A Note on Stabilization of Discrete Nonlinear Systems

Abstract

We deal with the stabilization problem of discrete nonlinear systems. We construct a control Lyapunov function on discrete nonlinear systems. Then, we present a new method to construct a continuous state feedback law.

1. Introduction

For nonlinear dynamical systems, Lyapunov function-based methods play a vital role in stability analysis since the Lyapunov functions have been used to design stabilizing feedback laws that render asymptotically controllable systems ISS to actuator errors and small observation noise [1]. This is an important and challenging problem for the general class of nonlinear systems. Artstein [2] proves that, for smooth control-affine systems evolving on 𝑅^𝑛 with controls in 𝑅^𝑚, there exists a globally stabilizing feedback control law continuously on 𝑅^𝑛⧵{0} if the system has a smooth control Lyapunov function. Sontag [3] provided an explicit proof of Artstein's Theorem which uses a “universal formula” for the stabilizing controller. Shahmansoorian et al. [4] presented a new stabilizing control law with respect to a control Lyapunov function. Many researchers dealt with the effects of control Lyapunov functions and obtained considerably meaningful results, just as in [5–12].

The stabilization of discrete systems is studied by means of Lyapunov functions in this paper. Xiushan et al. [13] also studied discrete system, but the control Lyapunov function and the control law they give are not easy to get. We will present an explicit control Lyapunov function and feedback law stabilizing a discrete-time control system. Furthermore, a new way to construct a continuous state feedback law is designed.

2. Main Results

The problem amounts to find a function 𝑢∶𝑅^𝑛→𝑅^𝑚 with 𝑢(0)=0, such that the closed-loop system𝑥_𝑘+1=𝑓𝑥_𝑘,𝑢𝑥_𝑘 (2.1) is asymptotically stable at the origin. Of course we assume that 𝑓(0,0)=0, but the origin is not asymptotically stable equilibrium for the unforced system 𝑥_𝑘+1=𝑓(𝑥_𝑘,0).

Equation (2.1) describes the behavior of the system in the sense that its evolution depends not only on an initial state, but also on a sequence of input signal. In this paper, we address the following discrete-time nonlinear control systems of the form:𝑥_𝑘+1=𝑓𝑥_𝑘+𝑢_𝑘𝑔𝑥_𝑘, (2.2) which is said to be affine-in-control, where 𝑓 and 𝑔 are continuous on 𝑅^𝑛 and 𝑓(0)=0, {𝑢_𝑘}∈𝑈⊂𝑅 (a sequence of input signal). The system (2.2) is globally asymptotically stable at the origin if there exists a map 𝑥→𝑢(𝑥), such that the system𝑥_𝑘+1=𝑓𝑥_𝑘+𝑢𝑥_𝑘𝑔𝑥_𝑘 (2.3) is globally asymptotically stable at 0∈𝑅^𝑛.

Definition 2.1. A smooth, proper, and positive definite function 𝑉 mapping 𝑅^𝑛 into 𝑅 is said to be a control Lyapunov for the discrete system (2.2) if and only if inf_{𝑢𝑘⊂𝑈}Δ𝑉𝑥_𝑘=𝑉𝑥_𝑘+1−𝑉𝑥_𝑘<0, (2.4) for all 𝑥∈𝑅^𝑛⧵{0}.

Remark 2.2. Suppose that 𝑉(𝑥)=𝑥^𝑇𝑃𝑥, then Δ𝑉𝑥_𝑘=𝑓𝑥_𝑘+𝑢_𝑘𝑔𝑥_𝑘^𝑇𝑃𝑓𝑥_𝑘+𝑢_𝑘𝑔𝑥_𝑘−𝑥^𝑇_𝑘𝑃𝑥_𝑘 (2.5) is a function about 𝑥_𝑘 and 𝑢_𝑘.

Theorem 2.3. Suppose that 𝑓=^𝑓1_𝑓2, 𝑔=^𝑔1_𝑔2, and let Ω₀=𝑥∈𝑅²⧵{0}(𝑥),Ω∣𝑔=0₁=𝑥∈𝑅²⧵{0}∣𝑔(𝑥)≠0. (2.6) Assume that there exist constants 𝐴>0,𝐵>0, such that sup_𝑥∈𝑅2(‖𝑓‖/‖𝑥‖)=𝐴<1, and sup_𝑥∈𝑅2𝑓₁𝑔₂−𝑓₂𝑔₁𝑔²₁+𝑔²₂𝑥²₁+𝑥²₂=𝐵<+∞, (2.7) then there exists a positive definite matrix 𝑃, such that 𝑉(𝑥)=𝑥^𝑇𝑃𝑥 is a control Lyapunov function of system (2.2).

Proof. For (𝑥,𝑢)∈𝑅²×𝑅, let ℎ_𝑥(𝑢)=𝑢²𝑔^𝑇𝑃𝑔+2𝑢𝑓^𝑇𝑃𝑔+𝑓^𝑇𝑃𝑓−𝑥^𝑇𝑃𝑥. (2.8) Obviously, Δ𝑉=𝑉(𝑥_𝑘+1)−𝑉(𝑥_𝑘)=ℎ_𝑥𝑘(𝑢_𝑘).
(i) If 𝑥_𝑘∈Ω₀, Δ𝑉 does not depend on 𝑢_𝑘, then we just need 𝑓^𝑇𝑃𝑓−𝑥^𝑇𝑃𝑥<0. (2.9) Suppose 𝜆_max,𝜆_min>0 are the eigenvalues of 𝑃, then 𝑓^𝑇𝑃𝑓−𝑥^𝑇𝑃𝑥≤𝜆_max𝑓²−𝜆_min‖𝑥‖²≤𝐴𝜆_max−𝜆_min‖𝑥‖²<0. (2.10) So, it is easy for us to give a suitable positive definite matrix 𝑃 such that which satisfies the relationship.
(ii) If 𝑥_𝑘∈Ω₁, 𝑔^𝑇(𝑥_𝑘)𝑃𝑔(𝑥_𝑘)>0, for the same reason, there exists 𝑢_𝑘∈𝑅 such that ℎ_𝑥𝑘(𝑢_𝑘)<0. Let 𝑢_𝑘𝑓=−^𝑇𝑥_𝑘𝑃𝑔𝑥_𝑘𝑔^𝑇𝑥_𝑘𝑃𝑔𝑥_𝑘. (2.11) An easy computation shows that Δ𝑉<0 is equal to 𝑓^𝑇𝑃𝑔𝑓^𝑇−𝑓𝑔^𝑇𝑃𝑔+𝑔^𝑇𝑃𝑔𝑥^𝑇𝑃𝑥>0, (2.12) and let 𝑃=^𝑝1_𝑝3^𝑝3_𝑝2, through a simple computation, we can get 𝑔^𝑇𝑃𝑔𝑥^𝑇𝑃𝑥>𝜆⁴_min𝑔²‖𝑥‖²>𝑓₁𝑔₂−𝑓₂𝑔₁²𝑝₁𝑝₂−𝑝²₃=𝑓^𝑇𝑃𝑓𝑔^𝑇−𝑔𝑓^𝑇𝑃𝑔. (2.13) Thus, we obtain 𝜆⁴_min𝜆_min⋅𝜆_max>𝑓₁𝑔₂−𝑓₂𝑔₁²𝑔²‖𝑥‖². (2.14) So, if sup_𝑥∈𝑅2𝑓₁𝑔₂−𝑓₂𝑔₁𝑔²₁+𝑔²₂𝑥²₁+𝑥²₂=𝐵<+∞, (2.15) we can choose suitable positive definite matrix 𝑃.
Then, we can choose a suitable positive definite matrix 𝑃 satisfies (i) and (ii), such that 𝑉(𝑥)=𝑥^𝑇𝑃𝑥 is a control Lyapunov function of system (2.2). The proof is end.

Corollary 2.4. If 𝑉(𝑥) is a control Lyapunov function for the discrete-time control system (2.2), then the control −𝑓𝑢=^𝑇𝑃𝑔𝑔^𝑇𝑃𝑔𝑥∈Ω₁0𝑥∈Ω₀ (2.16) globally asymptotically stabilizes the equilibrium 𝑥=0 of the system (2.2).

Proposition 2.5. Suppose that 𝑔(𝑥)≠0 for any 0≠𝑥∈𝑅². 𝑉(𝑥) is a control Lyapunov function for the discrete-time system (2.1), then the control Law (2.16) is smooth and globally asymptotically stabilizes the equilibrium 𝑥=0 of the system (2.1).

Theorem 2.6. If there exists a quadratic control Lyapunov function for the discrete-time system (2.2), there exists a continuous stabilizing feedback law 𝑢∶𝑅^𝑛→𝑅, and 𝑢(0)=0.

Proposition 2.7. Consider 𝐻_𝑥(𝑢)=𝑔²(𝑥)𝑢²+2𝑓(𝑥)𝑔(𝑥)𝑢+𝑓²(𝑥)−𝑥²=0, (2.17) where 𝑥∈𝑅,𝑓,𝑔∶𝑅→𝑅,𝑓²(𝑥)−𝑥²≤0.
Assume that 𝑢_1,2=−𝑓(𝑥)±𝑥sgn𝑥𝑔(𝑥)𝑔(𝑥) (2.18) are the roots of (2.17). If 𝑢_1,2 only have one discontinuity of the second kind 𝑥=𝑥₀, then one can construct one continuous function 𝑢(𝑥), such that 𝑢(0)=0 and 𝐻_𝑥(𝑢(𝑥))<0.

Proof. Since 𝑥₀ is the only second category discontinuous point of the roots of 𝑢₁,𝑢₂, so for any 𝜀>0, there must exist 𝛿>0, such that (1/|𝑢₁|),(1/|𝑢₂|)<𝜀 for any 𝑥∈(𝑥₀−𝛿,𝑥₀+𝛿) thus, we can let 𝑢^⋆₁=𝑢₁𝑥∉𝑥₀−𝛿,𝑥₀,𝑢+𝛿₁𝑥₀+𝑢+𝛿¹𝑥₀+𝛿−𝑢₁𝑥₀−𝛿𝑥−𝑥₀+𝛿𝑥2𝛿𝑥∈₀−𝛿,𝑥₀+𝛿. (2.19) Define 𝑢^⋆₂ in the same way and let 𝑢(𝑥)=𝑢^⋆₁+𝑢^⋆₂2, (2.20) then it is easy for us to verify that 𝑢(𝑥) is what we want to get.

Proposition 2.8. Consider 𝐻_𝑥(𝑢)=𝑔^𝑇𝑃𝑔𝑢²+2𝑓^𝑇𝑃𝑔𝑢+𝑓^𝑇𝑃𝑓−𝑥^𝑇𝑃𝑥=0, (2.21) where 𝑥∈𝑅^𝑛, 𝑓,𝑔∶𝑅^𝑛→𝑅^𝑛, 𝑓^𝑇𝑃𝑓−𝑥^𝑇𝑃𝑥≤0.
If 𝑥₀ is the only second category discontinuous point of the roots 𝑢_1,2=−𝑓^𝑇𝑃𝑔±𝑓^𝑇𝑃𝑔𝑓^𝑇𝑃𝑔−𝑔^𝑇𝑃𝑔𝑓^𝑇𝑃𝑓+𝑔^𝑇𝑃𝑔𝑥^𝑇𝑃𝑥𝑔^𝑇𝑃𝑔 (2.22) of (2.21), then one can construct one continuous function 𝑢(𝑥), such that 𝑢(0)=0 and 𝐻_𝑥(𝑢(𝑥))<0.

Proof. For all 𝜀>0, there must exist 𝛿>0, such that (1/|𝑢₁|)<𝜀 for any 𝑥∈𝐵_𝛿(𝑥₀). Let 𝑚=inf_{𝑥∈𝐵𝛿(𝑥0)}(1/𝑢₁(𝑥)), 𝑀=sup_{𝑥∈𝐵𝛿(𝑥0)}(1/𝑢₁(𝑥)). Obviously, there exists 𝑥₁,𝑥₂∈𝐵_𝛿(𝑥₀) satisfying 𝑢₁(𝑥₁)=(1/𝑚), 𝑢₁(𝑥₂)=(1/𝑀), then we can let 𝑢^⋆₁=𝑢₁𝑥∉𝐵_𝛿𝑥₀,1𝑀+𝑥−𝑥₂(1/𝑀)−(1/𝑚)𝑥₂−𝑥₁𝑥∈𝐵_𝛿𝑥₀. (2.23) Using the same way, we can get 𝑢^⋆₂, then we can get what we want to get.
By Proposition 2.8, it is easy for us to prove Theorem 2.6.

3. Example

Consider a discrete-time nonlinear system𝑥_𝑘+1=𝑎𝑥²_𝑘+𝑥_𝑘+𝑏𝑢_𝑘, (3.1) where 0<𝑎<1, 𝑏>0. Let 𝑉(𝑥)=𝑥², then it is easy for us to obtain𝐻_𝑥(𝑢)=(𝑥+𝑏)²𝑢²+2𝑎𝑥²(𝑥+𝑏)𝑢+𝑎²𝑥⁴−𝑥²,𝑢_1,2=−𝑎𝑥²±|𝑥|sgn(𝑥+𝑏).𝑥+𝑏 (3.2) By Theorem 2.6, we can get that a continuous control law𝑢(𝑥)=−3−3𝑎𝑏−5𝑎𝑥𝑥∈2𝑏𝑏,−2−𝑎𝑥²−3𝑥+1𝑥∉2𝑏𝑏,−2 (3.3) globally asymptotically stabilizes the equilibrium 𝑥=0 of the system (2.21).