Stability analysis for a class of nonlinear switched systems using variational principle

Author’s Accepted Manuscript Stability Analysis for a Class of Nonlinear Switched Systems Using Variational Principle A. Karami, M.J. Yazdanpanah, B. Moshiri

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To appear in: Journal of the Franklin Institute Received date: 10 November 2015 Revised date: 20 July 2016 Accepted date: 2 August 2016 Cite this article as: A. Karami, M.J. Yazdanpanah and B. Moshiri, Stability Analysis for a Class of Nonlinear Switched Systems Using Variational Principle, Journal of the Franklin Institute, http://dx.doi.org/10.1016/j.jfranklin.2016.08.001 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting galley proof before it is published in its final citable form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

Stability Analysis for a Class of Nonlinear Switched Systems Using Variational Principle A. Karamia , M.J. Yazdanpanaha,b,∗, B. Moshiria,b a

School of Electrical and Computer Engineering, College of Engineering, University of Tehran, Tehran, Iran b The Control and Intelligent Processing Center of Excellence (CIPCE), School of Electrical and Computer Engineering, College of Engineering, University of Tehran, Tehran, Iran

Abstract This paper presents a new theorem for stability of all possible trajectories starting at a given initial condition of a class of nonlinear switched systems with an arbitrary switching law. By utilizing the idea of variational principle, it is shown that a nonlinear switched system is asymptotically stable if and only if a critical trajectory is asymptotically convergent to the origin. In this regard, a necessary and sufficient condition is obtained. In order to investigate the stability property, the critical trajectory, characterized as a solution that maximizes a cost function, should be calculated. To this end, an affine control system, instead of the original switched system, and optimal control techniques are utilized. Afterwards, to calculate the critical trajectory, the initial condition x(0) is assumed given, and an algorithm is introduced for stability of all possible trajectories starting at a given initial condition of n-dimensional nonlinear switched systems under arbitrary switching with several operating modes. Simulation results reveal the efficiency of proposed technique. Keywords: Hybrid systems, Switched systems, Nonlinear systems, Stability analysis, Variational principle 1. Introduction Generally, the systems that experience both discrete and continuous dynamic behavior are called “hybrid systems” [1]. Hybrid systems have been seriously considered by many researchers during the past decade. This importance comes from the main role of these systems in practical applications and modeling of physical systems. A special and main ∗

Corresponding author, Email: [email protected]

Preprint submitted to Journal of the Franklin Institute

August 5, 2016

class of hybrid systems is switched systems, which are suitable for modeling natural and human-made systems with several operation modes through which any mode has a different dynamic behavior [2, 3]. Also, in switched systems, as a special class of hybrid systems, the behavior in any mode is described by a continuous subsystem and transition between the modes is described as a discrete-time event. The function that determines which mode is active at each point in time is called switching signal. In other words, the switched system consists of subsystems that at each time, only one of them is active [4, 5]. As an example of hybrid and switched systems is a vehicle which has a different dynamic behavior in each gear transmission and this behavior is changed by automatic gearbox. Also communication systems in [6, 7], power electronic in [8], flight control [9], chemical processes in [10] and automated highway systems [11] are other applications of hybrid systems. Definitively, stability of switched systems is one of the main issues that should be analyzed and investigated for utilizing such systems. So far, several works have been done on stability analysis of these systems and different approaches have been developed. The primary idea is to consider a nonlinear switched system as a nonlinear timevarying system and use the idea of time-varying Lyapunov function for stability analysis of the systems [12]. However, this idea needs a time-varying Lyapunov function satisfying the stability conditions for time-varying nonlinear systems. In the meantime, the switching signal should be known. Finding such a Lyapunov function is very difficult and almost impossible. Also, the assumption of switching signal being known is very restrictive. Therefore, this idea is not practically helpful and instrumental. Until now, the main developed approaches for stability analysis of switched systems are “Common Lyapunov Function (CLF)”, “Multiple Lyapunov Functions (MLF)” and also “Dwell-time” stability approaches. In the methods based on CLF a sufficient condition has been presented for uniformly globally asymptotically stability (UGAS) of an equilibrium point of switched systems. It has been shown that for a switched system consists of a finite family of subsystems, globally asymptotically stability (GAS) of any subsystem and binary commutation of subsystems are sufficient conditions for globally asymptotically stability [13]. Also, it has been demonstrated that by combining these results with the inverse Lyapunov theorem,

2

the conditions are sufficient for existence of a common Lyapunov function [13]. In [14], stability analysis for a special and limited class of switched nonlinear systems based on construction of a CLF is presented and sufficient conditions to guarantee the existence of a CLF are expressed in terms of linear inequalities. The results obtained from the methods based on MLF determine a sufficient condition for stability as follows: All subsystems must be stable and a slow-switching condition must be held between subsystems. In fact, this slow-switching condition is in the form of an upper bound on the probability mass function of the number of switching which happens between the first instance and current time [15]. In fact, the main idea of this method is that if there is a separated Lyapunov function for any subsystem, then it is sufficient to apply some limitations to switching signal to guarantee the general system stability. In the method based on MLF, finding Lyapunov functions and satisfying the conditions are easier than the method based on CLF; thus, this method is more customary than the CLF method. And, in the methods based on Dwell-time, stability condition is applied to the switching signal so that any subsystem must stay in a mode for a sufficient time. In other words, the condition is applied to the switching frequency or switching time between the two modes [16]. Also, [17] studied stability analysis of switched linear systems with nonlinear disturbances using generalized Gronwall–Bellman inequalities and presented sufficient conditions for stability analysis of special class of switched systems with constrained conditions on both linear and nonlinear sections. Also, in [18], necessary and sufficient conditions for stability of nonlinear switched systems with predefined and known switching path are presented where the switching signal is not arbitrary. In [19], stability analysis for switched systems with an arbitrary switching signal, however only for linear systems, is investigated. As can be observed, in CLF method, it is difficult and restrictive to find such a function or to solve linear matrix inequalities. In the meantime, only sufficient conditions for stability can be obtained. In the other two approaches, a condition is applied to the switching signal in addition to the existing conditions on subsystems. Since, switching signal may be time-dependent, state- dependent, pre-determined or random, applying conditions to the switching signal in order to guarantee stability is not practical in many applications and systems. So, it is desirable to investigate and analyse

3

stability of switched systems independent from switching signal; in other words, stability analysis of switched systems with an arbitrary switching signal is demanded. The only restriction is that the switching signal must have finite number of switching points in any bounded time interval. That is a standard assumption in switched systems and does not violate the generality of results. Variational principle is based on the following simple idea: If the “most unstable solution” or “critical trajectory” of switched system is convergent, then all other solutions of system are also convergent [4]. This idea has been previously introduced in [4, 20, 21, 22]. In this approach, by finding the critical solution of system, the complexity of stability problem is reduced from the behavior analysis of a system to behavior analysis of a specific and single trajectory. In [4], this idea has been used for stability analysis of linear switched systems and also related theorems have been presented only for linear switched systems with n = 2 theoretically and n = 3 numerically. Additionally, the variational approach is utilized to derive the stability analysis of positive linear switched systems with two subsystems [20, 21, 22]. In this paper, using the mentioned idea, a stability criterion is introduced that obtains a necessary and sufficient condition for stability analysis of nonlinear switched systems with an arbitrary switching signal. On this view, a nonlinear switched system is asymptotically stable if and only if a critical trajectory is asymptotically convergent to the origin. In order to calculate the critical trajectory, the scope of presented stability theorem is restricted to a class of nonlinear switched systems with a given continuous state initial condition and the critical trajectory is calculated utilizing an affine control system instead of the original switched system. So, a stability theorem and an algorithm are presented for the stability of all trajectories starting at a given initial condition of a class of n-dimension nonlinear switched systems with an arbitrary switching signal. Therefore, the basic idea of variational approach is utilized to obtain a necessary and sufficient condition for stability of all trajectories starting at a given initial state of nonlinear switched systems and an algorithm is introduced for this purpose. The contribution of this paper is in the line of presenting and proving a new stability criterion for nonlinear switched systems (Theorem 1 ) and also, introducing an algorithm for stability of all possible trajectories starting at a given initial condition of a class of nonlinear switched systems under arbitrary switching (Theorem 2 ) through which the

4

optimal control techniques is utilized to find a worst-case switching signal that maximizes a cost function while an affine nonlinear control system is used instead of the original nonlinear switched system (Lemma 1 ). The rest of this paper is organized as follows. In Section 2, stability of switched systems using the variational approach is presented. Section 3 presents calculation of the critical trajectory of switched system. The main result of this paper and details of the presented new method for stability analysis of hybrid systems is described in Section 4. In Section 5 and Section 6, the simulation and comparisonal results and conclusion are mentioned. Notations: The notations used in this paper are standard. The vectors are column vectors, xn×1 . The superscript ‘T’ used for matrix transpose. For a scalar f , ∂f /∂x = [∂f /∂x1 , · · · , ∂f /∂xn ]T is a column vector. The notation P  0 ( 0) means that matrix P is positive (semi) definite and P ≺ 0 ( 0) means that matrix P is negative (semi) definite. k· k represents the Euclidean norm. 2. Stability of nonlinear switched systems A mathematical model for switched systems can be considered as follows: x(t) ˙ = fσ(t) (x(t))

(1)

where x(t) ∈ Rn , fi : Rn → Rn are vector fields and σ : [0, ∞) → {0, 1, · · · , m} is a piecewise constant function of time which is called the switching signal. Any operation mode is related to a specific subsystem as x(t) ˙ = fi (x(t)) ∀i ∈ {0, 1, · · · , m}

(2)

and the switching signal determines which subsystem is active at any time. It is usually assumed that the switching signal has a finite number of switches on any limited time interval, i.e., possibility of infinity fast switching is excluded [4]. Since in many cases, exact details of switching mechanism is unknown and also this mechanism may depend on uncontrolled, unpredictable and uncertain conditions, applying the conditions to the switching signal in order to guarantee stability of the switched system is not possible and practical. So, the important point in stability analysis of these systems is the arbitrarily assumption of switching signal. Therefore, the following 5

important problem is raised:

Problem 1: Stability analysis of nonlinear switched systems under arbitrary switching; specifically, acquiring necessary and sufficient conditions for assuring the asymptotic stability of nonlinear switched system (1) under arbitrary switching.

Obviously, a necessary condition for asymptotic stability of switched systems is asymptotic stability of any subsystem, however, it is not a sufficient condition, as an example, see Example 2 of [4]. Stability analysis of switched systems due to infinite number of solutions for any initial condition x(0) = x0 of the system (1) is very difficult and challenging. As mentioned in [4], the variational approach is based on the following idea: “The critical trajectory of switched system (1) would be determined, if the solution is convergent; then, all other solutions are also convergent.” By using variational approach and finding the critical trajectory, behavior of a specific and single trajectory is analyzed instead of behavior of a complex system with an infinite number of solutions [4].

Assumptions 2.1: • The solution of system (1) for the initial state x(0) and any switching signal exists and is unique and continuous [23]. • Without loss of generality, the origin is equilibrium point for any subsystems and the switched system (1). Also, conception of system stability concerns stability of origin as the equilibrium point, as mentioned in [12]. Asymptotic stability of the system (1) means that the system is stable and in the meantime, all corresponding trajectories converge to the origin asymptotically for any switching law. In some literature on switched systems, this property is called uniform asymptotic stability (UAS), so that uniformity concept is with respect to the arbitrary switching signal. Definition 2.1: x∗ (t) for a given matrix Q(x) is called the critical trajectory, if it maximizes the following cost function J among all feasible solutions in terms of switching signal, σ(t), and initial continuous state condition, x(0): Z ∞ J(x, Q) = xT (t)Q(x)x(t)dt 0

6

(3)

x∗ (t) = arg max J x(t)

where Q(x) is a continuous positive definite matrix (Q(x)  0 , ∀x) that satisfies the following condition: kx1 k < kx2 k ⇒ xT1 Q(x1 )x1 < xT2 Q(x2 )x2

(4)

that k· k is Euclidean norm of vector. Theorem 1: The nonlinear switched system (1) under arbitrary switching is globally (locally) asymptotically stable, GAS (LAS), if and only if the the critical trajectory x∗ (t) is asymptotically convergent to the origin, i.e., limt→∞ x∗ (t) = 0 , ∀ x(0) ∈ Rn (D ⊂ Rn )

(5)

where D is a compact and convex domain which includes the origin.

Proof: Necessity: If the switched system (1) be asymptotically stable (A.S.) then all solutions of (1) including x∗ (t) are asymptotically convergent, i.e., limt→∞ x(t) = limt→∞ x∗ (t) = 0.

(6)

Sufficiency: By defining the set Ω = {t : kx(t)k >  > 0} and µ as its length 1 ; for an asymptotically stable system:

∀ x(t) and ∀  > 0 : µ(Ω) = M ≤ T < ∞,

(7)

where T is the maximum t for which kx(t)k > .

Now, assume that limt→∞ x∗ (t) = 0 and in the meantime the switched system (1) is

not asymptotically stable, i.e., at least a solution of system (1) exist such that that don’t converge to the origin. Therefore; ∃ x(t) and ∃  > 0 1

s.t. µ(Ω) = ∞

As an example, if Ω = [a, b] ∪ [c, d) then µ(Ω) = (b − a) + (d − c)), also µ can be ∞.

7

(8)

As a result,

R∞ 0

kx(t)k dt = ∞.

Now, two possible cases may happen: 1- x(t) is the critical trajectory, i.e., x(t) = x∗ (t). This means that limt→∞ kx(t)k 6= 0, i.e., x∗ (t) is not asymptotically convergent to the

origin and the assumption limt→∞ kx∗ (t)k = 0 is violated. 2- x(t) is not the critical trajectory, i.e., x(t) 6= x∗ (t).

Because of asymptotic convergence of x∗ (t) and also noting to the fact that kx(t)k >  > 0 ∀ t ∈ Ω, one has: ∃ T1 ≥ 0 s.t. ∀ t > T1 : kx(t)k > kx∗ (t)k

(9)

T

Therefore, with regard to (4), xT (t)Q(x)x(t) > x∗ (t)Q(x∗ )x∗ (t) for t > T1 and Z ∞ Z ∞ T T x Q(x)x dt > x∗ Q(x∗ )x∗ dt. T1

(10)

T1

Now, for t < T1 : RT RT T If 0 1 xT Q(x)x dt ≥ 0 1 x∗ Q(x∗ )x∗ dt, then regarding (10) one has: Z ∞ Z ∞ T T x Q(x)x dt > x∗ Q(x∗ )x∗ dt. 0

(11)

0

otherwise, time interval(s) exist(s) that for t in which x(t) < x∗ (t), i.e., times ti , i = 1, · · · , L exist such that x(ti ) = x∗ (ti ), (tL := T1 ).

Now, we define x0 (t) as maximum of x(t) and x∗ (t) in any time interval between ti

and ti+1 (i = 0, · · · , L − 1) for 0 ≤ t ≤ T1 : x0 (t) = max {x(t) , x∗ (t)}, i = 0, · · · , L − 1. t∈[ti ,ti+1 ]

So, x0 (t) ≤ x∗ (t), ∀ t ≤ T1 and regarding (4): Z T1 Z 0T 0 0 x Q(x )x dt ≥ 0

T

T

x∗1 Q(x∗ )x∗ dt

0

From the above inequality and (10), one has; Z T Z ∞ 0T 0 0 x Q(x )x dt + xT Q(x)x dt > 0 T Z T Z ∗T ∗ ∗ x Q(x )x dt + 0

(12) ∞

T

x∗ Q(x∗ )x∗ dt

T

Now, noting to the arbitrary nature of switching signal and the facts that x(ti ) = x∗ (ti ), i = 1, · · · , L, it becomes clear that x00 (t) defined below is a continuous solution for system (1): 8

It is clear that

Z

∞

  x0 (t) , 0 ≤ t < T 00 x (t) =  x(t) , t ≥ T 00T

00

00

x Q(x )x dt >

0

Z

∞

T

x∗ Q(x∗ )x∗ dt

(13)

0

therefore, regarding (11) and (13), x∗ (t) is not the critical trajectory. Therefore, in both cases, the trajectory x∗ (t) is not asymptotically convergent to the origin or is not the critical trajectory. This completes the proof.

By utilizing Theorem 1, the variational approach may be used for stability analysis of nonlinear switched systems by considering the following steps: 1. Stability analysis of all subsystems. 2. Determining the critical trajectory, x∗ (t). 3. The switched system (1) is asymptotically stable, if only if the critical trajectory x∗ (t) is asymptotically convergent to the origin. It is obvious that important step is characterization of the critical trajectory, x∗ (t). If one can calculate the x∗ (t) for a nonlinear switched system, Theorem 1 states the necessary and sufficient condition for global (local) asymptotically stability for a class of nonlinear switched systems with an arbitrary switching signal. However, calculation of the critical trajectory, x∗ (t), in the general case, i.e., maximization function J in terms of both the switching signal, σ(t), and the initial condition, x(0), is challenging. So, in this paper, the initial condition x(0) is assumed known and maximization is done in terms of the switching signal, σ(t). Assumption 2.2: In system (1), the initial condition x(0) is known and given. This assumption can be used in a wide range of applications and actual systems. However, stability analysis of switched systems with a given initial condition but under arbitrary switching (due to infinite number of solutions for any initial condition) is challenging. In summary, to calculate the critical trajectory using optimal control tools, the initial condition x(0) is required and stability analysis of nonlinear switched systems with a given initial condition under arbitrary switching is limited for a class of nonlinear switched

9

systems. So, regarding Assumption 2.2, a special version of stability definition is used as follows: Definition 2.2: The nonlinear switched system (1) with a given initial condition x(0) is called “asymptotically stable for x(0)”, i.e., the origin is an asymptotically stable equilibrium point for the switched system (1) with a given initial condition x(0), if the family of trajectories (1) with the initial condition x(0) -due to different switching signalsare asymptotically convergent to the origin. Remark 2.1: This definition is similar to the well-known definition of asymptotic stability; but the initial condition x(0), here, is fixed and given. Note that, due to various switching signals (different sequences and different times of switching), several trajectories are possible to take place [23]. Definition 2.3: Consider system (1) with a given initial condition x(0). x∗ (t) for a given matrix Q(x) is called the critical trajectory, if it maximizes the following cost function J among all feasible solutions in terms of the switching signal, σ(t): Z ∞ xT (t)Q(x)x(t)dt J(x, Q) =

(14)

0

x∗ (t) = arg max J x(t)

Theorem 2: The nonlinear switched system (1) with a given initial condition x(0) and arbitrary switching is “asymptotically stable for x(0)”, i.e., all possible trajectories starting at a given initial condition x(0)” are asymptotically convergent if and only if the corresponding critical trajectory x∗ (t) is asymptotically convergent to the origin, i.e., limt→∞ x∗ (t) = 0. In the sequel, notions of stability and the critical trajectory x∗ (t) are based on Definition 2.2 and Definition 2.3. Therefore, the critical trajectory is not determined for nonlinear switched system with any initial condition in a region around the equilibrium point and is characterized for a given initial condition. Hence, the proposed approach has a limitation that the initial condition of continuous state of switched system, x(0), must be given, and this limitation results in obtaining the necessary and sufficient stability condition for all possible trajectories starting at a given initial condition instead of local or global stability. In the next section, in order to characterize the critical trajectory (x∗ (t)), first, an affine nonlinear control system instead of the nonlinear switched system (1) is used and 10

then, optimal control tools are utilized to calculate the switching signal related to the critical trajectory. 3. Calculation of the critical trajectory As mentioned, a switched system with an arbitrary switching signal has an infinite number of responses for any initial condition; so, a basic idea is based on determination of the critical trajectory. If this solution is convergent to the origin, then all other solutions are convergent and thus, the nonlinear switched system would be asymptotically stable and vice versa. In this approach, finding the critical trajectory, x∗ (t), is necessary. In the following subsection, an affine nonlinear control system will be used instead of the nonlinear switched system (1) and then, in the next subsection, optimal control techniques will be used to calculate the critical trajectory. 3.1. Introducing switched systems as affine control systems For simplicity, first, the case m = 1 (the switched system with two subsystems) is considered. The results can be generalized for m > 1 . Lemma 1: The critical trajectory, x∗ (t), of the nonlinear switched system (1) with a given initial condition x(0) is equal to the solution of following nonlinear affine control system that maximizes the cost function (14): x(t) ˙ = f (x) + g(x)u(t)

(15)

where f (x) := f0 (x), g(x) := f1 (x) − f0 (x) and acceptable control input for (15) will be a measurable function of time which belongs to the set U = [0, 1]; i.e., control u(t) ∈ U for all t.

Proof: To determine the solution of system (15) that maximizes the cost function (14), the optimal control u∗ (t) should be calculated. This constrained optimal control problem can be summarized as follows: ∗

F ind u (t) to M aximize J =

Z

0

11

∞

xT Q(x)x dt

(16)

s.t. x(t) ˙ = f (x) + g(x)u(t); The control input is constrained as follows: u ∈ U = [0, 1] The maximized cost function is Z Z ∞ T ∗ x Q(x)x dt = J = max u∈U

∞

T

x+ Q(x+ )x+ dt

(17)

0

0

where x+ = x(t, x0 , u∗ ), The solution of constrained optimal control problem is obtained using Hamiltonian function H(x, u, λ) = xT Q(x)x + λT f (x) + λT g(x)u

(18)

and Maximum Principle H(x∗ , u∗ , λ∗ ) ≥ H(x∗ , u, λ∗ ), ∀ u ∈ U

(19)

The Hamiltonian equations for optimality are: ∂H(·) = f (x) + g(x)u; x(0) = x0 , ∂λ ∂H(·) ˙ λ(t) =− , λ(tf ) = λtf , ∂x  H(x∗ , u∗ , λ∗ ) = max x∗ T Q(x∗ )x∗ + λ∗ T f (x∗ ) + λ∗ T g(x∗ )u x(t) ˙ =

(20)

u∈[0,1]

Accordingly,

 u∗ (t) = arg max λ∗ T g(x∗ )u u∈[0,1]

(21)

By defining the switching function γ(x, λ) := λT g(x), i.e., the coefficient of u(t), optimal control law u∗ (t) is obtained as:    0, γ ∗ (t) < 0   u∗ (t) = 1, γ ∗ (t) > 0     U ndef ined, γ ∗ (t) = 0 where γ ∗ (t) = γ(x∗ , λ∗ ).

12

(22)

To investigate the Bang-Bang property of optimal control, zeroes of function γ(t) should be considered. If the switching function γ ∗ (t) has only a finite number of zeroes (i.e., S < ∞) on any bounded time interval, i.e., γ(t0k ) = 0,

d γ(t0 ) 6= 0 dt k

∀ k = 1, 2, · · · , S

(23)

in other words, γ ∗ (t) 6≡ 0 , ∀ t, then optimal control u∗ (t) has “Strictly Bang-Bang property” and takes values in only {0, 1}.

For singular intervals [t1 , t2 ] such that γ ∗ (t) = hλ∗ (t), g(x∗ (t)i ≡ 0, the switching

function γ ∗ (t) and its derivatives must be zero along optimal trajectories:               

γ ∗ (t) = 0 γ˙∗ (t) = 0 .. .

(24)

γ ∗ (n−1) (t) = 0

(n : order of system)

So, for a Normal system (without any singular interval), it is enough to have γ ∗ (t) 6= 0 or γ˙∗ (t) 6= 0 or · · · or γ ∗ (n−1) (t) 6= 0 for any time.

For n = 2 as example, the system (15) is Normal if the following set of equations are not satisfied:

  

where

γ ∗ (t) = 0 γ˙∗ (t) = 0

γ ∗ (t) = λ∗ (t)T g(x∗ (t)) = hλ∗ , g(x∗ )i d ˙ gi|∗ + hλ, gi| γ˙∗ (t) = γ ∗ (t) = hλ, ˙ ∗ dt ∂H(·) d λ˙ = − = − (xT Q(x)x) − fxT λ − gxT λu ∂x dx T ˆ ˆ − fxT λ − gxT λu = −Q(x) − fx λ − gxT λu = −Q ˆ where Q(x) =

d (xT Q(x)x). dx

So,

ˆ − fxT λ − gxT λu, gi|∗ + hλ, gx (f + gu)i|∗ γ˙∗ (t) = h−Q ˆ gi|∗ − hf T λ, gi|∗ − hg T λu, gi|∗ + hλ, gx f i|∗ + hλ, gx gui|∗ = −hQ, x x ˆ gi|∗ − hλ, fx gi|∗ + hλ, gx f i| = −hQ, ˆ gi|∗ + hλ, gx f − fx gi| = −hQ, ˆ gi|∗ + hλ, [f, g]i|∗ = −hQ, 13

(25)

In other words, the system (15) is Normal, if for every time, at least one of following inequalities is satisfied: hλ, gi|∗ 6= 0

ˆ gi|∗ hλ, [f, g]i|∗ 6= hQ,

(26)

where [f, g] is the Lie bracket of vector fields f and g. With regard to λ∗ (t) 6= 0 (Section 5.6 of [24]), g(x) = f1 (x) − f0 (x) 6= 0 and also liberty in the selection of Q(x) as design parameter by user, without loss generality, the system (15) can be assumed Normal. Therefore, considering the system (15) as Normal (Non-singular ), the switching function, γ ∗ (t), has finite number of zeroes and u∗ (t) has finite number of switching points on any bounded time interval, i.e., the optimal control input u∗ (t) has a Bang-Bang property. Now, science u∗ (t) takes values only {0, 1}, by considering σ ∗ (t) = u∗ (t), the critical

trajectory x∗ (t) of switched system (1) and the solution related to optimal control u∗ (t) in the system (15) are equal. 

3.2. Calculation of optimal control / Critical switching signal By utilizing Lemma 1, the critical trajectory x∗ (t) can be calculated by finding the optimal control u∗ (t) of system (15) and considering σ ∗ (t) = u∗ (t). So, this subsection is focused on calculation of the optimal control u∗ (t). The optimization problem is restated as follows: Problem 2: Find optimal control u∗ (t) that maximizes the cost function J as Z ∞ J= xT Q(x)x dt (27) 0

s.t. x(t) ˙ = f (x) + g(x)u(t) , x(0) = x0 ; u ∈ U = [0, 1] With regard to the maximum principle, Hamiltonian equations and other matters in the proof of Lemma 1, the optimal control u∗ (t) has the Bang-Bang control that takes

14

only values 0 or 1 based on the value of switching function γ ∗ (t) = λ∗ (t)T g(x∗ (t)):   0, γ ∗ (t) < 0 ∗ u (t) = (28)  1, γ ∗ (t) > 0 The main problem now, is calculation of λ∗ (t), because according to Hamiltonian

equations for optimality, the co-state λ∗ (t) should be calculated backward in time; however, in order to do backward calculation of λ(t), optimal state vector x∗ (t) or optimal control signal u∗ (t) should be characterized. In other words, in switched systems formulation, the critical switching signal σ ∗ (t) should be specified. Nevertheless, calculation of the optimal control u∗ (t) needs optimal co-state vector λ∗ (t). In brief, calculation of x∗ (t) with the initial condition x(0) = x0 , needs λ∗ (t) with the final condition λ∗ (tf ) = λtf , and vice versa. This problem with both initial and final conditions is called Two Point Boundary Value Problem (TPBVP). The above mentioned developed solutions for these problems have numeric and approximate nature. Since the system under control is nonlinear, the relevant numerical algorithms have a large computational cost, their convergence rate is slow and their results are very sensitive to the initial guess of response. Therefore, these approximate and trial and error methods are not useful and reliable for stability analysis of switched systems based on the so-far mentioned results. Consequently, in order to obtain a precise and explicit response, it is necessary to use other methods of optimal control. In order to solve the optimal control problem, the cost function (14) and (27) can be generalized as follows: J=

Z

∞

(xT Q(x)x + uT R(x)u) dt

(29)

0

where R(x) = εR0 (x) ≺ 0 such that R0 (x)  0 ∀x , ε− → 0 (as time goes to infinity)

¯ such that R0 (x) ≺ R ¯ , ∀x. (Note that u ∈ [0, 1] and and there exists constant matrix R also from [25], in order to exist the maximum value of cost function J, R(x) should be negative definite.) So, the Problem 2 can be restated as an “Input-affine and Constrained Infinite Time Nonlinear Optimal Control (ITNOC)” problem:

J

∗

s.t.

= max u∈U

Z

∞

(xT Q(x)x + uT R(x)u) dt

0

x(t) ˙ = f (x) + g(x)u, x(0) = x0 ; 15

u ∈ U = [0, 1]

(30)

where f (x), g(x), Q(x) and R(x) are sufficiently smooth matrix functions of x(t). Using techniques of calculus of variations, Hamiltonian for this problem is as follows: H(x, u, λ) = xT Q(x)x + uT R(x)u + λT (f (x) + g(x)u)

(31)

H(x, u∗ , λ∗ ) = max{xT Qx + uT Ru + λ∗ T (f (x) + g(x)u)}

(32)

and

u∈U

In addition, similar to [24] (Section 3.11), the HJB equation is in the form below ∂J T ∂J T T + max{x Qx + u Ru + (f (x) + g(x)u)} = 0 u∈U ∂t ∂x

(33)

In the infinite horizon case, J(x) is only a function of the initial condition of state and hence, is time invariant; therefore, ∂J/∂t = 0 and Eq. (33) can be simplified as: max{xT Q(x)x + uT Ru + u∈U

∂J T (f (x) + g(x)u)} = 0 ∂x

(34)

From [24, 26] and also Eq. (32) and Eq. (34): λ=

∂J ∂x

(35)

From HJB equation and also ∂H/∂u = 0, the optimal control u∗ (t) obtained as: 1 u∗ (t) = − R−1 (x)g T (x)λ∗ ; 0 ≤ u∗ (t) ≤ 1 2

(36)

Since the matrix ∂ 2 H/∂u2 = R(x) is negative definite and J is quadratic in u, this control maximizes J globally. Equivalently, for optimal control u = u∗ (t): ∂J ∗ T 1 ∂J ∗ T ∂J ∗ −1 T f (x) − g(x)R (x)g (x) =0 x Q(x)x + ∂x 4 ∂x ∂x T

(37)

In fact, J(x) is the value function of infinite horizon optimal control problem, which is a function of initial condition of x0 . Using State Dependent Riccati Equation (SDRE) approach [26], finding the optimal control u∗ (t) is based on the two following generalizations:

16

1. Without loss of generality, the vector function f (x) can be stated as the multiplication of a state dependent matrix, A(x) and state vector x. Therefore, the system can be stated as follows: f (x) = A(x)x, g(x) = B(x)

(38)

x(t) ˙ = A(x)x + B(x)u It should be mentioned that this expansion is not unique. For example, if A0 (x) ˜ ˜ exists such that f (x) = A0 (x)x, then for any matrix A(x) that A(x)x = 0, f (x) = ˜ ˆ [A0 (x) + A(x)].x = A(x)x. 2. The second generalization is related to the infinite horizon value function of J(x). Again, without loss of generality, J(x) can be stated as J(x) = xT P (x)x, where P (x) is a symmetric matrix of state x [26]. Now, 1 ∂P (x) ∂J = 2(P (x) + xT )x. ∂x 2 ∂x

(39)

where xT ∂P (x)∂x is a matrix with row i, column j elements defined by xT ∂pij (x)∂x (pij (x) is the ith row, jth column of the matrix P (x))[26]. By defining Π(x) := P (x) + 21 xT ∂P∂x(x) , Eq. (37) can be rewritten as follows: xT [Π(x)A(x) + AT (x)Π(x) + Q(x) − Π(x)B(x)R−1 (x)B T (x)Π(x)]x = 0

(40)

The sufficient condition for holding this equation is that Π(x) satisfies the following state dependent equation: Π(x)A(x) + AT (x)Π(x) + Q(x) − Π(x)B(x)R−1 (x)B T (x)Π(x) = 0

(41)

Under appropriate conditions, it can be guaranteed that for a given x, the abovementioned equation has a unique and positive definite solution Π(x). Implicit used in this approach is the fact that

∂J ∂x

= 2Π(x)x, i.e., 2Π(x)x be the gradient of scalar function

J(x). According to [12], v(x) = Π(x)x should satisfy “Curl” condition, Curl(Π(x)x) = 0, i.e., ∂vi ∂vj = , ∀i, j ∂xj ∂xi

(42)

The drawback of this approach is that for any arbitrary expansion of nonlinear system, f (x) = A(x)x, satisfying the Curl condition is not guaranteed. 17

As mentioned before, expansion of f (x) = A(x)x is not unique and has a degree of freedom. Also, it has been proven that for a ITNOC problem with given f (x), g(x), Q(x) and R(x), there exists at least one expansion of f (x) such that the obtained Π(x) satisfies the Curl condition [28]. However, the presented proof cannot be used for determining an analytic expansion of f (x) or an appropriate A(x) which satisfies the Curl condition. Assumption 3.1: A(x) is such that f (x) = A(x)x and also both matrix functions A(x) and B(x) are continuous and the pair (A(x), B(x)) is stabilizable (in the linear system sense) for all x. Assumption 3.2: Positive definite solution of Eq. (41), Π(x), satisfies the Curl condition for all x; i.e., curl(Π(x)x) = 0. Similar to results in [26] and with regard to Q(x)  0 and R(x) ≺ 0 for all x, Assumption 3.1 guarantees the existence of a positive definite solution for Eq. (41) for all x and Assumption 3.2 is a necessary and sufficient condition for the existence of a scalar function J(x) such that satisfies the Curl condition, i.e., ∂J/∂x = 2Π(x)x. So, the optimal control signal u∗ (t) = −R−1 (x)B T (x)Π(x)x . satisfying necessary conditions for optimality of Eq. (37). Since R(x) → 0, therefore u∗ (t) → ±∞, but with regard to the constraint on the

control signal, u ∈ U = [0, 1], the optimal control u∗ (t) has the Bang-Bang property, i.e., u∗ (t) ∈ {0, 1} for all t.

In summary, from Bang-Bang property as well as considering above calculations, the control signal to maximize the cost function (29) is as follows:   0, γ ∗ (t) < 0 u∗ (t) =  1, γ ∗ (t) > 0

(43)

where the switching function γ(x, λ) := λT g(x) = λT B(x), γ ∗ (t) = γ(x∗ , λ∗ ) and λ(t) = ∂J ∂x

= 2Π(x)x that Π(x) is solution of equation (41). Now, with regard to calculated optimal control, u∗ (t), i.e., the switching signal related

to the critical trajectory, x∗ (t) can be found from (1) or (15).

Although in this paper, mathematical relations for the switched system with twosubsystems (case of m = 1) are mentioned completely, the proposed approach can be 18

generalized for m > 1. For the general case of switched systems with m subsystems (m > 1); the Theorems 1, 2 are applicable for both cases of m = 1 and m > 1. Also, the control system (15) in Lemma 1 reads as follows: x(t) ˙ = f (x) +

m X

gk (x)uk (t);

k=1

= f (x) + G(x)¯ u(t) where f (x) := f0 (x), gk (x) := fk (x) − f0 (x), k = 1, · · · , m, G(x) = [g1 (x), · · · , gm (x)] and u¯ = (u1 , · · · , um )T ∈ U where U is the standard simplex ( ) m X U = u¯ ∈ Rm : uk ≥ 0; uk ≤ 1 . k=1

and other relations of the critical trajectory calculation can be generalized by using matrix/vector functions of G(x) and u¯(t), respectively, instead of vector/scalar functions of g(x) and u(t). As general, the Hamiltonian reads  H(x∗ , u¯∗ , λ∗ ) = max x∗ T Q(x∗ )x∗ + λ∗ T f (x∗ ) + λ∗ T G(x∗ )¯ u u ¯∈U

Therefore,

 u¯∗ (t) = arg max λ∗ T G(x∗ )¯ u u ¯∈U

(44)

with appropriate dimensions of λn×1 , Gn×m , u¯m×1 . By defining the switching function γ1×m (x, λ) := λT G(x), and ∗ u¯∗ (t) = arg max {γ1∗ .u1 + γ2∗ .u2 , · · · , γm .um } u ¯∈U

the optimal control law, u¯∗ (t) is obtained as:   u∗ (t) = 0 ; γj∗ (t) < 0 , ∀j = 1, 2, · · · , m j  u∗ (t) = 1, u∗ (t) = 0 (j : 1, · · · , m) ; k = arg max ∗ l=1,···,m {γl (t)} k j6=k

(45)

(46)

Also, the optimal control relations can be systematically generalized, (R(x) is an

m × m negative definite matrix).

19

4. Main result As mentioned in previous sections, with regard to Theorem 2 in Section 2, the “critical” trajectory, x∗ (t), should be determined for stability of all trajectories starting at a given initial condition of nonlinear switched systems with an arbitrary switching signal. To characterize x∗ (t), by utilizing the Lemma 1, the Problem 2 should be solved. Then, according to Section 3, this problem can be restated as a “Input-affine and Constrained Infinite Time Nonlinear Optimal Control” problem, i.e.; Find the optimal control u∗ (t) that maximizes the following cost function Z ∞ (xT Q(x)x + uT R(x)u) dt J = 0

s.t.

x(t) ˙ = f (x) + g(x)u, x(0) = x0 ;

(47)

u ∈ U = [0, 1] where f (x) := f0 (x) , g(x) := f1 (x) − f0 (x) and Q(x)  0 : kx1 k < kx2 k ⇒ xT1 Q(x1 )x1 < xT2 Q(x2 )x2 ¯ : Constant s.t. R0 (x) ≺ R ¯ , ∀x R(x) = εR0 (x) ≺ 0 : R0 (x)  0 , ε− → 0 , ∃ R

(48)

(49)

the above conditions are exactly (4) and condition on R(x) in (29). With regard to the constraint on control signal and according to Section 3, the optimal control u∗ (t) has the Bang-Bang property, i.e.,   0, γ ∗ (t) < 0 u∗ (t) =  1, γ ∗ (t) > 0

(50)

where γ ∗ (t) = λ∗ T (t)(f1 (x∗ ) − f0 (x∗ )).

As mentioned in Section 3, in order to find the switching function used in the above Bang-Bang control, co-state λ(t) is needed. However, with regard to switching and nonlinearity nature of the system, explicit calculation of co-state λ(t) is very difficult and even impossible in most problems. In this paper, for finding λ(t), a new method is proposed. According to the discussion in Section 3, in brief; λ(t) =

∂J = 2Π(x)x(t) ∂x 20

(51)

where Π(x)A(x) + AT (x)Π(x) + Q(x) − Π(x)B(x)R−1 (x)B T (x)Π(x) = 0

(52)

So, only Π(x) is required to be obtained.

To solve the above Riccati equation, the following methods can be used: 1. Let Q(x) and R(x) be given as the design parameters that satisfy conditions (48), (49) and also for given f (x), g(x), obtain A(x) such that Assumptions 3.1 and 3.2 are held. Now, by solving State-Dependent Riccati Equation (SDRE), Eq. (52), state matrix Π(x) can be calculated. Therefore, according to Eq. (51), λ(t) and thus the switching function of optimal control signal can be determined. 2. Assume R(x), A(x), Π(x) matrices be design parameters such that R(x) satisfies condition (49) and Assumption 3.1 and 3.2 are held. So, Q(x) can be calculated from SDRE as: Q(x) = Π(x)B(x)R−1 (x)B T (x)Π(x) + AT (x)Π(x) − Π(x)A(x)

(53)

where f (x) = A(x)x and Π(x) > 0 : Curl(Π(x)x) = 0. Thus, it is only sufficient to investigate holding the condition (48) for Q(x). Note, regarding liberty in the selection of Q(x) and R(x) in the first method or selection of R(x), A(x), Π(x) in the second method as input parameters, solving the SDRE or HJB equation is very simpler than solving the regular SDRE or HJB equations. After solving the Riccati equation and finding Π(x) or Q(x), the signal control u∗ (t) can be calculated from (50) and “critical trajectory”, x∗ (t), can be determined as x(t) ˙ = f0 (x) + (f1 (x) − f0 (x))u∗ (t), x∗ (t) = x(t) . Remark 4.1 : Regarding the curl condition, one has ∂J/∂x = 2Π(x)x, i.e., Π(x)x is the derivative of continuous and smooth function of J: Z ∞ ∂ 2Π(x)x = ( xT Q(x)x + uT R(x)u dt) ∂x 0 Z ∞ ∂ ∂ T (x Q(x)x) + uT R(x)u dt = ∂x ∂x Z0 ∞ ∂ ∂ = xT (Q(x) + QT (x))x + uT R(x)u dt ∂x ∂x 0 21

and also due to continuity of x, Q(x) and R(x), the function under integral does not include any impulse and so the matrix Π(x) is continuous. Also, as mentioned, to solve the Riccati equation, two methods are usable. In first method, Q(x) is defined as the design parameter and the Riccati equation shall be solved. In second method, Π(x) is defined as the design parameter and the obtained Q(x) shall satisfy the mentioned conditions. So, a continuous Π(x) with desired properties can be defined or even obtained. Remark 4.2 : As mentioned (Section 3.2: λ∗ (t) 6= 0 and g(x) 6= 0) and regarding liberty in selection of Π(x) or Q(x), the system (15) can be considered Normal (Nonsingular ). So, the switching function, γ ∗ (t), has finite number of zeroes and u∗ (t) has finite number of switching on any bounded time interval. Finally, the algorithm of stability analysis of all possible trajectories starting at a given initial condition of nonlinear switched systems under arbitrary switching can be summarized as follows: 1. Stability analysis of any subsystems (Asymptotic stability condition for all subsystems is necessary). 2. Define an affine nonlinear control system instead of the switched system. (as stated in Section 3) 3. Calculate the “optimal control” u∗ (t) for the system obtained in Stage 2 using optimal control methods (as stated in this section: methods 1 or 2 mentioned in above). 4. Characterizing the “critical trajectory”, x∗ (t), corresponding to the optimal control u∗ (t). 5. All possible trajectories starting at a given initial condition x(0) are asymptotically convergent if and only if the “critical trajectory” x∗ (t) is asymptotically convergent. Generally, as mentioned in [4], several important advantages of the idea of variational approach compared to other stability analysis methods used for switched systems are: • This approach allows the usage of powerful and developed tools of optimal control theory in stability analysis of switched systems. • Since the worst-case analysis introduces an exact mechanism that leads to instability via switching signal design, this approach results in a necessary and sufficient

22

condition for stability analysis of switched systems. This ability is not possible in other methods based on finding sufficient conditions for stability. • In this approach, numerical methods can be used and results of numerical algorithms for optimal control can be used for stability analysis. 5. Simulation results In this section, in order to show ability of the presented method in extracting the necessary and sufficient condition for stability of all possible trajectories starting at a given initial condition of switched systems, four examples are simulated. Comparing its generality and simplicity with other methods exhibits capabilities of the proposed approach. Example 1- Consider a linear switched system with n = 2 as follows [4]:

where

  f (x), M ode = 0 0 x(t) ˙ =  f (x), M ode = 1 1 

f0 (x) = A0 x, A0 =  

f1 (x) = A1 x, A1 = 

0

1

−2 −1 0

(54)

 

1

−(2 + k) −1

 

According to [4] and using generalized first integral, geometric concepts and other computations, k ∗ = 6.985 is determined such that for k > k ∗ the switched system is unstable and for k < k ∗ that is stable. This method is applicable only for two-dimensional linear switched systems (n = 2, m + 1 = 2). Using the proposed approach with the following parameters:   2 1  f (x) = A0 (x)x, R(x) = 0.001, Π(x) =  0 1

the Q(x) satisfies the condition (48) and also the k ∗ = 6.985 can be found simply according to figures 1, 2 and 3.

23

Thus, thethe proposed approach precisely confirms results obtained in in [4],[4], however thisthis Thus, proposed approach precisely confirms results obtained however method is is useful forfor stability analysis of of all all possible trajectories starting at at a given ini-inimethod useful stability analysis possible trajectories starting a given tialtial condition of of both n-dimensional linear andand nonlinear switched systems with several condition both n-dimensional linear nonlinear switched systems with several operating modes and arbitrary switching signal. operating modes and arbitrary switching signal. Moreover, oneone of of thethe advantages of of this method is its application in linear andand nonlinMoreover, advantages this method is its application in linear nonlinearear switched systems under arbitrary switching with different orders with a given initial switched systems under arbitrary switching with different orders with a given initial useuse an an analytical method only forfor a linear condition. While thethe methods mentioned in in [4] [4] condition. While methods mentioned analytical method only a linear system with n n == 2 and this analytical method cannot be be used forfor linear system with system with 2 and this analytical method cannot used linear system with nn == 3; 3; thus, a numerical method and search algorithm is used forfor linear systems with thus, a numerical method and search algorithm is used linear systems with n= 3. 3. n= However, thethe proposed approach cancan be be applied forfor all all linear andand nonlinear systems However, proposed approach applied linear nonlinear systems with different orders without using thethe numerical algorithms. with different orders without using numerical algorithms.

Figure 1: System trajectory for k=5. Figure 1: System trajectory for k=5.

Example 2- 2In In this example, thethe conservativeness of method presented in [27], “LieExample this example, conservativeness of method presented in [27], “Liealgebraic Stability Condition forfor Nonlinear Switching Systems” is shown. algebraic Stability Condition Nonlinear Switching Systems” is shown. Stability condition forfor a nonlinear switched system with twotwo modes is as Stability condition a nonlinear switched system with modes is as [f0 , [f0 , f1 ]](x) = [f1 , [f0 , f1 ]](x) = 0, ∀x ∈ Rn . n [f0 , [f0 , f1 ]](x) = [f1 , [f0 , f1 ]](x) = 0, ∀x ∈ R .

(55) (55)

that [·, ·] is Lie-bracket operator. Now, we consider system mentioned in example (1) that [·, ·] is Lie-bracket operator. Now, we consider system mentioned in example (1) 24

24

Figure 2: System trajectory for k=8.

Figure 2: System trajectory for k=8. Figure 2: System trajectory for k=8.

Figure 3: System trajectory for k=6.985

Figure 3: System trajectory for k=6.985

Figure 3: System trajectory for k=6.985 25

25

25

with k = 5, i.e., 

f0 (x) = A0 x, A0 =  

f1 (x) = A1 x, A1 = 

0

1

−2 −1 0

 

1

−(2 + 5) −1

To investigate the stability condition using [27]: 

[f0 , [f0 , f1 ]](x) =  

[f1 , [f0 , f1 ]](x) = 

5

10

15 −5 5

10

65 −5

 



 x 6= 0. 

 x 6= 0.

This system based on the results of previous example is stable (k = 5 < k ∗ ) while stability condition (55) is not satisfied for such this linear and 2-dimension stable switched system. In other words, this can be satisfied only by a few systems and so generality of this sufficient condition is not enough. Therefore, it is clear that this sufficient condition cannot be useful for stability analysis of linear and nonlinear systems. Example 3- In this example, the proposed approach is used for stability analysis of all possible trajectories starting at a given initial condition of a nonlinear switched system and the ability of this method for nonlinear switched systems is illustrated. Consider the following nonlinear switched system from [15]:   f (x), M ode = 0 0 x(t) ˙ =  f (x), M ode = 1

(56)

1

where



f0 (x) = 

− 23 x1

+ x2

(x1 + x2 )sin(x1 ) − 3x2  3 −2x1 − x1  f1 (x) =  x1 − x2 

 

Using the proposed approach, this system is stable. It can be simply found according to figures 4 and 5. Also, in [15] it is shown that this system is stable. 26

Figure 4: System trajectory (Initial condition=[2;-2])

Figure 4: System trajectory (Initial condition=[2;-2]) Figure 4: System trajectory (Initial condition=[2;-2])

Figure 5: Optimal Control Signal Figure 5: Optimal Control Signal

Figure 5: Optimal Control Signal

27 27

27

Example 4- As posed in [17], the following switched system with initial state x(0) = 4- As posed in [17], of theproposed followingapproach switched system state x(0) = [0.8,Example 0.5]T is stable. The results confirmwith the initial mentioned statement T [0.8, 0.5] is 6stable. The results of proposed approach confirm the mentioned statement (see figures and 7): (see figures 6 and 7):

  A0 x(t) + f0 (t, x(t)), M ode = 0 x(t) ˙ = A x(t) + f (t, x(t)), M ode = 0 0 0  A 1 x(t) + f1 (t, x(t)), M ode = 1 x(t) ˙ =  A x(t) + f (t, x(t)), M ode = 1 1 1

(57) (57)

     −2 1   −3 2  A0 = −2 1  , A1 =  −3 2     A0 = 2 −3 1 −2 , A1 = 1 −2 2 −3 2 T f0 (t, x) = (0.5sinx1 −−2t e−2t , 0.25x 2 T2 ) , f0 (t, x) = (0.5sinx1 − e , 0.25x2 ) ,

f1 (t, x) = (0.5x1 x2 , 0.5x2 +−2t e−2t )T . f1 (t, x) = (0.5x1 x2 , 0.5x2 + e )T .

Figure 6: System trajectory (Initial condition=[0.8;0.5]) Figure 6: System trajectory (Initial condition=[0.8;0.5])

6. Conclusions 6. Conclusions A new method for stability of all possible trajectories starting at a given initial conA new method for stability of all possible trajectories starting at a given initial condition of nonlinear switched systems under arbitrary switching is presented. While the dition of nonlinear switched systems under arbitrary switching is presented. While the 28 28

Figure 7: Optimal Control Signal

Figure 7: Optimal Control Signal

method presented in [4] can be used only for stability analysis and acquires global stabil-

method presented in [4] can be used only for stability analysis and acquires global stability property of two-dimension linear switched systems (analytically) and three-dimension

ity linear property of two-dimension linear switched systems (analytically) and three-dimension switched systems (numerically), the proposed approach here can analyze analytilinear systems (numerically), proposed approach here initial can analyze analyticallyswitched the stability property of all possiblethe trajectories starting at a given condition cally stability property all possible trajectories starting at a given initial condition of the n-dimensional nonlinear of switched systems. As mentioned, “Common Lyapunov Function” method only presents sufficient stabilof n-dimensional nonlinear switched systems. ity in this method finding a common Lyapunov is sufficient difficult instabilAs conditions, mentioned,and “Common Lyapunov Function” method onlyfunction presents practice. In addition, “Multiple Lyapunov Functions” and “Dwell-time” methods only

ity conditions, and in this method finding a common Lyapunov function is difficult in introduce sufficient stability conditions including constraints on the switching signal.

practice. In addition, “Multiple Lyapunov Functions” and “Dwell-time” methods only Therefore, regarding the limitation of these conventional stability analysis methods, the

introduce sufficient stability conditions including constraints on the switching signal. proposed approach for stability analysis of all possible trajectories starting at a given

Therefore, regarding the limitation of these conventional stability analysis methods, the initial condition of nonlinear switched systems has several advantages in many real approposed approach analysis of all switching possible trajectories starting at a given plications especiallyfor in stability systems with an arbitrary signal. initial These condition of nonlinear switched of systems has several advantages in many real apadvantages are “Presentation the necessary and sufficient condition for stability ofespecially all possibleintrajectories starting at a given switching initial condition”, plications systems with an arbitrary signal. “Independence of the stability condition the switchingofsignal”, and “Applicability for various nonlinThese advantages arefrom “Presentation the necessary and sufficient condition for staear switched systems”. Also, this approach has a limitation that the initial condition bility of all possible trajectories starting at a given initial condition”, “Independence of of continuous state of switched system, x(0), must be given, and this limitation results

the stability condition from the switching signal”, and “Applicability for various nonlinin obtaining the necessary and sufficient stability condition for all possible trajectories

ear switched systems”. Also, this approach has a limitation that the initial condition 29 must be given, and this limitation results of continuous state of switched system, x(0),

in obtaining the necessary and sufficient stability condition for all possible trajectories 29

starting at a given initial condition instead of local or global stability. In this paper, using conventional concepts of nonlinear optimal control such as “HJB equations”, “Bang-Bang Property” and “Infinite Time Nonlinear Optimal Control”, the critical trajectory is computed for the stability analysis of a class of nonlinear switched systems (regarding Assumptions 3.1 and 3.2 ) with a given initial state condition. It is shown that, all possible trajectories starting at a given initial condition of a nonlinear switched system under arbitrary switching are asymptotically convergent if and only if the critical trajectory is asymptotically convergent to the origin. The simulation results confirm the efficiency of proposed technique. Acknowledgement The authors would like to thank members of Advanced Control Systems Lab. of the University of Tehran for the fruitful discussion on the subject. [1] Liberzon, D.: ‘Switching in systems and control’, Birkhauser, Boston, 2003. [2] Xiang, W., Xiao, J., Iqbal, M.N.: ‘Robust observer design for nonlinear uncertain switched systems under asynchronous switching’, Nonlinear Analysis: Hybrid Systems, 2012, 6, pp. 754-773. [3] Yang, H., Jiang, B., Cocquempot, V.: ‘A survey of results and perspectives on stabilization of switched nonlinear systems with unstable modes’, Nonlinear Analysis: Hybrid Systems, 2014, 13, pp. 45-60. [4] Margalioat, M.: ‘Stability analysis of switched systems using variational principles: An introduction’, Automatica, 2006, 42, pp. 2059-2077. [5] Gomez-Gutierrez, D., Celikovsky, S., Ramirez-Trevino, A., Castillo- Toledo, B.: ‘On the observer design problem for continuoustime switched linear systems with unknown switchings’, Journal of the Franklin Institute, 2015, 352, (4), pp. 1595-1612. [6] Johansson, K.H., Santucci, F.: ‘On hybrid control problems in communication systems’, Proc. 44th IEEE Conf. on Decision and Control and European Control Conf., 2005, 1, pp. 5624-5629.

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[7] Parriaux, J., Millrioux, G.: ‘Designing self-synchronizing switched linear systems: An application to communications’, Nonlinear Analysis: Hybrid Systems, 2013, 7, pp. 68-79. [8] Geyer,T., Papafotiou, G., Morari, M.: ‘Model predictive control in power electronics: A hybrid systems approach’, Proc. 44th IEEE Conf. on Decision and Control and European Control Conf., 2005, 1, pp. 5606-5611. [9] Jin, Y., Fu, J., Zhang, Y., Jing, Y.: ‘Reliable control of a class of switched cascade nonlinear systems with its application to flight control’, Nonlinear Analysis: Hybrid Systems, 2014, 11, pp. 11-21. [10] Cho, Y.C., Cassandras, C.G., Know, W.H.: ‘Optimal control for steel annealing processes as hybrid systems’, Control Engineering Practice, 2004, 12, (10), pp. 13191328. [11] Malloci, I., Lin, Z., Yan, G.: ‘Stability of interconnected impulsive switched systems subject to state dimension variation’, Nonlinear Analysis: Hybrid Systems, 2012, 6, pp. 960-971. [12] Khalil, H.K.: ‘Nonlinear Systems’, Third Edition, Prentice Hall, 2002. [13] Mancilla-Aguilar, J.L.: ‘A condition for the stability of switched nonlinear systems’, IEEE Transactions on Automatic Control, 2000, 45, (11), pp. 2077-2079. [14] Aleksandrov, A.Yu., Chen, Y., Platonov A.V., Zhang, L.: ‘Stability analysis for a class of switched nonlinear systems’, Automatica, 2011, 47, pp. 2286-2291. [15] Chatterjee, D., Liberzon, D.: ‘On s tability of randomly switched nonlinear systems’, IEEE Transactions on Automatic Control, 2007, 52, (12), pp. 2390-2394. [16] Hespanha, J.P.: ‘Uniform stability of switched linear systems: extensions of Lasalle’s invariance principle’, IEEE Transactions on Automatic Control, 2004, 49, (4), pp. 470-482. [17] Dong, J. G.: ‘Stability analysis of switched systems with general nonlinear disturbances’, Mathematical and Computer Modelling, 2013, 58, pp. 1563-1567.

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[18] Zhu, L., Feng, G.: ‘Necessary and sufficient conditions for stability of switched nonlinear systems’, Journal of the Franklin Institute, 2015, 352, (1), pp. 117-137. [19] Li, Z., Xu, Y., Fei, Z., Agarwal, R.: ‘Exponential stability analysis and stabilization of switched delay systems’, Journal of the Franklin Institute, 2015, 352, (11), , pp. 4980-5002. [20] Margaliot, M., Branicky, M. S.: ‘Stability analysis of positive linear switched systems: A variational approach’, IFAC Proceedings Volumes, 2009, 42, (6), , pp. 31-35. [21] Fainshil, L., Margaliot, M.: ‘A maximum principle for the stability analysis of positive bilinear control systems with applications to positive linear switched systems’, SIAM Journal on Control and Optimization, 2012, 50, (4), , pp. 2193-2215. [22] Hochma, G., Margaliot, M.: ‘High-order maximum principles for the stability analysis of positive bilinear control systems’, Optimal Control Applications and Methods, 2015. [23] Sun,Z., Ge, S.S.: ‘Stability Theory of Switched Dynamical Systems’, Communications and Control Engineering, Springer, London, 2011. [24] Kirk, D.: ‘Optimal control theory: An introduction’, Englewood Cliffs: PrenticeHall, 1970. [25] Naidu, D.S.: ‘Optimal control Systems’, CRC Press, 2003. [26] Manousiouthakis, V., Chmielewski, D.J.: ‘On constrained infinite-time nonlinear optimal control’, Chemical Engineering Science, 2002, 57, pp. 105-114. [27] Margalioat, M., Liberzon, D.: ‘Lie-algebraic stability conditions for nonlinear switched systems and differential inclusions’, Systems and Control Letters, 2006, 55, pp. 8-16. [28] Hung, Y., Lu, W.M.: ‘Nonlinear optimal control: Alternatives to Hamilton-Jacobi equation’, Proceeding of the 35th IEEE Conf. on Decision and Control, Kobe, Japan, 1996, pp. 3942-3947.

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