Ultimately Exponentially Bounded Estimates for a Class of Nonlinear Discrete−Time Stochastic Systems

Abstract

In this paper, the ultimately exponentially bounded estimate problem of nonlinear stochastic discrete−time systems under generalized Lipschitz conditions is considered. A new sufficient condition making the estimation error system uniformly exponentially bounded in the mean square sense is given. The gain matrix can be obtained by solving matrix inequality. In the last section, numerical examples are provided verify the effectiveness of the conclusions.

1. Introduction

Stochastic factors are very common in engineering practices, and the classical deterministic equations cannot describe the random objects properly. In fact, the error between real systems and deterministic models is inevitable; therefore, how to control systems of parameter disturbance and random disturbance to maintain good performance has become a valuable topic. As early as the beginning of the 20th century, random factors have been widely studied by scholars. Gibbs studied the Hamilton Jacobi differential system in 1902, set the initial state of random in the analysis, and then had the rudiment of stochastic differential equation. Later, when Langevin studied Brown motion in 1908, he found that the random collisions between molecules and particles in the medium would affect the state change of the system, and the famous Langevin equation was obtained from this change trend. In 1951, Itô first put forward the concept of stochastic differential equation and gave the Itô formula of stochastic integral. Since then, the stochastic system has attracted more and more scholars’ attention, and many excellent results have been achieved [1,2,3,4,5,6].

In recent years, for stochastic systems satisfying some special nonlinearities, the state estimation problem has become very important due to the fact that nonlinearities are inherent characteristics of many dynamic systems. The discrete Razumikhin−type theorems on the pth moment exponential stability were established in [7], and the stability criteria are given by utilizing the pth moment stability. Ref. [8] studied the stability analysis problem of stochastic nonlinear discrete−time systems. Some difficulties have been solved in using Lyapunov’s method and fixed−point theory. The asymptotic stability in probability was investigated for nonlinear stochastic discrete systems in [9], where asymptotic stability criteria based on linear matrix inequalities are presented for a class of quasi−linear discrete−time stochastic systems. In [10], the authors studied the security tracking control problem with quadratic cost under network attacks for discrete stochastic systems. By transforming the security tracking problem into the state stability problem in the sense of probability of the closed−loop system, the controller parameters were given in terms of matrix inequalities. Exponential stability and ultimate boundedness for stochastic functional differential equations were studied in [11], and the same conditions for pth moment exponential stability were given when the original system was unbounded and unstable. A sufficient condition for the existence of the H−infinity filtering of nonlinear discrete stochastic systems was obtained by applying a Hamilton–Jacobi inequality in [12]. The first− and second−order conditions under weak assumptions for stochastic systems in discrete case were investigated in [13]. In addition, there are many important achievements on correlated problems related to exponential ultimate boundedness and stability, such as [14,15]. Inspired by the above studies, we developed a new ultimately exponentially bounded criterion in terms of matrix inequality by applying generalized Lipschitz conditions for the studied system.

2. Model Description

Consider discrete stochastic system:

$$\left\{\begin{array}{l}x(i+1)=Ax(i)+\phi (x(i),u(i),\theta )+D\omega (i)\\ h(i+1)=Cx(i)\end{array}\right.$$

(1)

here,

$$\phi (x(i),u(i),\theta )={\phi }_1(x(i),u(i))+B{\phi }_2(x(i),u(i))\theta$$

(2)

where $x\in {ℝ}^n$ is the state vector; $u\in {ℝ}^p$ is the input vector; $h(i)\in {ℝ}^r$ is a measurable output vector; $\theta$ is a constant parameter; $A,B,C$ are the known constant matrices; $\omega (i)$ is an independent normal distribution random variable on a complete probability space; and $\phi (x(i),u(i))\in {ℝ}^n$ is a real valued nonlinear vector function related to state $x(i)$ and input $u(i)$.

In this paper, it is assumed that the following generalized Lipschitz condition for the nonlinear function $\phi (x(i),u(i),\theta )$ is satisfied:

$${\tilde{\phi }}^{\prime }Q\tilde{\phi }\le {(x-\widehat{x})}^{\prime }R(x-\widehat{x})$$

(3)

where $\tilde{\phi }=\phi (x(i),u(i),\theta )-\phi (\widehat{x}(i),u(i),\theta ),$ and $Q,R$ are two symmetric positive definite matrices.

The main purpose of this paper is to design the adaptive observer of the system (1) under the following assumptions, i.e.,

$$\left(\mathrm{I}\right)\Vert \theta \Vert \le {\gamma }_{\theta },\Vert {\phi }_2{\Vert }_F\le {\gamma }_{\phi }$$

$$\left(\mathrm{II}\right)B^{T}P=TC$$

where, ${\gamma }_{\theta },{\gamma }_{\phi }$ are real constants, $P$ is a positive definite matrix, and $T$ is a real matrix. Before giving the main conclusions, we first introduce the following lemmas.

Lemma 1

([16]).Consider discrete stochastic process $x(k)$. Suppose there is a stochastic function $V_k(x(k))$ and constants $0<k_1\le 1,k_2,k_3>0$, so that

$$k_3\Vert x(k){\Vert }^2\le V_{k}x(k)$$

and

$$\mathbb{E}[V_{k+1}(x(k+1))|x(k)-V_k(x(k))]\le -k_1{V}_k(x(k))+k_2\hspace{1em}\hspace{1em}\mathrm{a}.\mathrm{s}.$$

is true; then, the process $x(k)$ is ultimately exponentially bounded in the mean square. In addition,

$$\mathbb{E}[k_3\Vert x(k){\Vert }^2]\le {(1-k_1)}^k+k_2/k_1$$

Lemma 2 

([17]).(Schur Complement) The following conditions are equivalent:

(1) 

$X<0;$

(2) 

$X_{11}<0,X_{22}-X_{12}^T{X}_{11}^{-1}{X}_{12}<0;$

(3) 

$X_{22}<0,X_{11}-X_{12}^{}{X}_{22}^{-1}{X}_{12}^T<0,$

where $X=\left[\begin{array}{cc}{X}_{11}& X_{12}\\ X_{12}^T& X_{22}\end{array}\right]$ is the symmetric matrix.

3. Main Results

In the following, an adaptive state observer is designed to estimate parameter $\theta$. The observer of nonlinear stochastic system is

$$\widehat{x}(i+1)=A\widehat{x}(i)+\phi (\widehat{x}(i),u(i),\widehat{\theta })+L(y(i)-C\widehat{x}(i))$$

(4)

$$\widehat{\theta }(i+1)=\mathrm{\Gamma }\widehat{\theta }(i)+\mathrm{\Gamma }{\phi }_2^T(\widehat{x}(i),u(i))T(y(i)-C\widehat{x}(i))$$

(5)

where $L$ and $T$ are undetermined matrices with appropriate dimensions.

According to the Formulae (1), (4) and (5), Error vector $\tilde{x}(i)=x(i)-\widehat{x}(i)$ and $\tilde{\theta }(i)=\theta -\widehat{\theta }(i)$ can be expressed as

$$\tilde{x}(i+1)=(A-LC)\tilde{x}(i)+\tilde{\phi }+B{f}_2(\widehat{x}(i),u(i))\tilde{\theta }(i)+Dx(i)\omega (i)$$

(6)

$$\tilde{\theta }(i+1)=\theta -\mathrm{\Gamma }\widehat{\theta }(i)-\mathrm{\Gamma }{\phi }_2^T(\widehat{x}(i),u(i))TC\tilde{x}(i)$$

(7)

Theorem 1.

Consider stochastic systems (1), the adaptive observer satisfies formulae (4) and (5). If there are positive definite matrices $P>0,Q>0,R>0,$ constants ${\sigma }_1,{\sigma }_2$ and gain matrix $L$ making the following matrix inequalities solvable:

$$\left[\begin{array}{ccccc}2R-P& A^{T}P-C^T{Y}^T& A^{T}P-C^T{Y}^T& A^{T}P-C^T{Y}^T& C^T{Y}^T\\ \ast & -P& 0& 0& 0\\ \ast & \ast & -Q& 0& 0\\ \ast & \ast & \ast & -I& 0\\ \ast & \ast & \ast & \ast & I-2P\end{array}\right]<{\sigma }_{1}I$$

(8)

$$\left[\begin{array}{cccc}{\gamma }_{\phi }^2{B}^{T}B-\mu I+I& {\gamma }_{\phi }{B}^{T}P& {\gamma }_{\phi }{B}^T& \\ \ast & -Q+P& 0& \\ \ast & \ast & -P& \end{array}\right]<{\sigma }_{2}I$$

(9)

where

$$\mu I=\mathrm{\Gamma }-{\mathrm{\Gamma }}^{-1}<0$$

Then, the error estimation systems (6) and (7) are ultimately exponentially bounded in the mean square sense.

Proof of Theorem 1.

Set $\xi (i)=\left[\begin{array}{c}\tilde{x}(i)\\ \tilde{\theta }(i)\end{array}\right]$, considering Lyapunov function

$$V_i={\xi }^T(i)\overline{P}\xi (i)$$

where $\overline{P}=\left[\begin{array}{cc}P& 0\\ 0& {\mathrm{\Gamma }}^{-1}\end{array}\right],$ then, the difference operator

$$\begin{array}{l}\mathrm{\Delta }V=\mathbb{E}[V_{i+1}|\xi (i)-V_i]\\ \hspace{1em} =\mathbb{E}\{-V_i+[(A-LC)\tilde{x}(i)+\tilde{\phi }\\ \hspace{1em}\hspace{1em}+B{\phi }_2(\widehat{x}(i),u(i))\tilde{\theta }(i)+Dx(i)\omega (i){]}^T\\ \hspace{1em}\hspace{1em}\hspace{1em}P[(A-LC)\tilde{x}(i)+\tilde{\phi }+B{\phi }_2(\widehat{x}(i),u(i))\tilde{\theta }(i)+Dx(i)\omega (i)]\\ \hspace{1em}\hspace{1em}+{[\theta -\tilde{\theta }(i)-\mathrm{\Gamma }{\phi }_2^T(\widehat{x}(i),u(i))TC\tilde{x}(i)]}^T\\ \hspace{1em}\hspace{1em}\hspace{1em}{\mathrm{\Gamma }}^{-1}[\theta -\tilde{\theta }(i)-\mathrm{\Gamma }{\phi }_2^T(\widehat{x}(i),u(i))TC\tilde{x}(i)]\}\end{array}$$

can be obtained.

Noticing conditions (I) and (II), we can obtain

$$\begin{array}{l}\mathrm{\Delta }V={\tilde{x}}^T{(i)(A-LC)}^{T}P(A-LC)\tilde{x}(i)\\ \hspace{1em} +2{\tilde{x}}^T(i)P{Q}^{-1/2}{Q}^{1/2}\tilde{\phi }\\ \hspace{1em} +2{\tilde{x}}^T(i)(A-{LC)}^{T}PB{f}_2(\widehat{x}(i),u(i))\tilde{\theta }\\ \hspace{1em} +{\tilde{\phi }}^{T}P\tilde{\phi }+2{\tilde{\phi }}^T{(Q-P)}^{1/2}{(Q-P)}^{-1/2}\\ \hspace{1em} PB{\phi }_2(\widehat{x}(i),u(i))\tilde{\theta }(i)\\ \hspace{1em} +{\tilde{\theta }}^T(i){\phi }_2^T(\widehat{x}(i),u(i))B^{T}PB{\phi }_2(\widehat{x}(i),u(i))\tilde{\theta }(i)\\ \hspace{1em} +{\tilde{\theta }}^T(i)\mathrm{\Gamma }\tilde{\theta }(i)-2{\tilde{\theta }}^T(i)\mathrm{\Gamma }{\phi }_2^T(\widehat{x}(i),u(i))TC\tilde{x}(i)\\ \hspace{1em} +{\tilde{x}}^T(i)C^T{T}^T{\phi }_2(\widehat{x}(i),u(i))\mathrm{\Gamma }{\phi }_2^T(\widehat{x}(i),u(i))TC\tilde{x}(i)\\ \hspace{1em} -{\tilde{x}}^T(i)P\tilde{x}(i)-{\tilde{\theta }}^T(i){\mathrm{\Gamma }}^{-1}\tilde{\theta }(i)\\ ={\xi }^T(i)\left[\begin{array}{cc}(1,1)& 0\\ \ast & (2,2)\end{array}\right]\xi (i)+a_1\end{array}$$

where

$$\begin{array}{l}(1,1)={(A-LC)}^{T}P(A-LC)+2R+{(A-LC)}^{T}P{Q}^{-1}P(A-LC)\\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}+{(A-LC)}^{T}PP(A-LC)+C^T{L}^{T}LC-P\\ (2,2)={\gamma }_{\phi }^2{B}^{T}B+{\gamma }_{\phi }^2{B}^{T}P{(Q-P)}^{-1}PB+{\gamma }_{\phi }^2{B}^{T}PB+\mathrm{\Gamma }-{\mathrm{\Gamma }}^{-1}+I\\ a_1=\mathrm{tr}(D^{T}PD)\end{array}$$

Note that if the conditions

$$\begin{array}{l}{(A-LC)}^{T}P(A-LC)+2R+{(A-LC)}^{T}P{Q}^{-1}P(A-LC)\\ +{(A-LC)}^{T}PP(A-LC)+C^T{L}^{T}LC-P<-{\sigma }_{1}I\\ {\sigma }_1>0\end{array}$$

$$\begin{array}{l}{\gamma }_{\phi }^2{B}^{T}B+{\gamma }_{\phi }^2{B}^{T}P{(Q-P)}^{-1}PB+{\gamma }_{\phi }^2{B}^{T}PB+\mu I+I<-{\sigma }_{2}I\\ {\sigma }_2>0\end{array}$$

$$\mu I=\mathrm{\Gamma }-{\mathrm{\Gamma }}^{-1}<0$$

are satisfied, then

$$\mathbb{E}[V_{i+1}|\xi (i)-V_i]<-\max \{{\sigma }_1,{\sigma }_2\}\Vert \xi (i){\Vert }^2+a_1$$

$$\le -\max \{{\sigma }_1,{\sigma }_2\}/{\lambda }_{\max }(\overline{P})V_i+a_1$$

(10)

In this way, using Lemma 1, we can directly deduce that the error dynamic systems (4) and (5) are exponentially bounded in the mean square sense.

In addition, setting $L=P^{-1}Y$, according to the Schur lemma and inequality, it is easy to see that inequalities (10) are equivalent to linear matrix inequalities, respectively, (8) and (9).

Thus, using YALMIP toolbox in Matlab and solving matrix inequalities (8) and (9), the undetermined matrices $L$ and $T$ in adaptive observers (4) and (5) can be obtained. □

4. Numerical Examples

Numerical examples are given below in order to illustrate the feasibility of the conclusions.

Example 1.

Consider nonlinear stochastic system (1), and the system parameters are as follows:

$$\begin{gathered} A=\left[\begin{array}{cc}-2.7& 1\\ 0.5& -10\end{array}\right],B=\left[\begin{array}{c}2.1\\ 1.5\end{array}\right] \\ D=\left[\begin{array}{cc}0.1& 0\\ 0& 0.1\end{array}\right],C=\left[\begin{array}{cc}1& 0.5\end{array}\right] \\ \phi (x(i),u(i),\theta )=0.01\cos (x_2(i))\theta ,\theta =0.5 \end{gathered}$$

Then, there are${\gamma }_{\theta }=0.5,{\gamma }_{\phi }=0.01$. If the parameter $\mathrm{\Gamma }$ is selected as $1/(1+\sqrt{2})$, then using YALMIP toolbox in Matlab to solve LMIs (11), (12) and the matching condition (II), we can obtain

$$\begin{gathered} P=\left[\begin{array}{cc}158.4679& 67.4292\\ 67.4292& 50.2413\end{array}\right] \\ Q=\left[\begin{array}{cc}1.9985& 1.1203\\ 1.1203& 1.9823\end{array}\right]\times {10}^3 \\ \hspace{1em}\hspace{1em}\hspace{1em}R=\left[\begin{array}{cc}1.0791\times {10}^3& -5.971\times {10}^2\\ -5.971\times {10}^2& 1.0407\times {10}^3\end{array}\right] \\ \hspace{1em}\hspace{1em}Y=\left[\begin{array}{c}-87.4140\\ -36.9471\end{array}\right], T=433.9264 \\ \hspace{1em}\hspace{1em}\hspace{1em} {\sigma }_1=3.1355\times {10}^3,{\sigma }_2=1.991\times {10}^3 \\ \hspace{1em}\hspace{1em}L=P^{-1}Y=\left[\begin{array}{c}-0.5565\times {10}^{-1}\\ 1.15\times {10}^{-2}\end{array}\right] \end{gathered}$$

Thus, matrices $L,T$ are solved easily using the YALMIP toolbox, and then the error estimation systems are exponentially bounded in the mean square sense. Error vector and parameter response trajectories are provided in Figure 1, which shows that error system is ultimately exponentially bounded, and the estimation effect is also relatively good.

Example 2.

Considering a single−link flexible−joint robot system [18], the system parameters are taken as

$$\overline{A}=\left[\begin{array}{cccc}0& 1& 0& 0\\ -48.6& -1.26& 48.6& 0\\ 0& 0& 0& 10\\ 1.95& 0& -1.95& 0\end{array}\right],B=\left[\begin{array}{c}1\\ \begin{array}{l}0\\ 0\\ 0\end{array}\end{array}\right]$$

$C$is a four−dimensional identity matrix. Using the Euler discretization method with the sampling time 0.01, the system for (1) is given, where $A=I+0.01\overline{A},\phi (x(i),u(i),\theta )=0.01{[\begin{array}{cccc}10\sin x_4\theta & 0& 0& 0\end{array}]}^T.$ In addition, we set  $\theta =0.5,\mathrm{\Gamma }=0.5,{\gamma }_{\phi }=0.3,{\gamma }_{\theta }=0.5,$ so that by solving matrix inequalities (8), (9) and the matching condition (II), we can obtain

$$\begin{array}{l}P=\left[\begin{array}{cccc}6.667\times {10}^{-1}& 3\times {10}^{-4}& -1\times {10}^{-4}& 0\\ 3\times {10}^{-4}& 6.752\times {10}^{-1}& -4.1\times {10}^{-3}& 2\times {10}^{-4}\\ -1\times {10}^{-4}& -4.1\times {10}^{-3}& 6.687\times {10}^{-1}& -1\times {10}^{-4}\\ 0& 2\times {10}^{-4}& -1\times {10}^{-4}& 6.667\times {10}^{-1}\end{array}\right]\\ Q=\left[\begin{array}{cccc}4.3664& 2.6698& 2.6698& 2.6698\\ 2.6698& 4.5265& 2.7041& 2.7041\\ 2.6698& 2.7041& 4.5265& 2.7041\\ 2.6698& 2.7041& 2.7041& 4.5265\end{array}\right]\times {10}^6\\ R=\left[\begin{array}{cccc} 3.0647& -0.8319& -0.8319& -0.8228\\ -0.8319& 3.0647& -0.8319& -0.8228\\ -0.8319& -0.8319& 3.067& -0.8228\\ -0.8228& -0.8228& -0.8228& 3.0895\end{array}\right]\times {10}^6\\ Y=\left[\begin{array}{cccc}3.333& 2.208& 2.382& 2.390\\ 2.208& 3.333& 2.696& 2.381\\ 2.382& 2.696& 3.333& 2.423\\ 2.390& 2.381& 2.423& 3.333\end{array}\right]\times {10}^{-1}\\ T=\left[\begin{array}{cccc}6.667\times {10}^{-1}& 3\times {10}^{-4}& -1\times {10}^{-4}& 0\end{array}\right]\hfill \\ {\sigma }_1=7.7036\times {10}^6,{\sigma }_2=3.6102\times {10}^6\\ L=\left[\begin{array}{cccc}4.999& 3.310& 3.572& 3.584\\ 3.288& 4.959& 4.021& 3.546\\ 3.585& 4.064& 5.011& 3.648\\ 3.585& 3.570& 3.635& 4.999\end{array}\right]\times {10}^{-1}\end{array}$$

Figure 2 gives the trajectory of the state $\tilde{x}(i)$ of error system (6) and (7). It shows that the error system is ultimately exponentially bounded. In addition, estimation error of state is fluctuating near 0, which means that the observer (4), (5) has a good estimation effect on the system (1).

5. Conclusions

This paper studied the ultimately exponentially bounded estimate problem for nonlinear discrete systems based on generalized Lipschitz conditions. A new condition making the error system uniformly exponentially bounded in the mean square sense was established. The obtained condition was based on some matrix inequalities, so that the gain matrix can be easily solved by YALMIP toolbox in Matlab. Two numerical examples were given to verify the effectiveness of our method.