State Estimation on Switching Systems via High-Order Sliding Modes

Abstract

In this chapter, the problem of continuous and discrete state estimation for switched nonlinear systems is solved using High-Order Sliding-Mode techniques. In the first part of this chapter, the systems with exogenous switchings are studied. The solvability of the observation problem, for the continuous and discrete states, is proposed using structural properties of the system. This structural properties are exploited to design the observers. The high-order sliding-mode techniques are introduced to guarantee finite time convergence to zero of the estimation error for the continuous state. The discrete state is reconstructed using the information of the equivalent output injection. The continuous state observation, discrete state and unknown input reconstruction for switched systems with autonomous switchings are realized in the second part of the chapter. In this part, the structural properties of the systems are exploited to ensure the observability of the continuous and discrete states. Finite-time convergence to zero of the estimation error, for continuous and discrete states, are achieved. The information of the equivalent output injection is used for unknown input reconstruction. Simulation results are provided to illustrate the methods exposed in the chapter.

Appendix

6.1.1 Proof of Theorem 6.1

System (6.14), under Assumption 6.7, can be represented, on new coordinates $z$, as:

$$\begin{aligned} \dot{z}&=Az+B\varphi _{\lambda ^{*}}(z),\nonumber \\ y_{z}&=Cz, \end{aligned}$$

(6.58)

where

$$\begin{aligned} \begin{array}{ccc} A=\left[ \begin{array}{cccc}0 &{} 1 &{} \cdots &{} 0 \\ \vdots &{} \vdots &{} \ddots &{} \vdots \\ 0 &{} 0 &{} \cdots &{} 1 \\ 0 &{} 0 &{} \cdots &{} 0\end{array}\right] _{n\times n},&{} B=\left[ \begin{array}{c} 0 \\ \vdots \\ 0 \\ 1\end{array}\right] _{n\times 1},&C=\left[ \begin{array}{cccc}1&0&\cdots&0\end{array}\right] _{1\times n}, \end{array} \end{aligned}$$

(6.59)

$$\begin{aligned} \varphi _{\lambda ^{*}}(z)=\left( L_{f_{\lambda ^{*}}(x)}^{n}h(x)\right) \vert _{x=\varPhi _{\lambda ^{*}}^{-1}(z)}. \end{aligned}$$

(6.60)

Notice that the functions $\varphi _{i^{*}}(z)$ contain the unknown input that acts on the system. On the other hand, the observers (6.8), under Assumption 6.7, can be represented as:

$$\begin{aligned} \dot{\hat{z}}_{j}&=A\hat{z}_{j}+B{\varphi }_{j}(\hat{z}_{j})+\nu _{j},\nonumber \\ y_{\hat{z}_{j}}&=C\hat{z}_{j}, \end{aligned}$$

(6.61)

where ${\varphi }_{j}(\hat{z}_{j})=\left( L_{{f}(\hat{x}_{j})}^{n}h(\hat{x}_{j})\right) \vert _{x_{j}=\varPhi _{j}^{-1}(\hat{z}_{j})}$.

Define the state estimation errors as

$$\begin{aligned} e_{{z}_{j}}=\hat{z}_{j}-z. \end{aligned}$$

(6.62)

Thus, the state estimation error dynamics takes the following form:

$$\begin{aligned} {\dot{e}}_{{z}_{j}}=Ae_{{z}_{j}}+B\varPsi _{j}(\hat{z}_{j},z)+\nu _{j}, \end{aligned}$$

(6.63)

where $\varPsi _{j}(\hat{z}_{j},z)={\varphi }_{j}(\hat{z}_{j})-\varphi _{\lambda ^{*}}(z)$. If it is possible to find appropriate correction terms $\nu _{j}$, which can steer the vector $e_{{z}_{j}}$ to zero, then equality $\hat{z}_{j}=z$, will be satisfied only for the case when $j=\lambda ^{*}$. Therefore, only one of the observers can be associated with the corresponding state observation error in the time interval $t\in [0,t_{1})$.

On the other hand, it is not desirable to design the correction terms in the coordinates $\hat{z}_{j}$ but in the coordinates $\hat{x}_{j}$. Therefore, defining the following output error vector:

$$\begin{aligned} \mathbf {\varepsilon }_{j}=\left[ \begin{array}{c}\varepsilon _{{j},1} \\ \vdots \\ \varepsilon _{{j},n}\end{array}\right] =\left[ \begin{array}{c}e_{{y}_{j}} \\ \vdots \\ e_{{y}_{j}}^{(n-1)}\end{array}\right] . \end{aligned}$$

(6.64)

The state observation error dynamics (6.63) turns into output observation error dynamics as follows:

$$\begin{aligned} \dot{\varepsilon }_{j}=A{\varepsilon _{j}}+B\varPsi _{j}(\varPhi _{j}(\hat{x} _{j}),\varPhi _{\lambda ^*}(x))+\nu _{j}. \end{aligned}$$

(6.65)

In an extended structure

$$\begin{aligned} \begin{array}{l} \dot{\varepsilon }_{j,1}=\varepsilon _{j,2}+\nu _{j,1}, \\ \dot{\varepsilon }_{j,2}=\varepsilon _{j,3}+\nu _{j,2}, \\ \ \ \ \ \ \ \vdots \\ \dot{\varepsilon }_{j,n}=\varPsi _{j}(\varPhi _{j}(\hat{x}_{j}),\varPhi _{\lambda ^{*}}(x))+\nu _{j,n}. \end{array} \end{aligned}$$

(6.66)

Notice that the dynamic structures (6.66) are very similar to the high-order sliding-mode differentiator (6.12). Thus, if a variable change is realized in the structure (6.13), it is possible to obtain the following sliding-mode differentiator form:

$$\begin{aligned} \dot{\varepsilon }_{j,1}=&\varepsilon _{j,2}-\alpha _{j,1}M_{j}^{\frac{1}{n}}\left| \varepsilon _{j,1} \right| ^{\frac{n-1}{n}}\text {sign}\left( \varepsilon _{j,1}\right) , \nonumber \\ \dot{\varepsilon }_{j,2}=&\varepsilon _{j,3}-\alpha _{j,2}M_{j}^{\frac{1}{n-1}}|\varepsilon _{j,2} -\dot{\varepsilon }_{j,1}|^{\frac{n-2}{n-1}}\text {sign}\left( \varepsilon _{j,2} -\dot{\varepsilon }_{j,1}\right) , \nonumber \\ \vdots&\nonumber \\ \dot{\varepsilon }_{j,n}=&\varPsi _{j}(\varPhi _{j}(\hat{x}_{j}), \varPhi _{\lambda ^*}(x))-\alpha _{j,n}M_{j}\text {sign}\left( \varepsilon _{j,n}-\dot{\varepsilon }_{j,n-1}\right) . \end{aligned}$$

(6.67)

Let the parameters $\alpha _{j,k}$ be chosen recursively and sufficiently large according to the high-order sliding-mode differentiator properties described in [22]. In view of the Assumption 6.10, the following equality is satisfied in finite time when $j=\lambda ^{*}$:

$$\begin{aligned} \left[ \varepsilon _{\lambda ^*,1},\varepsilon _{\lambda ^*,2},\ldots ,\varepsilon _{\lambda ^*,n}\right] \equiv \left[ 0,0,\ldots ,0\right] . \end{aligned}$$

(6.68)

The condition $\varepsilon _{\lambda ^*,1}\equiv 0$, $\forall t\in [t_{\lambda ^*},t_{1})$ implies that $\left[ \varepsilon _{\lambda ^*,2},\ldots ,\varepsilon _{\lambda ^*,n}\right] \equiv \left[ 0,\ldots ,0\right]$, $\forall t \in [t_{\lambda ^*},t_{1})$ with $t_{1}\ge T_{\delta }$. To prove that, assume that the condition $\varepsilon _{\lambda ^*,1}\equiv 0$ is satisfied in a nonzero time interval. This condition implies that $\dot{\varepsilon }_{\lambda ^*,1}\equiv 0$ in the same time interval. Thus, from the first row of (6.67) it is obtained that $\varepsilon _{\lambda ^*,2}\equiv 0$. Then, since $\varepsilon _{\lambda ^*,2}\equiv 0$ and $\dot{\varepsilon }_{\lambda ^*,1}\equiv 0$ from the second row of (6.67) it is obtained that $\varepsilon _{\lambda ^*,3}\equiv 0$. If the same procedure is iterated the following expressions are obtained

$$\begin{aligned} \varepsilon _{\lambda ^*,k}\equiv 0,\quad \forall k=1,\ldots ,n. \end{aligned}$$

(6.69)

Given Assumption 6.10, the last row of (6.67) defines the following differential inclusion

$$\begin{aligned} \dot{\varepsilon }_{\lambda ^*,n}\in \left[ -M_{\lambda ^*},M_{\lambda ^*}\right] -\alpha _{\lambda ^*,n} M_{\lambda ^*}\text {sign}\left( \varepsilon _{\lambda ^*,n}-\dot{\varepsilon }_{\lambda ^*,n-1}\right) , \end{aligned}$$

(6.70)

where $\varPsi _{\lambda ^*}(\cdot )\in \left[ -M_{\lambda ^*},M_{\lambda ^*}\right]$. Therefore, according to [22], the dynamics (6.67), for $j=\lambda ^{*}$, converges to zero after a finite time, i.e., $\varepsilon _{\lambda ^*}\equiv 0 \ \forall t>t_{\lambda ^*}$, and according to Assumption 6.7, it is ensure that the state estimation error $e_{x_{\lambda ^*}}=\hat{x}_{\lambda ^*}-x$ also converge to zero in finite time in spite of the unknown input. Notice that it is always possible to select the gain $M_{\lambda ^*}$ sufficiently large such that $t_{\lambda ^*}<t_{1}\ge T_{\delta }$. In this way, the theorem is proven.

6.1.2 Proof of Proposition 6.1

Let us consider the dynamics (6.67) on the time instants before and after the switching time $t_{1}$, i.e.,

$$\begin{aligned} \dot{\varepsilon }_{\varDelta _{\lambda ^*\kappa ,1}}&= \varepsilon _{\varDelta _{\lambda ^*\kappa ,2}}-\alpha _ {\varDelta _{\lambda ^*\kappa ,1}}M_{\varDelta _{\lambda ^*\kappa }}^{\frac{1}{n}}\left| \varepsilon _ {\varDelta _{\lambda ^*\kappa ,1}} \right| ^{\frac{n-1}{n}}\text {sign}\left( \varepsilon _{\varDelta _{\lambda ^*\kappa ,1}}\right) , \nonumber \\ \dot{\varepsilon }_{\varDelta _{\lambda ^*\kappa ,2}}&= \varepsilon _{\varDelta _{\lambda ^*\kappa ,3}} -\alpha _{\varDelta _{\lambda ^*\kappa ,2}}M_{\varDelta _{\lambda ^*\kappa }}^{\frac{1}{n-1}}\left| \varepsilon _{\varDelta _{\lambda ^*\kappa ,2}} -\dot{\varepsilon }_{\varDelta _{\lambda ^*\kappa ,1}}\right| ^{\frac{n-2}{n-1}}\text {sign}\left( \varepsilon _{\varDelta _ {\lambda ^*\kappa ,2}}-\dot{\varepsilon }_{\varDelta _{\lambda ^*\kappa ,1}}\right) , \nonumber \\&\vdots&\nonumber \\ \dot{\varepsilon }_{\varDelta _{\lambda ^*\kappa ,n}}&= \varPsi _{\varDelta _{\lambda ^*\kappa }}(\cdot )-\alpha _ {\varDelta _{\lambda ^*\kappa ,n}}M_{\varDelta _{\lambda ^*\kappa }}\text {sign}\left( \varepsilon _{\varDelta _ {\lambda ^*\kappa ,n}}-\dot{\varepsilon }_{\varDelta _{\lambda ^*\kappa ,n-1}}\right) , \end{aligned}$$

(6.71)

where $\lambda ^*$ is the current operating mode, $\kappa \in \fancyscript{Q}$, with $\kappa \ne \lambda ^*$, represents the next operating mode, and

$$\begin{aligned}&\varepsilon _{\varDelta _{\lambda ^*\kappa ,k}}=\varepsilon _{\varDelta _{\lambda ^*,k}}(t_{1}^{-})- \varepsilon _{\varDelta _{\kappa ,k}}(t_{1}^{+}), \ \forall k=1,\ldots ,n, \\&\varPsi _{\varDelta _{\lambda ^*\kappa }}(\cdot ) = \varPsi _{\varDelta _{\lambda ^*}}(t_{1}^{-})-\varPsi _{\varDelta _{\kappa }}({t_{1}^{+}}). \end{aligned}$$

If the reset equations (6.31) are applied on each switching time, then the trajectories of the system (6.71) always remain in the sliding surface, i.e., $\varepsilon _{\varDelta _{\lambda ^*\kappa ,k}}=0$. Thus, the state estimation is maintained in spite of the switchings. Notice that due to the nature of the system, the transformations $\varPhi _{\lambda (x)}$ do not present jumps in the switching times. Nevertheless, the jumps can appear in the trajectories generated by the observers. However, the reset equations (6.31) avoid this happening.