(𝜔𝑡)}=1/2 and 𝐸{cos⁴(𝜔𝑡)}=3/8, inequality (3.30) provides the condition for almost sure asymptotic stability from corollary as 𝐸𝑓²(𝑡≤())321−𝛼³𝜔²8−8𝛼+3𝛼². (3.31) As expected, condition (3.31) is weaker than condition (3.24). Figure 1 displays the result obtained from (3.31) for 𝛼 in the range of 0 to 1. A comparison of conditions yielding from the theorem and corollary is shown in Figure 1.

Figure 1 

Stability results for example 1 obtained by the Theorem and Corollary.

4. L-F Transformation Approach for Quasiperiodic System

In the previous section, we presented the theorem and corollary that provide the bounds on the quasiperiodic term so that the system described by (1.2) is stable. Alternatively, one can use L-F transformation type approach to ascertain the stability of quasiperiodic system. Unlike the theorem and corollary, this approach does not need 𝐀 to have negative real parts (c.f. (3.1)).

Consider a quasiperiodic system given in second order form [14] by𝑥̈+(𝛿+𝛼(cos𝑡+cos𝜔𝑡))𝑥=0, (4.1) where 𝛿,𝛼 are constants, quasiperiodicity 𝜔≈𝑝/𝑞 where 𝑝,𝑞 are integers and 𝑥̈=𝑑²𝑥/𝑑𝑡². Equation (4.1) can be written in the state space form as𝐱̇=𝐀(𝑡)𝐱, (4.2) where 𝐱={𝑥𝑥̇}^𝑇, 𝐀=⁰¹_{−(𝛿+𝛼(cos𝑡+cos𝜔𝑡))0}.

Now using the transformation 𝑡=2𝑞𝜏, (4.1) can be transformed to14𝑞²𝑑²𝑥𝑑𝜏²+cos2𝑞𝜏+cos2𝑝𝜏𝑥=0. (4.3) It can be noted that (4.3) is a time periodic system with principle period 𝜋. The stability of (4.3) is governed by the Floquet theory. It is possible to find out the State Transtion Matrix (STM) Φ(𝜏) at the end of the principle period (also called as the Floquet Transition Matrix (FTM) Φ(𝜏=𝜋)) numerically [14, 15] or analytically using Picard iterations [16, 17]. For the details on the analytical computation of STM using Picard iteration approximation, we refer to reference [16]. It is noted that the Picard iteration approach yields an approximate closed form symbolic expression of the STM for time periodic system.

If eigenvalues of the FTM are inside the unit circle, then (4.3) is asymptotically stable. If the eigenvalues are on the unit circle then the system is simply stable, and if the eigenvalues are outside the unit circle, then the system is unstable. The stability (or instability) of the time periodic system given by (4.3) implies the stability (or instability) of (4.1). It can be noted that the Floquet theory states that the STM (Φ(𝜏)) of (4.3) can be partitioned as𝚽(𝜏)=𝐐(𝜏)𝐞^𝐁𝜏, (4.4) where 𝐐(𝜏) is the time periodic L-F transformation matrix and 𝐁 is the constant matrix of appropriate dimensions. The eigenvalues of 𝐁 are called the Floquet exponent and govern the stability of the time periodic system given by (4.3). For computation of the L-F transformation matrix via Chebyshev polynomials, we refer the reader to reference [18].

5. Conclusion

In this paper, simple and efficient computational techniques to guarantee sufficient conditions for almost sure asymptotic stability of periodic quasiperiodic systems have been presented. First, the L-F transformation has been utilized to convert the periodic part of time-periodic system to a time-invariant form. For the linear periodic-quasiperiodic system, a theorem and related corollary have been suggested using the results previously obtained by Infante [8]. In order to apply the theorem and the corollary successfully, it is observed that the eigenvalues of matrix 𝐀, which governs the stability of the system, must have negative real parts and matrix 𝐁 must be positive definite. One example is presented to show the application. Another apporach pressed here is based on the Floquet-type approach, where a quasiperiodic system is approximated as a periodic system and the Floquet theory can be applied to investigate the stability. Unlike the Infante type approch, the Floquet approach does not require eigenvalues of matrix 𝐀 to have negative real parts. In certain cases, Floquet type decomposition for quasiperiodic system can be used to reduce quasiperiodic system to LTI system. It is expected that these methodology would be useful in studying stability and designing controllers for a number of MEMS, where governing differential equations have time periodic quasiperiodic coefficients. The approaches presented in this paper can be extended to study stability and robustness of a linear time-periodic system subjected to random perturbations.