Backstepping-based state estimation for a class of stochastic nonlinear systems

1. INTRODUCTION

Since state space has been widely used to present the model of the dynamic system, state estimation is a key research problem to characterize the system properties as the internal states are mostly unmeasurable. It is a challenging technical problem because the unmeasurable states are normally subjected to random noises. For example, process noise and measurement noise widely exist in the state space model. Therefore, it is also a filtering problem for state estimation if the random noises have been considered in the design procedure. Nowadays, state estimation and filtering have been adopted in many applications such as the robotics system^[1], intelligent manufacturing^[2], transportation monitoring^[3], industrial performance optimization^{[4, 5]}, etc.

As a well-developed solution for estimation problem, Kalman^[6] filter was firstly designed in the 1960s, and it has been considered as a standard design method. However, the traditional Kalman filter was given for a discrete-time linear dynamic system. To deal with the nonlinearities of the system dynamics, many variants have been proposed successfully based on Kalman filter framework, such as extended Kalman filter^[7] and unscented Kalam filter^[8]. However, the problem formulation for these filters still use discrete-time models. To deal with the continuous-time model, the Kalman-Bucy filter^[9] was proposed by solving linear Riccati equation. Notice that the Kalman-Bucy filter has a linear structure. It is difficult to extend the result to nonlinear systems directly, while it is difficult to solve the associated Riccati equation analytically for the nonlinear system model. Following the Kalman filter framework, robust Kalman filter^[10] and unscented Kalman filtering^[11] have been extended to continuous-time models. Different from the Kalman filter framework, the numerical solution has been developed as particle filter^[12], which considers the distribution of the particle for each sampling instant. The main issue of particle filter is the convergence analysis. It is still an open question currently. The minimum entropy filtering has also been presented^{[13, 14]}, which considers non-Gaussian noises in the system model. Inspired by the probability density function (PDF) in particle filtering and the minimum entropy filtering, the Fokker-Planck-Kolmogorov (FPK) equation is used in this paper to produce the continuous-time solution.

FPK-based state estimation^[15] has been presented recently. However, this paper focuses on the filtering design based on linearization. Trying to eliminate the nonlinear effects in the closed-loop, in this paper, the backstepping design is adopted for the estimation error dynamics as the system model is represented by stochastic differential equation. In particular, the structure of the filter can be confirmed using the system model. Thus, the dynamics of the estimation error can be produced following the stochastic differential equation. To stabilize the estimation error, the backstepping design is adopted. As a result, the stochastic differential equation in terms of the estimation error can be further converted to a linear Ornstein-Uhlenbeck process^[16], while the virtual tracking error in backstepping is close to zero. In the ideal case, the estimation error is subjected to the Brownian motion. Note that the system states are Gaussian due to the fact that the system dynamics are converted to being linear, while the stochastic differential equation is subjected to Brownian motion. Then, we can use variance to characterize the randomness of the estimation error. Following the variance assignment method^[17], the desired value of variance results in the optimal parametric selection for the presented filter. In addition, the presented framework can be used for stochastic distribution control^[18] where the tracking error can be optimized similarly to the estimation error in this paper. As one possible application, neural membrane potential estimation can be taken into account, as the nonlinear dynamics^[19] are affected by complex random neural interaction^[20] and the membrane potential is difficult to measure directly. Then, the proposed estimator is considered as a possible solution.

The remainder of the paper is organized as follows. The formulation is given in Section 2 including the system model and some preliminaries. The filter structure design is given in Section 3 with backstepping procedure. The optimal parameter is obtained in Section 4 using the variance assignment. The simulation results are shown in Section 5 to indicate the performance of the presented filtering algorithm. A discussion for multivariate system is given in Section 6, and the conclusions are summarized in Section 7.

2. FORMULATION

To illustrate the main idea of the estimation method, let us start from a simple univariate stochastic nonlinear system which is described by the Itô process:

where $$ f\left(\cdot\right) $$ stands for a known smooth nonlinear function, $$ W_t $$ denotes the Wiener process, and $$ \sigma > 0 $$ is given as a real constant. $$ x_t $$ and $$ y_t $$ stand for the system state and system output, respectively. $$ c $$ denotes a real constant. Note that the measurement equation is presented in linear form, which is widely used in practice to describe the property of the sensor.

To deliver the main result, the preliminaries, such as definitions^[21] and lemma^[22], are recalled here.

Definition 1For any given $$ V\left(x_t, t\right)\in\mathscr C^{1, 2} $$$$ \left(\mathbb R^n\times\mathbb R_+;\mathbb R_+\right) $$ associated with the stochastic differential Equation (1), the differential operator $$ \mathscr L $$ is defined as follows:

Definition 2The solution process $$ \left\{x\left(t\right), t\ge0\right\} $$ of the stochastic system in Equation (1) is said to be bounded in probability if $$ \lim_{c\to\infty\sup_{0\le t<\infty}}P\left\{\|x\left(t\right)\|>c\right\} = 0 $$, where $$ P\left\{\cdot\right\} $$ denotes probability operator and $$ c $$ is a real positive constant.

Lemma 1Consider the stochastic nonlinear system model in Equation (1) and assume that $$ f\left(x_t\right) $$ is $$ \mathscr C^1 $$ in the arguments and $$ f\left(0\right) $$ is bounded uniformly in $$ t $$. If there exist nonnegative functions $$ V\left(x_t, t\right)\in\mathscr C^{1, 2} $$$$ \left(\mathbb R^n\times\mathbb R_+;\mathbb R_+\right) $$, $$ \mu_1\left(\cdot\right) $$, $$ \mu_2\left(\cdot\right) $$$$ \in \mathscr K_\infty $$, constants $$ c_1>0 $$ and $$ c_2\ge0 $$, such that

where $$ c_2 = 0 $$ implies that the system is stable probability sense and $$ c_2>0 $$ means the system is second-moment stable.

3. FILTER STRUCTURE

Based on the system model in Equation (1), the state estimation scheme can be produced by the following filtering structure.

where $$ \hat x_t $$ denotes the estimated system state $$ x_t $$. $$ v_t $$ stands for the compensative signal for estimating correction.

Since the estimation error is defined as $$ e_t = x_t-\hat x_t $$, the dynamics of $$ e_t $$ can be reflected by the following equations.

where $$ \bar f_t = f\left(x_t\right)-f\left(\hat x_t\right) $$.

It is shown above that the estimation problem can be further converted as the estimation error stabilization using the signal $$ v_t $$. Next, we can introduce an integrator into the system motivated by the backstepping design.

where $$ u_t $$ denotes the new filtering compensative signal. Note that the signal $$ v_t $$ in Equation (5) is the integral of the signal $$ u_t $$.

Following the backstepping design approach, the virtual signal is given first:

where $$ \theta $$ denotes a real positive constant as a design parameter.

To stabilize Equation (6), the virtual error variable is further defined as follows:

Substituting the virtual error $$ z_t $$ into Equation (6), we have

The Itô's lemma can be used here to obtain the following equation:

Then, a Lyapunov function candidate is selected as follows:

which leads to

Based on Definition 1, Lemma 1, and Young's inequality, the following result is obtained:

and

Thus, the compensative signal $$ u_t $$ is further designed as

where $$ G $$ denotes a design parametric function.

Substituting the designed signal $$ u_t $$ to $$ \mathscr L V_z $$, Equation (12) can be rewritten as follows:

Using Lemma 1, the estimation error $$ e_t $$ with the designed signal $$ u_t $$ is bounded in probability sense. To simplify the expression of $$ \mathscr LV_t $$, $$ G $$ can be selected as follows, and then the $$ z^4 $$ term can be further eliminated.

which results in

Thus, it is shown that the estimation error is bounded in probability sense when $$ \theta\ge\frac{1}{2} $$.

Note that, as the linear dynamical measurement equation is known, the estimation error can be approximated by the measurement error signal.

which indicates that the presented compensative signal is implementable.

4. PARAMETER SELECTION

Substituting the compensative signal into $$ \mathscr LV_z $$ shows that the virtual error $$ z_t $$ is also bounded in probability sense. Therefore, the estimation error with the designed signal can be represented by the linear Ornstein-Uhlenbeck process.

For the obtained Ornstein-Uhlenbeck process, the Fokker-Planck-Kolmogorov equation can be obtained as follows:

where $$ p(x, t) $$ and $$ x $$ stand for the probability density function and the random variable of $$ e_t $$, respectively.

Solving the presented FPK equation in analytical form, we have

where $$ x_0 $$ denotes the initial value of $$ e_t $$ at $$ t_0 $$.

Following the aforementioned discussion in introduction, the designed signal governs the PDF of estimation error $$ e_t $$ following the Gaussian distribution. Moreover, the mean value converges to zero and the variance value can be calculated as follows:

where the variable is governed by the design parameter $$ \theta $$.

To achieve the filtering performance, the design parameter $$ \theta $$ should be selected properly. Note that, if we have the ideal case for the system state estimation, $$ \bar f_t = 0 $$ holds. Thus, the estimation error dynamics can be further described as follows:

where the associated FPK equation is obtained as the following heat equation.

Based on Green's function and heat kernel, the optimal variance $$ r\left(t\right) $$ can be obtained.

Then, the parameter $$ \theta $$ can be formulated by analytically solving the following equation.

which can be rewritten as follows.

Based on Lambert W function, we can further solve this equation and its solution is given as follows:

where $$ W_0\left(\cdot\right) $$ stands for Lambert W function.

Note that the real solution of equation above does not always exist, and we can consider the stationary solution to simplify the calculation.

We can further select $$ \theta = \frac{1}{2} $$ when $$ t>1 $$ in order to satisfy the estimation error stabilization condition.

Remark 1Note that variance is equivalent to entropy for linear stochastic system, thus the proposed filtering algorithm can be considered as a special case for minimum entropy filtering.

To summarize the design procedure, the following pseudo-code is demonstrated here as Algorithm 1.

5. SIMULATION

As a validation, we consider the following numerical system model as an example:

To achieve the objective of filtering design, the system state $$ x_t $$ subjected to noise should be approximated using the measured signal $$ y_t $$. Using the proposed algorithm, the designed filter can be obtained as follows:

Using the designed nonlinear filter, the estimation error $$ e_t $$ can be produced in a nonlinear form. Then, the nonlinear dynamics of this estimation error and the virtual signal $$ z_t $$ can be further formulated. Based on the presented backstepping design, the simulation results are developed in Figures 1-5, where the initial value of the compensative signal is 0 and the initial value of the system state is -0.1. Figure 1 demonstrates the estimation performance of the presented method where the measured signal and the estimated one are shown separately as a comparison. It is shown that the randomness in the measured signal is attenuated, while the estimated signal is close to the ideal signal without noise. A comparative study is also given in this figure using a high-gain observer. It is shown that the high-gain design achieves the state estimation with filtering effects. However, the performance is sensitive regarding to the gain value. Figure 2 indicates the estimation error signal. As in the aforementioned analysis, the estimation error would be described as a Brownian motion in ideal condition. However, in the computational simulation, the discretization has to be used where the trajectory is the increment of the Brownian motion. In particular, it is described as Gaussian white noise. Note that error $$ e_true $$ is given in the figure where the value is given between the estimated value and the true value without noise. $$ e_{HG} $$ and $$ e_t $$ are the values between the measured value and the estimated values. $$ e_{HG} $$ implies that the high-gain design is closer to the measured value as an observer and the presented filtering algorithm leads to a result close to the true value without noise. Figures 3-5 illustrate the designed filtering compensative signal $$ v_t $$, $$ u_t $$, and the tracking error of virtual signal $$ z_t $$. In addition, Figures 6 and 7 show the mean values of $$ z_t $$ and $$ e_t $$. Notice that the mean values of both signals are bounded and close to zero. In addition, Figures 8 and 9 show the variance values of $$ z_t $$ and $$ e_t $$, where the variance value of $$ e_t $$ is close to the assigned value 0.0001. Basically, in the assigned value there exists an error due to the influence of the filtering compensative signal. Based on these results, the design estimation objective for the investigated system is achieved.

Figure 1. The estimation of the system state $$ x_t $$.

Figure 2. The estimation error of the system state $$ x_t $$.

Figure 3. The compensative signal $$ v_t $$ of filter with integrator.

Figure 4. The compensative signal u_t of filter before integral.

Figure 5. The trajectory of the virtual error $$ z_t $$.

Figure 6. The mean value of the virtual error signal $$ z_t $$.

Figure 7. The mean value of the estimation error $$ e_t $$.

Figure 8. The variance of the virtual error signal $$ z_t $$.

Figure 9. The variance of the estimation error $$ e_t $$.

6. DISCUSSION

Only the single variable estimation is considered above to simplify the backstepping design. The main challenge of extending the presented algorithm to the multivariate case is basically the multi-variable backstepping design. Inspired by multi-variable controller design^[23], the block backstepping design is one of the potential solutions. The system model in Equation (1) can be extended to vector-valued form as follows:

where $$ \bar f\left(\cdot\right) $$ stands for a known smooth nonlinear function $$ \bar f: \mathbb R^{n\times1}\to\mathbb R^n $$, $$ \bar W_t $$ denotes n-dimensional Wiener process, and $$ \Sigma $$ denotes a given as a real positive square matrix with n dimensions. $$ \bar x_t\in \mathbb R^n $$ and $$ y_t\in \mathbb R^1 $$ stand for the system state vector and system output, respectively. $$ C\in \mathbb R^n $$ denotes a vector-valued coefficient.

Then, the filter structure can be confirmed in vectorized form.

where $$ \hat {\bar x}_t $$ denotes the estimated system state $$ \bar x_t $$. $$ \bar v_t\in\mathbb R^n $$ denotes the vector-based filtering compensative signal.

Similar to the design procedure, the Lyapunov functions can also be reused where the vector-value variables will be used. Since Lemma 2 holds for the multivariate system, the developed result in this paper can be extended directly following the block backstepping design. Notice that the linear Ornstein-Uhlenbeck process will be in the multi-dimensional form which leads to difficulty in solving the FPK equation as the joint probability density function has to be involved in the multivariate case. To avoid this problem, the design parameter $$ \theta $$ should be selected as the positive diagonal matrix. Then, a set of FPK equations can be obtained where the vector state can be decomposed as multiple single variables. Therefore, the presented parameter selection scheme can also be reused for the multivariate system.

In addition, the measurement noise cannot be ignored in practice. An extended model should be formulated with the measurement noise which can be expressed as another stochastic differential equation. Thus, the convergence of the estimation error cannot be simply converted as the proposed linear dynamics. This extension will be considered as a future work with other assumptions.

7. CONCLUSION

The state estimation problem is investigated for a class of stochastic nonlinear systems, where the system model is described by the stochastic differential equation. To achieve the design objective, a new nonlinear filtering approach is designed. In particular, the design scheme is divided into two components: (1) The filtering structure can be confirmed based on the system model while the nonlinear estimation error can be further formulated. Then, an integrator is introduced into the estimation error for matching the backstepping design procedure. After that, the nonlinear dynamics can be converted to a linear Ornstein-Uhlenbeck process, where the mean value and variance value of the obtained estimation error is adjustable. (2) Since the variance value can be formulated analytically, the parameter can be designed for the filter. Ideally, the Brownian motion can be considered to obtain the desired variance value. Then, the design parameter in backstepping can be confirmed in order to attenuate the randomness of the state estimation. The simulation results and the multivariate system extension are also given to show the effectiveness and the potential extension of the presented estimation method. To extend the presented results, some comparative studies will be produced as future work where high-order sliding mode design^{[24, 25]} and adaptive high-gain design^[26] will be further considered.

DECLARATIONS