Observability and State Estimation for a Class of Nonlinear Systems

Many important approaches have been presented in the literature concerning the state estimation problem for a given nonlinear control system (see for instance [1] - [19], [21] - [25], [27], [28] and relative references therein). Most of them are based on the existence of an observer system exhibiting state estimation. The corresponding hypotheses include observability assumptions and persistence of excitation. In [8], [9] and [10] Luenberger type observers and switching estimators are proposed for a general class of triangular systems under weaker assumptions than those adopted in the existing literature. In [16] and [24] the state estimation is exhibited for a class of systems by means of a hybrid observer.

The present note is inspired by the approach adopted in [16], where a ”hybrid dead-beat observer” is used, as well as by the methodologies applied in [20], [26] and [29], where, fixed point theorems are used for the establishment of sufficient conditions for observability and asymptotic controllability. Our main purpose is to establish that, under certain hypotheses, including persistence of excitation, the State Estimation Design Problem (SEDP) around a given fixed value of initial time is solvable for a class of nonlinear systems by means of a sequence of mappings $X_{\nu}$, exclusively dependent on the dynamics of the original system, the input $u$ and the corresponding output $y$, and further each $X_{\nu}$ is independent of the time-derivatives of $u$ and $y$. An algorithm for explicit construction of these mappings is provided.

We consider time-varying finite dimensional nonlinear control systems of the form:

where $u$ is the input of (1.1). Our main results establish sufficient conditions for the approximate solvability of the SEDP for (1.1). The paper is organized as follows. Section II contains definitions, assumptions, as well as statement and proof of our main result (Proposition 2.1) concerning the state estimation problem for the general case (1.1). We apply in Section III the main result of Section II for the derivation of sufficient conditions for the solvability of the same problem for certain subclasses of systems (1.1), whose dynamics have triangular structure (Proposition 3.1). According to our knowledge, the sufficient conditions proposed in Sections II, III are weaker than those imposed in the existing literature for the solvability of the observer design problem for the same class of systems. More extensions are discussed in Section IV of present work, concerning the solvability of the SEDP by means of a hybrid observer under certain additional assumptions (Proposition 4.1).

Notations: For given $x\in{{\mathbb{R}}^{n}}$, $\left|x\right|\text{ }$denotes its usual Euclidean norm. For a given constant matrix $A\in{{\mathbb{R}}^{m\times n}}$, ${A}^{\prime}$ denotes its transpose and $\left|A\right|:=\sup\left\{\left|Ax\right|,\left|x\right|=1\right\}$ is its induced norm. For any nonempty set $I\subset\mathbb{R}$ and map $I\ni t\to A(t)\in{{\mathbb{R}}^{m\times n}}$ we adopt the notation $\|A(\cdot){{\|}_{I}}\text{:=}\operatorname{ess}.\sup\{\left|A(t)\right|,t\in I\}$.

In this section we provide sufficient conditions for observability of (1.1) and solvability of the SEDP. We assume that for each $t\geq 0$ the mappings $A\left(t,\cdot,\cdot\right):$ ${{\mathbb{R}}^{k}}\times{{\mathbb{R}}^{m}}\to{{\mathbb{R}}^{n\times n}}$ , $C\left(t,\cdot\right):{{\mathbb{R}}^{m}}\to{{\mathbb{R}}^{k\times n}}$ and $f\left(t,\cdot,\cdot,\cdot\right):$ ${{\mathbb{R}}^{k}}\times{{\mathbb{R}}^{n}}\times{{\mathbb{R}}^{m}}\to{{\mathbb{R}}^{n}}$ are continuous and further $f$ is (locally) Lipschitz continuous with respect to $x$, i.e., for every bounded $I\subset{{\mathbb{R}}_{\geq 0}},X\subset{{\mathbb{R}}^{n}},U\subset{{\mathbb{R}}^{m}}$ and $Y\subset{{\mathbb{R}}^{k}}$ there exists a constant $C>0$ such that

Also, assume that for any $\left(x,y,u\right)\in{{\mathbb{R}}^{n}}\times{{\mathbb{R}}^{k}}\times{{\mathbb{R}}^{m}}$ the mappings $A\left(\cdot,y,u\right)$, $f\left(\cdot,y,x,u\right)$ and $C(\cdot,u)$ are measurable and locally essentially bounded in $\mathbb{R}_{\geq 0}$. Let ${{t}_{0}}\geq 0,\tau>{{t}_{0}}$ and let ${{U}_{[{{t}_{0}},\tau]}}$ be a nonempty set of inputs $u\in{{L}_{\infty}}\left(\left[{{t}_{0}},\tau\right];{{\mathbb{R}}^{m}}\right)$ of (1.1) (without any loss of generality it is assumed that ${{U}_{[{{t}_{0}},\tau]}}$ is independent of the initial state). Define by ${{Y}_{[{{t}_{0}},\tau],u}}$ the set of outputs $y$ of (1.1) defined on the interval $[{{t}_{0}},\tau]$ corresponding to some input $u\in{{U}_{[{{t}_{0}},\tau]}}$:

provided that ${{t}_{\max}}\geq\tau$ where $t_{\max}=t_{\max}(t_{0},x_{0},u)\leq+\infty$ is the maximum time of existence of the solution $x(\cdot;{{t}_{0}},{{x}_{0}},u)$ of (1.1) with initial $x({{t}_{0}};{{t}_{0}},{{x}_{0}},u)={{x}_{0}}$.

According to Definition 2.1, observability is equivalent to the existence of a (probably noncausal) functional $X(\cdot,\cdot,\cdot):\left\{{{t}_{0}}\right\}\times{{Y}_{[{{t}_{0}},\tau],u}}\times{{U}_{[{{t}_{0}},\tau]}}\to{{\mathbb{R}}^{n}}$, such that, for every ${{x}_{0}}\in{{\mathbb{R}}^{n}}$ for which (2.3) holds for certain $u\in{{U}_{[{{t}_{0}},\tau]}}$ and $y\in{{Y}_{[{{t}_{0}},\tau],u}}$, we have:

and $X$ is exclusively dependent on the input $u$ and the output $y$ of (1.1) and is in general noncausal. Knowledge of $X$ satisfying the previous properties guarantees knowledge of the initial state value, thus knowledge of the future values of the solution of (1.1), provided that the system is complete.

It is worthwhile to remark here that the approach proposed in [16] for the construction of a hybrid dead-beat observer for a subclass of systems (1.1) is based on an explicit construction of a (noncausal) map $X$ satisfying (2.4). However, for general nonlinear systems, the precise and direct determination of the functional $X$ is a difficult task. The difficulty comes from our requirements for the candidate $X$ to be exclusively dependent on $u$ and $y$ and the dynamics of system and, for practical reasons, it should be independent of the time-derivatives of $u$ and $y$. We next provide a weaker sequential type of definition of the solvability of SEDP, which is adopted in the present work, in order to achieve the state determination for general case (1.1) by employing an explicit approximate strategy.

It should be emphasised that uniqueness requirement in (2.6b) is not essential. We may replace (2.6b) by the assumption that there exists $x_{0}$ satisfying both (2.3) and (2.6a). Then uniqueness of such a vector $x_{0}$ is a consequence of (2.6a), definition (2.5) and the fact that each functional $X_{\nu}$ exclusively depends on $u$ and $y$.

Obviously, according to the definitions above, the following implications hold:

Solvability of SEDP $\Rightarrow$ Solvability of approximate SEDP $\Rightarrow$ Observability (over $I$).
For completeness, we note that the first implication follows by setting $X_{\nu}:=X,\nu=1,2,\ldots$ in (2.4). The second implication is a direct consequence of both assumptions (2.6a,b), definition (2.5) and the exclusive dependence of each $X_{\nu}$ from $u$ and $y$. The converse claims are not in general valid; particularly, observability does not in general imply solvability of the (approximate) SEDP, due to the additional requirements of Definitions 2.2 and 2.3 concerning the independence $X$, $X_{\nu}$, respectively, from the time-derivatives of $u$ and $y$.

From (2.6a) we deduce that, if the approximate SEDP is solvable for (1.1) over $I$, then for any $T>t_{0}$ for which $T\leq{{t}_{\max}}({{t}_{0}},{{x}_{0}},u)$ it holds:

In order to state and establish our main result, we first require the following notations and additional assumptions for the dynamics of (1.1). Consider ${{t}_{0}}\in I$, $\tau>{{t}_{0}}$, $u\in{{U}_{[{{t}_{0}},\tau]}}$, $y\in{{Y}_{[{{t}_{0}},\tau],u}}$ and $d\in{{C}^{0}}\left(\left[{{t}_{0}},\tau\right];{{\mathbb{R}}^{m}}\right)$. We denote by $\Phi(t,{{t}_{0}})$ the fundamental matrix solution of

and define the mappings:

We are in a position to provide our main assumptions together with the statement and proof of our main result.
A1. For system (1.1) we assume that there exists a nonempty subset $I$ of $\mathbb{R}_{\geq 0}$ in such a way that for all ${{t}_{0}}\in I$, $\tau>{{t}_{0}}$ close to ${{t}_{0}}$ and for each $u\in{{U}_{[{{t}_{0}},\tau]}}$ and $y\in{{Y}_{[{{t}_{0}},\tau],u}}$ it holds:

where the map $\Psi$ is given by (2.9);
A2. In addition, we assume that for every ${{t}_{0}}\in I$, $T>{{t}_{0}}$ close to ${{t}_{0}}$, $u\in{{U}_{[{{t}_{0}},T]}}$, $y\in{{Y}_{[{{t}_{0}},T],u}}$, $\ell\in(0,1)$ and constants $R,\theta>0$ there exists a constant $\tau\in\left({{t}_{0}},\min\left\{{{t}_{0}}+\theta,T\right\}\right)$ such that

Assumption A1 is a type of persistence of excitation and A2 is a type of contraction condition. Assumptions A1 and A2 are in general difficult to be checked, however, they are both fulfilled for a class of nonlinear triangular systems, under weak assumptions that are exclusively expressed in terms of system’s dynamics (see (3.1) in the next section). We are in a position to state and prove our main result. Our approach leads to an explicit algorithm for the state estimation.

The existence result of Proposition 2.1 does not in general determine explicitly the desired sequence of mappings ${{X}_{\nu}}$ exhibiting (2.27). The reason is that, although existence of the constant $k$ satisfying (2.19) is guaranteed from boundedness of $\left\{x(t),t{{\in}{[{{t}_{0}},\tau]}}\right\}$, its precise determination requires knowledge of a bound of the previous set, which, in general, is not available. The rest part of this section is devoted for the establishment of a constructive algorithm, exhibiting the state determination. The corresponding procedure is based on the approach given in proof of Proposition 2.1 plus some appropriate modifications.

Algorithm

To simplify the procedure, we distinguish two cases:
Case I: First, we assume that a bounded region of the state space is a priori known, where the unknown initial state of (1.1) belongs. Particularly, assume that for all ${{t}_{0}}\in I$, almost all $\tau>{{t}_{0}}$ near ${{t}_{0}}$ and input $u\in{{U}_{[{{t}_{0}},\tau]}}$, an open ball ${{B}_{R}}$ of radius $R>0$ centered at zero is known, such that the corresponding set of outputs of (1.1) is modified as follows:

For the case above we adopt a slight modification of the approach used for the proof of Proposition 2.1. Our proposed algorithm contains two steps:
Step 1: Define

where the latter is involved in (2.28). Notice that, due to the additional assumption (2.28), it follows that (2.19) holds with $k=1$ and for $\tau$ close to $t_{0}$. Next, consider a strictly increasing sequence $\left\{{{R}_{\nu}}\in{{\mathbb{R}}_{>0}},\nu=1,2,\ldots\right\}$ satisfying the first equality of (2.18), namely,

and with ${{R}_{1}}$ as above. We set $\ell=1/2$ and find a decreasing sequence ${{t}_{\nu}}\in{{\mathbb{R}}_{>0}},\nu=1,2,\ldots$ with ${{t}_{\nu}}\to{{t}_{0}}$ and in such a way that

Step 2: Consider the sequence ${{z}_{\nu+1}}(t):={{\mathcal{F}}_{{{{t}_{\nu}}}}}(t;{{t}_{0}},y,{{z}_{\nu}},u),t\in[{{t}_{0}},t_{\nu}]$ with arbitrary initial ${{z}_{1}}(\cdot)\in{{C}^{0}}\left([{{t}_{0}},{{t}_{k}}];{{\mathbb{R}}^{n}}\right)$ satisfying (2.23) with $k=1$ and set

It then follows that (2.27) holds with unique $x({{t}_{0}})\in{{B}_{R}}$ satisfying (2.3). Particularly, we have:

therefore, the sequence of mappings ${{X}_{\nu}}$, as defined by (2.32), exhibits the state determination. above satisfies the desired (2.5) and (2.6).
Case II (General Case): We now provide an algorithm, which exhibits the state determination for the general case, without any additional assumption. The algorithm contains two steps:
Step 1: Repeat the same procedure followed in Case I, with $R=1,2,3,\ldots$ and construct a sequences of mappings

Step 2: Define

Notice that, since the set $\left\{x(t),t{{\in}{[{{t}_{0}},\tau]}}\right\}$ is bounded, there exists an integer $k$ such that $\|x(\cdot){{\|}_{[{{t}_{0}},\tau]}}<k<k+1<k+2<\ldots$. The latter in conjunction with (2.33) yields:

which, due to selection $\ell=1/2$, implies ${{X}_{\nu}}({{t}_{0}},y,u):=\xi_{\nu}^{\nu}\to x({{t}_{0}})$. We conclude that for the general case the sequence of mappings ${{X}_{\nu}}$, as defined by (2.34), exhibits the state determination. Finally, it should be noted that, according to the methodology above, contrary to the approach adopted in the proof of Proposition 2.1, the specific knowledge of $k$ satisfying (2.35) is not required.

In this section we apply the results of Section II to triangular systems (1.1) of the form:

where we make the following assumptions:
H1 (Regularity Assumptions). It is assumed that for each $(x,y,u)\in{{\mathbb{R}}^{n}}\times\mathbb{R}\times{{\mathbb{R}}^{m}}$ the mappings ${{a}_{i}}(\cdot,y,u),\,\,i=2,\ldots,n$, ${{f}_{i}}(\cdot,{{x}_{1}},u)$, $i=1,\ldots,n-1$ and ${{f}_{n}}(\cdot,{{x}_{1}},\ldots,{{x}_{n}},u)$ are measurable and locally essentially bounded and for each fixed $t\geq 0$ and $u\in{{\mathbb{R}}^{m}}$ the mappings ${{a}_{i}}(t,\cdot,u),i=2,\ldots,n,{{f}_{i}}(t,\cdot,u),i=1,\ldots,n-1$ and ${{f}_{n}}(t,\ldots,u)$ are (locally) Lipschitz.

Obviously, (3.1) has the form of (1.1) with

We also make the following observability assumption:
H2. There exists a measurable set $I\subset{{\mathbb{R}}_{\geq 0}}$ with nonempty interior such that for all ${{t}_{0}}\in I$, $\tau>{{t}_{0}}$ close to ${{t}_{0}}$ and for each $u\in{{U}_{[{{t}_{0}},\tau]}}:={{L}_{\infty}}([{{t}_{0}},\tau];{{\mathbb{R}}^{m}})$ and $y\in Y_{[t_{0},\tau],u}$ it holds: