Full-State Linearization and Stabilization of SISO Markovian Jump Nonlinear Systems

Abstract

This paper investigates the linearization and stabilizing control design problems for a class of SISO Markovian jump nonlinear systems. According to the proposed relative degree set definition, the system can be transformed into the canonical form through the appropriate coordinate changes followed with the Markovian switchings; that is, the system can be full-state linearized in every jump mode with respect to the relative degree set {𝑛,…,𝑛}. Then, a stabilizing control is designed through applying the backstepping technique, which guarantees the asymptotic stability of Markovian jump nonlinear systems. A numerical example is presented to illustrate the effectiveness of our results.

1. Introduction

Markovian jump systems have been studied since the pioneering work on quadratic control of linear jump systems in the 1960s. It switches from one mode to another in a random way, and the switching between the modes is governed by a Markov process with discrete and finite state space. Due to the advantage of the probabilistic description through Markov process, Markovian jump systems are well suited to model changes induced by external causes, for example, random faults, unexpected events, and uncontrolled configuration changes; see [1–5]. Noticeable achievements have been made in the past three decades on stability analysis and controller design of linear Markovian jump systems (see [1, 2, 6–11] and the references therein). Reference [7] has investigated the problem of control for discrete time-delay linear systems with Markovian jump parameters. References [10, 11] discussed the sliding mode control problems for Markovian jump linear singular systems. Meanwhile, many authors have considered the results for linear Markovian jump time-delay systems; see [7, 12–15] and the references therein.

On the other hand, many authors have turned to study the stability and stabilization problems of Markovian jump nonlinear systems. Reference [16] especially, has established the framework of stability analysis for Markovian jump nonlinear system. Reference [17] has proposed the Kalman-Yacubovitch-Popov (KYP) Lemma for the passivity of Markovian jump nonlinear system. [18] discussed the 𝐻_∞ control design for Markovian jump nonlinear system. Reference [19] has discussed the stabilization problems for interconnected Markovian jump nonlinear system from the viewpoint of dissipativity theory. Reference [20] has discussed the stabilization problem for a class of Markovian jump nonlinear systems in the special canonical form. All these works extended results for Markovian jump nonlinear system from the traditional nonlinear control theory, such as [21–26].

The purpose of this paper is to fill some gaps on the geometric control theory for Markovian jump nonlinear systems. It is well known that the geometric control theory plays an important role in the stabilization of nonlinear systems; see [23, 25, 26]. However, as to the Markovian jump systems, there is still no related research work on this topic. The difficulty concretes on how to transform the Markovian jump nonlinear system into the canonical form? Motivated by this point, this paper investigates the linearization and stabilizing control design problems for Markovian jump nonlinear systems. More concretely, the main contribution of this paper contains the following aspects.(i)The definition of relative degree set is proposed for Markovian jump nonlinear systems. Because there exist many nonlinear subsystems corresponding to different jump mode, it would be more appropriate to define the relative degrees of the system in terms of set. (ii)The full-state linearization method for Markovian jump nonlinear system is proposed. For every subsystem, the corresponding diffeomorphism can be constructed to transform the system into the triangle form, which implies that all the subsystems can be transformed to the common triangle form, but in different coordinates. (iii)The backstepping technique is applied for the stabilizing control design of the transformed triangle form of Markovian jump nonlinear system.

The rest of this paper is organized as follows. Section 2 discusses the feedback linearization of Markovian jump systems. According to the proposed relative degree set definition, the appropriate coordinate is adopted under which the system can be transformed into the canonical form followed with the Markovian switchings. Section 3 discusses the stabilizing control design through applying the backstepping technique on the canonical form of Markovian jump nonlinear systems. Section 4 presents a numerical example to illustrate the effectiveness of our results. Section 5 concludes this paper.

For convenience, we adopt the following notation: 𝐴^′ is the transpose of the corresponding matrix 𝐴. 𝐴≥0 (𝐴>0) is the positive semi-definite (positive-definite) matrix. 𝑅^𝑛 is 𝑛-dimensional Euclidean space. ‖𝑥‖ is 2-norm of a vector 𝑥∈𝑅^𝑛. 𝒦 is the family of all increasing functions 𝑘, such that 𝑘(0)=0, 𝑘(𝑡)>0 for 𝑡>0. 𝒦_∞ is the family of all functions 𝑘∈𝒦 with 𝑘(∞)=∞.

2. Feedback Linearization

Consider the following controlled Markovian jump nonlinear system 𝑥̇=𝑓𝑥,𝑟_𝑡+𝑔𝑥,𝑟_𝑡𝑢,𝑦=ℎ𝑥,𝑟_𝑡, (1) in a fixed complete probability space (Ω,ℱ,𝑃), where 𝑥(𝑡)∈𝑅^𝑛 is the state vector, 𝑦(𝑡)∈𝑅 is the controlled output, and 𝑢(𝑡)∈𝑅 is the control input. {𝑟_𝑡,𝑡≥0} is the continuous-time Markov process taking values in a finite set 𝜑={1,2,…,𝑁}, which represents the switching between the different modes and its dynamics are described by the following transitions probabilities: 𝑃𝑟_𝑡+ℎ=𝑗∣𝑟_𝑡=𝜆=𝑖^𝑖𝑗ℎ+𝑜(ℎ),𝑖≠𝑗,1+𝜆_𝑖𝑖ℎ+𝑜(ℎ),𝑖=𝑗, (2) where 𝜆_𝑖𝑗 is the transition rate from mode 𝑖 to 𝑗 with 𝜆_𝑖𝑗≥0 when 𝑖≠𝑗 and 𝜆_𝑖𝑖=−∑^𝑁_{𝑗=1,𝑗≠𝑖}𝜆_𝑖𝑗 and 𝑜(ℎ) is such that lim_ℎ→0𝑜(ℎ)/ℎ=0. The Markovian switching parameter process is assumed to be completely measurable at any time instant. For simplicity of notations, take 𝑓_𝑖 for 𝑓(𝑥,𝑖) when 𝑟_𝑡=𝑖∈𝜑 and so forth. 𝑓_𝑖, 𝑔_𝑖, and ℎ_𝑖 are Lipschitz satisfying a linear growth condition, which guarantees that the system (1) has a unique strong solution [1].

Motivated by the nonlinear control theory for the deterministic systems (see [25, 26] and the references therein), we present the following definition related to the relative degree for Markovian jump nonlinear systems.

Definition 1 (relative degree set). The Markovian jump nonlinear system (1) is said to have the relative degree set {𝜌₁,𝜌₂,…,𝜌_𝑁}, if for every 𝑟_𝑡=𝑖∈𝜑, the corresponding system is with the relative degree 𝜌_𝑖, that is, 𝐿_𝑔𝑖𝐿_𝑓^𝑗−1_𝑖ℎ_𝑖(𝑥)𝐿=0,_𝑔𝑖𝐿^𝜌𝑖_𝑓⁻¹_𝑖ℎ_𝑖(𝑥)≠0, (3) where 𝑗=1,2,…,𝜌_𝑖−1.

Obviously, the above definition is a generalization of the relative degree definition in [25, 26]. Here, the relative degree set contains all the relative degree, of all the subsystem for every system mode 𝑖∈𝜑, which is determined by the characteristic of Markovian jump nonlinear systems.

Note that we start off with the case when all the subsystems are SISO, because it gives more insight into computations and helps understanding the concepts which can be generalized to multivariable systems. Moreover, we consider the system (1) with the relative degree set {𝑛,…,𝑛}, which satisfies the following condition according to Definition 1: 𝐿_𝑔𝑖𝐿_𝑓^𝑛−1_𝑖ℎ_𝑖(𝑥)𝐿=0,_𝑔𝑖𝐿_𝑓^𝑛−1_𝑖ℎ_𝑖(𝑥)≠0. (4)

Now, let us consider a set of new coordinates 𝜉^(𝑖)=𝜙_𝑖(𝑥),𝑖∈𝜑, (5) which indeed stands for the following coordinates: 𝜉^(𝑖)=𝜙_𝑖(𝑥)=𝜙_𝑖1(𝑥)𝜙_𝑖2(𝑥)⋮𝜙_𝑖𝑛(𝑥)=ℎ_𝑖(𝑥)𝐿¹_𝑓𝑖ℎ_𝑖(𝑥)⋮𝐿_𝑓^𝑛−1_𝑖ℎ_𝑖(𝑥). (6)

Take the new coordinate variables as follows: 𝑦₁^(𝑖)=ℎ_𝑖(𝑥),𝑦₂^(𝑖)=𝐿¹_𝑓𝑖ℎ_𝑖(𝑥),⋮𝑦_𝑛^(𝑖)=𝐿_𝑓^𝑛−1_𝑖ℎ_𝑖(𝑥). (7)

For every 𝑖∈𝜑, if the above 𝜙_𝑖 and its inverse 𝜙_𝑖⁻¹ are both smooth functions, 𝜙_𝑖 is called a diffeomorphism; see [25, 26]. Under this assumption, we have 𝑦̇₁^(𝑖)=𝑦₂^(𝑖),𝑦̇₂^(𝑖)=𝑦₃^(𝑖),⋮𝑦̇_𝑛^(𝑖)=𝐿^𝑛_𝑓𝑖ℎ_𝑖(𝑥)+𝐿_𝑔𝑖𝐿_𝑓^𝑛−1_𝑖ℎ_𝑖(𝑥)𝑢. (8) Due to the fact that 𝐿_𝑔𝑖𝐿_𝑓^𝑛−1_𝑖ℎ_𝑖(𝑥)≠0, we can set the control law as 𝑢=𝐿_𝑔𝑖𝐿_𝑓^𝑛−1_𝑖ℎ_𝑖(𝑥)⁻¹𝑣−𝐿^𝑛_𝑓𝑖ℎ_𝑖(𝑥), (9) then the system (8) is feedback equivalent to the following linearized form: 𝜉̇^(𝑖)=𝐴𝜉^(𝑖)+𝐵𝑣,𝑦=𝐶𝜉^(𝑖), (10) where 𝐴=010⋯0001⋯0⋮⋱⋮⋮010⋯00_𝑛×𝑛00⋮01,𝐵=_𝑛×1,𝐶=10⋯00_1×𝑛, (11) which can be regarded as the canonical form for Markovian jump nonlinear system.

Through the above discussion, we have transformed the Markovian jump system into the triangle form corresponding to different 𝜉^(𝑖)-coordinate, respectively. Hence, we have established the relationship between the proposed relative degree set definition and the full-state linearization for Markovian jump nonlinear systems as follows.

Theorem 2. If the system (1) has a relative set {𝑛,…,𝑛}, then the system is full-state feedback linearizable.

Remark 3. The above result can be regarded as a generalized work from deterministic nonlinear control theory to Markovian jump nonlinear system. The key point concentrates on taking the appropriate coordinate changes followed with the Markovian switchings. In this way, we can transform the subsystems into the canonical form (10), which have the same triangle structure but within different coordinates.

Following with the update of the Markovian jump mode 𝑟_𝑡=𝑖, we can transform the 𝑥-system into the corresponding 𝜉^(𝑖)-system, which also works for the stability conditions. Considering the system (1), for any nonnegative functions 𝑈(𝑥,𝑖)∈𝐶(𝑅^𝑛)∶𝑅^𝑛→𝑅⁺, with 𝑈(0,𝑖)=0, 𝑖∈𝜑, we have ℒ𝑈(𝑥,𝑖)=𝜕𝑈^′(𝑥,𝑖)𝑓𝜕𝑥^𝑖+^𝑁_𝑗=1𝜆_𝑖𝑗𝑈𝑥,𝑗, (12) where ℒ is the infinitesimal generator of the system; see [1, 17]. Through the coordinate transformation, denoting and 𝑉𝜉^(𝑖),𝑖=𝑉𝜙_𝑖(𝑥)(),𝑉𝜉,𝑖=𝑈𝑥,𝑖_(𝑖),𝑗=𝑉𝜉^(𝑗),𝑗, (13)(14) we have ℒ𝑉𝜉^(𝑖)𝑥,𝑖=ℒ𝑈^(𝑖),𝑖=𝜕𝑈^′(𝑥,𝑖)𝑓𝜕𝑥^𝑖+^𝑁_𝑗=1𝜆_𝑖𝑗𝑈=𝑥,𝑗𝜕𝑈^′(𝑥,𝑖)𝑓𝜕𝑥^𝑖+𝜆_𝑖1𝑈(𝑥,1)+⋯+𝜆_𝑖𝑁𝑈()=𝑥,𝑁𝜕𝑉^′𝜉^(𝑖),𝑖𝜕𝜉^(𝑖)𝐴+𝜆_𝑖1𝑉𝜉⁽¹⁾,1+⋯+𝜆_𝑖𝑁𝑉𝜉^(𝑁)=,𝑁𝜕𝑉^′𝜉^(𝑖),𝑖𝜕𝜉^(𝑖)𝐴+𝜆_𝑖1𝑉𝜉^(𝑖),1+⋯+𝜆_𝑖𝑁𝑉𝜉^(𝑖)=,𝑁𝜕𝑉^′𝜉^(𝑖),𝑖𝜕𝜉^(𝑖)𝐴+^𝑁_𝑗=1𝜆_𝑖𝑗𝑉𝜉^(𝑖),𝑗. (15)

Due to the fact that the stability of the system (1) is equivalent to the system (10), the stability condition for the system (1) in [1] can be transformed with respect to the system (10) as follows.

Lemma 4. The unforced system (10) is globally asymptotically stable in probability (GASP), if there is a set of positive-definite functions 𝑉(𝜉^(𝑖),𝑖)∈𝐶(𝑅^𝑛) with 𝑉(𝜉^(𝑖),𝑖)→∞ as ‖𝜉^(𝑖)‖→∞, 𝑐∈𝒦, 𝑖∈𝜑, such that ℒ𝑉𝜉^(𝑖),𝑖=𝜕𝑉^′𝜉^(𝑖),𝑖𝜕𝜉^(𝑖)𝐴+^𝑁_𝑗=1𝜆_𝑖𝑗𝑉𝜉^(𝑖)𝜉,𝑗≤−𝑐^(𝑖). (16)

Proof. According to Definition 5.34 and Theorem 5.36 of [16], the asymptotical stability of the system can be guaranteed by 𝜕𝑉^′𝜉^(𝑖),𝑖𝜕𝜉^(𝑖)𝐴+^𝑁_𝑗=1𝜆_𝑖𝑗𝑉𝜉^(𝑗),𝑗≤−𝑐𝜉^(𝑖). (17) Then, considering the transformation of (14), the above condition can be represented in the form of (16). The proof of Lemma 4 is completed.
Obviously, the above results also hold for the closed-loop system under the control law. Although with the relative degree set {𝑛,…,𝑛}, the Markovian jump nonlinear system (1) can be feedback equivalent to the triangle form (10), but how to construct the control 𝑣 such that the 𝜉^(𝑖)-system can be stabilized following with the Markovian switchings, which is an interesting problem and will be discussed in the next section.

3. Stabilizing Control Design

In this section, we consider the backstepping control design of the Markovian jump nonlinear system (1). Through the discussion in Section 2, the system (1) can be transformed into the following 𝜉^(𝑖)-system: 𝑦̇₁^(𝑖)=𝑦₂^(𝑖),𝑦̇₂^(𝑖)=𝑦₃^(𝑖),⋮𝑦̇_𝑛^(𝑖)=𝑣. (18)

Now, we continue to take the following coordinate change: 𝑧₁^(𝑖)=𝑦₁^(𝑖),𝑧₂^(𝑖)=𝑦₂^(𝑖)−𝑐₁^(𝑖),⋮𝑧_𝑛^(𝑖)=𝑦_𝑛^(𝑖)−𝑐^(𝑖)_𝑛−1, (19) where the known function 𝑐_𝑘^(𝑖) is smooth and satisfying 𝑐_𝑘^(𝑖)(0)=0, 𝑘∈{1,…,𝑛−1}, 𝑖∈𝜑. Equivalently, we have 𝑦₁^(𝑖)=𝑧₁^(𝑖),𝑦₂^(𝑖)=𝑧₂^(𝑖)+𝑐₁^(𝑖),⋮𝑦_𝑛^(𝑖)=𝑧_𝑛^(𝑖)+𝑐^(𝑖)_𝑛−1. (20)

Furthermore, for every 𝑖∈𝜑, between any two of the following coordinates 𝑧₁^(𝑖)⋮𝑧_𝑛^(𝑖)⟺𝑦₁^(𝑖)⋮𝑦_𝑛^(𝑖)⟺𝑥₁⋮𝑥_𝑛⟺𝑦₁^(𝑗)⋮𝑦_𝑛^(𝑗)⟺𝑧_𝑗^(𝑖)⋮𝑧_𝑛^(𝑗) (21) or 𝑍_𝑛^(𝑖)⟺𝜉^(𝑖)⟺𝑥⟺𝜉^(𝑗)⟺𝑍_𝑛^(𝑗), (22) it can be found that there exists the corresponding inverse transformation, which is also a diffeomorphism. For simplicity of notations, take 𝑍_𝑛^(𝑖)=𝑇_𝑖(𝑥), then we have 𝑍_𝑛^(𝑗)=𝑇_𝑗𝑇_𝑖⁻¹𝑍_𝑛^(𝑖). (23) Denote 𝜉_𝑘^(𝑖)=𝑦₁^(𝑖),…,𝑦_𝑘^(𝑖)′,𝑍_𝑘^(𝑖)=𝑧₁^(𝑖),…,𝑧_𝑘^(𝑖)′,𝐹_𝑘^(𝑖)=𝑦₂^(𝑖),…,𝑦^(𝑖)_𝑘−1^′. (24)

As follows, we discuss the control design for the system (18). Firstly, for every 𝑖∈𝜑, taking the following set of Lyapunov functions 𝑉𝜉^(𝑖)=1,𝑖2^𝑛_𝑘=1𝑦_𝑘^(𝑖)−𝑐^(𝑖)_𝑘−1²=12^𝑛_𝑘=1𝑧_𝑘^(𝑖)2. (25)

Motivated by (16), the infinite small operator for the system (18) is given as follows: ℒ𝑉𝜉^(𝑖),𝑖=𝑦₁^(𝑖)𝑦₂^(𝑖)+^𝑛−1_𝑘=2𝑦_𝑘^(𝑖)−𝑐^(𝑖)_𝑘−1𝑦^(𝑖)_𝑘+1−𝜕𝑐^(𝑖)_𝑘−1𝜕𝜉^(𝑖)_𝑘−1𝜉̇^(𝑖)_𝑘−1+𝑦_𝑛^(𝑖)−𝑐^(𝑖)_𝑛−1𝑣−𝜕𝑐^(𝑖)_𝑛−1𝜕𝜉^(𝑖)_𝑛−1𝜉̇^(𝑖)_𝑛−1+12^𝑁𝑛_𝑗=1𝑘=1𝜆_𝑖𝑗𝑦_𝑘^(𝑗)−𝑐^(𝑖)_𝑘−1²=𝑧₁^(𝑖)𝑧₂^(𝑖)+𝑐₁^(𝑖)+^𝑛−1_𝑘=2𝑧_𝑘^(𝑖)𝑧^(𝑖)_𝑘+1+𝑐_𝑘^(𝑖)−𝜕𝑐^(𝑖)_𝑘−1𝜕𝜉^(𝑖)_𝑘−1𝐹^(𝑖)_𝑘−1+𝑧_𝑛^(𝑖)𝑣−𝜕𝑐^(𝑖)_𝑛−1𝜕𝜉^(𝑖)_𝑛−1𝐹^(𝑖)_𝑛−1+12^𝑁_𝑗=1𝜆_𝑖𝑗^𝑛_𝑘=1𝑧_𝑘^(𝑗)2. (26) Using Young's inequality, we have ^𝑛−1_𝑘=1𝑧_𝑘^(𝑖)𝑧^(𝑖)_𝑘+1≤^𝑛−1_𝑘=112𝛽²_𝑘𝑧_𝑘^(𝑖)2+12𝛽²_𝑘𝑧^(𝑖)_𝑘+1²=^𝑛_𝑘=112𝛽²_𝑘+12𝛽²_𝑘−1𝑧_𝑘^(𝑖)2, (27) where 𝛽_𝑘>0 are designed parameters. Then, (26) reduces to ℒ𝑉𝜉^(𝑖),𝑖≤𝑧₁^(𝑖)12𝛽²₁+12𝛽²₀𝑧₁^(𝑖)+𝑐₁^(𝑖)+^𝑛−1_𝑘=2𝑧_𝑘^(𝑖)12𝛽²_𝑘+12𝛽²_𝑘−1𝑧_𝑘^(𝑖)+𝑐_𝑘^(𝑖)𝐿−𝜕𝑐^(𝑖)_𝑘−1𝜕𝜉^(𝑖)_𝑘−1𝐹^(𝑖)_𝑘−1+𝑧_𝑛^(𝑖)𝑣−𝜕𝑐^(𝑖)_𝑘−1𝜕𝜉^(𝑖)_𝑘−1𝐹^(𝑖)_𝑛−1+^𝑁_𝑗=1𝜆_𝑖𝑗12^𝑛_𝑘=1𝑧_𝑘^(𝑗)2. (28) Taking 𝑐₁^(𝑖)=−𝛼₁𝑧₁^(𝑖)−12𝛽²₁+12𝛽²₀𝑧₁^(𝑖),𝑐_𝑘^(𝑖)=𝛼_𝑘𝑧_𝑘^(𝑖)−12𝛽²_𝑘+12𝛽²_𝑘−1𝑧_𝑘^(𝑖)+𝜕𝑐^(𝑖)_𝑘−1𝜕𝜉^(𝑖)_𝑘−1𝐹^(𝑖)_𝑘−1, (29)(30) where 𝛼_𝑘>0 are also designed parameters. Note that, the final value of the designed control 𝑣 can be regulated through adjusting the parameters 𝛼_𝑘 and 𝛽_𝑘, which can benefit many practical control design requirements. With fixed appropriate 𝛼_𝑘 and 𝛽_𝑘, the transformation of (19) and (23) can be constructed directly.

Considering the last term of (29), through the transformation (22), we have 𝑉𝜉^(𝑖),𝑗=𝑉_𝑛𝑍_𝑛^(𝑖)=1,𝑗2^𝑛_𝑘=1𝑧_𝑘^(𝑗)2_{𝑧𝑛(𝑖)=𝑇𝑗(𝑇𝑖−1(𝑍𝑛(𝑖)))}. (31) Denote 𝑉_𝑛−1𝑍^(𝑖)_𝑛−1,𝑗=𝑉_𝑛𝑍^(𝑖)_𝑛−1,0,𝑗,Δ𝑉_𝑛𝑍_𝑛^(𝑖),𝑗=¹₀𝜕𝑉_𝑛𝑍^(𝑖)_𝑛−1,𝜍^(𝑖),𝑗𝜕𝜍^(𝑖)_{𝜍(𝑖)=𝜏𝑧𝑛(𝑖)}𝑑𝜏, (32) then we have 𝑉_𝑛𝑍_𝑛^(𝑖),𝑗=𝑉_𝑛−1𝑍^(𝑖)_𝑛−1,𝑗+𝑧_𝑛^(𝑖)Δ𝑉_𝑛𝑍_𝑛^(𝑖),𝑗. (33) Moreover, repeating this procedure, we have V_𝑛𝑍_𝑛^(𝑖)=1,𝑗2^𝑛_𝑘=1𝑧_𝑘^(𝑗)2_{𝑍𝑛(𝑗)=𝑇𝑗(𝑇𝑖−1(𝑍𝑛(𝑖)))}=𝑉_𝑛−1𝑍^(𝑖)_𝑛−1,𝑗+𝑧_𝑛^(𝑖)Δ𝑉_𝑛𝑍_𝑛^(𝑖)⋮=𝑉,𝑗₁𝑍₁^(𝑖),𝑗+^𝑛_𝑘=2𝑧_𝑘^(𝑖)Δ𝑉_𝑘𝑍_𝑘^(𝑖),𝑗. (34) Note that 𝑉₁𝑍₁^(𝑖),𝑗=𝑉𝑧₁^(𝑖)=𝑉,0,…,0,𝑗0,…,0,𝑗+𝑧₁^(𝑖)Δ𝑉₁=𝑧₁^(𝑖)Δ𝑉₁. (35)

Finally, we have 𝑉𝜉^(𝑖),𝑗=^𝑛_𝑘=1𝑧_𝑘^(𝑖)Δ𝑉_𝑘𝑍_𝑘^(𝑖),𝑗. (36)

Through the above discussion on the coupled term, we can have ℒ𝑉𝜉^(𝑖),𝑖≤𝑧₁^(𝑖)12𝛽²₁+12𝛽²₀𝑧₁^(𝑖)+𝑐₁^(𝑖)+^𝑛−1_𝑘=2𝑧_𝑘^(𝑖)12𝛽²_𝑘+12𝛽²_𝑘−1𝑧_𝑘^(𝑖)+𝑐_𝑘^(𝑖)−𝜕𝑐^(𝑖)_𝑘−1𝜕𝜉^(𝑖)_𝑘−1𝐹^(𝑖)_𝑘−1+𝑧_𝑛^(𝑖)𝑣−𝜕𝑐^(𝑖)_𝑛−1𝜕𝜉^(𝑖)_𝑛−1𝐹^(𝑖)_𝑛−1+^𝑁_𝑗=1𝜆_𝑖𝑗^𝑛_𝑘=1𝑧_𝑘^(𝑖)Δ𝑉_𝑘𝑍_𝑘^(𝑖),𝑗=𝑧₁^(𝑖)12𝛽²₁+12𝛽²₀𝑧₁^(𝑖)+^𝑁_𝑗=1𝜆_𝑖𝑗Δ𝑉₁𝑍₁^(𝑖),𝑗+𝑐₁^(𝑖)+^𝑛−1_𝑘=1𝑧_𝑘^(𝑖)12𝛽²_𝑘+12𝛽²_𝑘−1𝑧_𝑘^(𝑖)+𝑐_𝑘^(𝑖)−𝜕𝑐_𝑘−1𝜕𝜉^(𝑖)_𝑘−1𝐹^(𝑖)_𝑘−1+^𝑁_𝑗=1𝜆_𝑖𝑗Δ𝑉_𝑘𝑍_𝑘^(𝑖)+𝑧,𝑗_𝑛(𝑖)𝑣−𝜕𝑐^(𝑖)_𝑛−1𝜕𝜉^(𝑖)_𝑛−1𝐹^(𝑖)_𝑛−1+^𝑁_𝑗=1𝜆_𝑖𝑗Δ𝑉_𝑛𝑍_𝑛^(𝑖),𝑗. (37)

Through updating 𝑐₁^∗(𝑖)=−𝛼₁𝑧₁^(𝑖)−12𝛽²₁+12𝛽²₀𝑧₁^(𝑖)−^𝑁_𝑗=1𝜆_𝑖𝑗Δ𝑉₁𝑍₁^(𝑖),𝑐,𝑗_{𝑘∗(𝑖)}=−𝛼_𝑘𝑧_𝑘^(𝑖)−12𝛽²_𝑘+12𝛽²_𝑘−1𝑧_𝑘^(𝑖)+𝜕𝑐^(𝑖)_𝑘−1𝜕𝜉^(𝑖)_𝑘−1𝐹^(𝑖)_𝑘−1−^𝑁_𝑗=1𝜆_𝑖𝑗Δ𝑉_𝑘𝑍_𝑘^(𝑖),𝑗, (38)(39) we can get the control as follows:𝑣^∗=−𝛼_𝑛𝑧_𝑛^(𝑖)+𝜕𝑐^∗(𝑖)_𝑛−1𝜕𝜉^(𝑖)_𝑛−1𝐹^(𝑖)_𝑛−1−^𝑁_𝑗=1𝜆_𝑖𝑗Δ𝑉_𝑛𝑍_𝑛^(𝑖),𝑗. (40) Moreover, by (9), we can design the original control 𝑢 as follows: 𝑢^∗=𝐿_𝑔𝑖𝐿_𝑓^𝑛−1_𝑖ℎ_𝑖(𝑥)⁻¹𝑣^∗−𝐿^𝑛_𝑓𝑖ℎ_𝑖(𝑥). (41)

Combined with the above discussion, we get to the following theorem.

Theorem 5. If the system (1) has a relative set {𝑛,…,𝑛}, then the closed-loop system under the control 𝑢^∗ is GASP.

Proof. Based on Lemma 4 and (37), sufficient condition (16) for the closed-loop system reduces to ℒ𝑉𝜉^(𝑖),𝑖≤−^𝑛_𝑘=1𝛼_𝑘𝑧_𝑘^(𝑖)2≤−𝑐𝑉𝜉^(𝑖),𝑖, (42) where 𝑐=2min{𝛼₁,𝛼₂,…,𝛼_𝑛}. According to Theorem 5.37 in [16], for every 𝑖∈𝜑, the function 𝑐𝑉(𝜉^(𝑖),𝑖)=(𝑐/2)‖𝜉^(𝑖)‖² belongs to 𝒦-function, then (42) indicates the system (1) to be GASP. The proof of this theorem is completed.

Remark 6. Obviously, the difficulty of applying the backstepping technique on Markovian jump nonlinear system is that the coupled term in the Lyapunov inequality (16) is determined by the characteristic of the system. However, this work has provided the above line to settle this problem. Future work could be concentrated on reducing the computing complexity, which is mainly caused by the following two aspects. Firstly, we need to check the relative degree set of the system through Definition 1. Secondly, we need to calculate the backstepping coefficients 𝑐_𝑖 containing the terms Δ𝑉_𝑘(𝑍_𝑘^(𝑖),𝑗) caused by the additional term (16). Both show the complexity of geometric control theory of Markovian jump nonlinear systems.

Remark 7. The above results only consider the case of Markovian jump nonlinear system with the relative degree set {𝑛,…,𝑛}. Indeed, we have discussed the full-state linearization problem for every subsystem corresponding to the jump mode 𝑟_𝑡=𝑖, respectively. However, if the subsystem has arbitrary relative degree set, that is, {𝜌₁,𝜌₂,…,𝜌_𝑁}, the Markovian jump nonlinear system cannot be full-state linearized, and we need to study the zero-output dynamics of the system. Hence, how to solve the input-output linearization and stabilization problems is very interesting and requires further studies.

4. Simulation Results

In this section, we present a numerical example for the nonlinear Markovian jump system (1). Let 𝑟(𝑡) be a right-continuous Markov chain taking values in 𝜑={1,2} with the generator Π=𝜆_𝑖𝑗=−112−2. (43) Consider the following Markovian jump nonlinear system: ̇Mode1∶𝑥=−2𝑥²₁+2𝑥₂𝑥₂+01𝑢,𝑦=𝑥₁,̇𝑥Mode2∶𝑥=²²𝑥₁−𝑥₂+10𝑢,𝑦=𝑥₂. (44)(45) Firstly, based on Definition 1, we discuss the relative set of the above system. 𝐿_𝑔1ℎ₁=𝜕ℎ₁𝑔𝜕𝑥¹=01𝐿10=0,_𝑔1𝐿_𝑓1ℎ₁=𝜕𝜕ℎ₁/𝜕𝑥𝑓₁𝑔𝜕𝑥¹=𝜕−2𝑥²₁+2𝑥₂𝑔𝜕𝑥¹=−4𝑥₁201𝐿=2≠0,_𝑔2ℎ₂=𝜕ℎ₂𝑔𝜕𝑥²=10𝐿01=0,_𝑔2𝐿_𝑓2ℎ₂=𝜕𝜕ℎ₂/𝜕𝑥𝑓₂𝑔𝜕𝑥²=𝜕𝑥₁−𝑥₂𝑔𝜕𝑥²=101−1=1≠0. (46) Obviously, the above conditions satisfy the assumption in Definition 1, that is, the relative degree set is {2,2}, which implies that system (44) and (45) can be, respectively, transformed into the following canonical form: 𝑦̇₁⁽¹⁾=𝑦₂⁽¹⁾,𝑦̇₂⁽¹⁾𝑦̇=𝑣,₁₍₂₎=𝑦₂⁽²⁾,𝑦̇₂⁽²⁾=𝑣. (47) From (29), taking 𝛼₁=1, 𝛼₂=1, 𝛽₀=1, and 𝛽₁=1, we have 𝑐₁=−2𝑧₁⁽¹⁾,𝑐₂=−2𝑧₁⁽²⁾. (48)

Moreover, we can construct the relationship (22) between the following coordinates: 𝑦₁⁽¹⁾=𝑥₁,𝑦₂⁽¹⁾=−2𝑥²₁+2𝑥₂,𝑧₁⁽¹⁾=𝑥₁,𝑧₂⁽¹⁾=−2𝑥²₁+2𝑥₂−2𝑥₁,𝑦₁⁽²⁾=𝑥₂,𝑦₂⁽²⁾=𝑥₁−𝑥₂,𝑧₁⁽²⁾=𝑥₂,𝑧₂⁽²⁾=𝑥₁−3𝑥₂, (49) which implies the following diffeomorphism between 𝑍₂⁽¹⁾ and 𝑍₂⁽²⁾: 𝑧₁⁽¹⁾=3𝑧₁⁽²⁾+𝑧₂⁽²⁾,𝑧₂⁽¹⁾=−18𝑧₁⁽²⁾²−2𝑧₂⁽²⁾²−12𝑧₁⁽²⁾𝑧₂⁽²⁾−4𝑧₁⁽²⁾−2𝑧₂⁽²⁾,𝑧₁⁽²⁾=𝑧₁⁽¹⁾²+𝑧₁⁽¹⁾+12𝑧₂⁽¹⁾,𝑧₂⁽²⁾=−3𝑧₁⁽¹⁾²−2𝑧₁⁽¹⁾−32𝑧₂⁽¹⁾. (50) Now, we discuss the coupled terms 𝑉𝑍_𝑛⁽¹⁾=1,12𝑧₁⁽¹⁾²+12𝑧₂⁽¹⁾²,𝑉𝑍_𝑛⁽¹⁾=1,22𝑧₁⁽¹⁾²+𝑧₁⁽¹⁾+12𝑧₂⁽¹⁾²+12−3𝑧₁⁽¹⁾²−2𝑧₁⁽¹⁾−32𝑧₂⁽¹⁾²,𝑉𝑍_𝑛⁽²⁾=1,123𝑧₁⁽²⁾+𝑧₂⁽²⁾²+12−18𝑧₁⁽²⁾²−2𝑧₂⁽²⁾²−12𝑧₁⁽²⁾𝑧₂⁽²⁾−4𝑧₁⁽²⁾−2𝑧₂⁽²⁾²,𝑉𝑍_𝑛⁽²⁾=1,22𝑧₁⁽²⁾²+12𝑧₂⁽²⁾². (51) From (34), we have that Δ𝑉₁𝑍₁⁽¹⁾=1,12𝑧₁⁽¹⁾,Δ𝑉₂𝑍₂⁽¹⁾=1,12𝑧₂⁽¹⁾,Δ𝑉₁𝑍₁⁽¹⁾,2=5𝑧₁⁽¹⁾³+7𝑧₁⁽¹⁾²+52𝑧₁⁽¹⁾,Δ𝑉₂𝑍₂⁽¹⁾,2=5𝑧₁⁽¹⁾²+2𝑧₁⁽¹⁾+54𝑧₂⁽²⁾,Δ𝑉₁𝑍₁⁽²⁾,1=162𝑧₁⁽²⁾³+72𝑧₁⁽²⁾²+252𝑧₁⁽²⁾,Δ𝑉₂𝑍₂⁽²⁾,1=216𝑧₁⁽²⁾³+2𝑧₂⁽²⁾³+108𝑧₁⁽²⁾²𝑧₂⁽²⁾+24𝑧₁⁽²⁾𝑧₂⁽²⁾²+84𝑧₁⁽²⁾²+4𝑧₂⁽²⁾²+32𝑧₁⁽²⁾𝑧₂⁽²⁾+11𝑧₁⁽²⁾+52𝑧₂⁽²⁾,Δ𝑉₁𝑍₁⁽²⁾=1,22𝑧₁⁽²⁾,Δ𝑉₂𝑍₂⁽²⁾=1,22𝑧₂⁽²⁾. (52) Then, from (40), we can construct the control law 𝑣^∗ as follows: 𝑣₁=−𝑧₁⁽¹⁾+𝜕𝑐₁⁽¹⁾𝜕𝑧₁⁽¹⁾𝐹₁⁽¹⁾−𝜆₁₁Δ𝑉₂𝑍_𝑛⁽¹⁾,1−𝜆₁₂Δ𝑉₂𝑍_𝑛⁽¹⁾𝑧,2=−5₁⁽¹⁾²+𝑧₁⁽¹⁾−14𝑧₂⁽¹⁾,𝑣₂=−𝑧₁⁽²⁾+𝜕𝑐₁⁽²⁾𝜕𝑧₁⁽²⁾𝐹₁⁽²⁾−𝜆₂₁Δ𝑉₂𝑍_𝑛⁽²⁾,1−𝜆₂₂Δ𝑉₂𝑍_𝑛⁽²⁾𝑧,2=−432₁⁽²⁾³−4𝑧₂⁽²⁾³−216𝑧₁⁽²⁾²𝑧₂⁽²⁾−48𝑧₁⁽²⁾𝑧₂⁽²⁾²−168𝑧₁⁽²⁾²−8𝑧₂⁽²⁾²−64𝑧₁⁽²⁾𝑧₂⁽²⁾−19𝑧₁⁽²⁾−6𝑧₂⁽²⁾. (53) Moreover, by (9), we can design the original control 𝑢 as follows: 𝑢^∗₁=−4𝑥³₁−94𝑥²₁+4𝑥₁𝑥₂+34𝑥₁−54𝑥₂,𝑢^∗₂=−4𝑥³₁−108𝑥³₂−12𝑥²₁𝑥₂−36𝑥₁𝑥²₂−8𝑥²₁−49𝑥²₂−16𝑥₁𝑥₂−5𝑥₁−2𝑥₂, (54) which guarantee the asymptotically stability of the closed-loop system.

Given initial values 𝑥₁(0)=2 and 𝑥₂(0)=0, the simulation results of the above discussion are presented as follows. Figure 1 shows the states of Markovian switchings. Correspondingly, the trajectories of 𝑦₁^(𝑖) and 𝑦₂^(𝑖) are shown in Figures 2–3, respectively. Obviously, the control law 𝑣^∗ asymptotically stabilizes the closed-loop system.

Figure 1 

State of Markov process 𝑟𝑡.

Figure 2 

Trajectories of state 𝑦1(𝑖).

Figure 3 

Trajectories of state 𝑦2(𝑖).

5. Conclusion

This paper has discussed the linearization and stabilization problems for a class of general Markovian jump nonlinear systems. Following with the Markovian switchings, the appropriate coordinate is adopted under which the system can be transformed into the canonical form corresponding to the relative degree set. Then, the backstepping technique is applied in the stabilizing control design of Markovian jump nonlinear systems. Further efforts could be concentrated on the study of the linearization and stabilization of Markovian jump nonlinear systems with arbitrary relative degree set {𝜌₁,𝜌₂,…,𝜌_𝑛}.

Acknowledgment

This work is partially supported by the National Basic Research Program of China (973 Program) (Grant no. 2012CB215203), the National Natural Science Foundation of China (no. 61203043 and no. 51036002) and the Fundamental Research Funds for the Central Universities.