Keywords

1 Introduction

An observer for a plant is a system which performs the reconstruction of the state vector from the available inputs. The problem of designing observers for linear control systems was first introduced and fully solved by Luenberger [9]. The problem of designing observers for nonlinear control systems was proposed by Thau [23]. Over the past three decades, significant attention has been paid in the control systems literature to the construction of observers for nonlinear control systems [1].

A necessary condition for the existence of a local exponential observer for nonlinear control systems was obtained by Xia and Gao [27]. On the other hand, sufficient conditions for nonlinear observers have been derived in the control literature from an impressive variety of points of view. Kou, Elliott and Tarn [6] derived sufficient conditions for the existence of local exponential observers using Lyapunov-like method. In [4, 7, 8, 26], suitable coordinate transformations were found under which a nonlinear control systems is transferred into a canonical form, where the observer design is carried out. In [24], Tsinias derived sufficient Lyapunov-like conditions for the existence of local asymptotic observers for nonlinear systems. A harmonic analysis method was proposed in [2] for the synthesis of nonlinear observers.

A characterization of local exponential observers for nonlinear control systems was first obtained by Sundarapandian [11]. In [11], necessary and sufficient conditions were obtained for exponential observers for Lyapunov stable continuous-time nonlinear systems and an exponential observer design was provided by Sundarapandian which generalizes the linear observer design of Luenberger [9] for linear control systems. In [14], Sundarapandian obtained necessary and sufficient conditions for exponential observers for Lyapunov stable discrete-time nonlinear systems and also provided a formula for designing exponential observers for Lyapunov stable discrete-time nonlinear systems. In [10], Sundarapandian derived new results for the global observer design for nonlinear control systems.

The concept of nonlinear observers for nonlinear control systems was also extended in many ways. In [12, 13], Sundarapandian derived new results characterizing local exponential observers for nonlinear bifurcating systems. In [15, 16, 21, 22], Sundarapandian derived new results for the exponential observer design for a general class of nonlinear systems with real parametric uncertainty. In [17–20], Sundarapandian derived new results and characterizations for general observers for nonlinear systems.

This work gives a discussion on recent results on general observers for nonlinear systems [19, 20]. This work is organized as follows. Section 2 gives a definition of general observers (asymptotic and exponential) for continuous-time nonlinear systems. Section 3 provides a necessary condition for general asymptotic observers for continuous-time nonlinear systems. Section 4 details a characterization of general exponential observers for continuous-time nonlinear systems. Section 5 gives a definition of general observers (asymptotic and exponential) for continuous-time nonlinear systems. Section 6 provides a necessary condition for general asymptotic observers for continuous-time nonlinear systems. Section 7 details a characterization of general exponential observers for continuous-time nonlinear systems. Section 8 gives a summary of the main results discussed in this work.

2 Definition of General Observers for Continuous-Time Nonlinear Systems

In this section, we consider a \(C^1\) nonlinear plant of the form

$$\begin{aligned} \left. \begin{array}{ccl} \dot{x} &{} = &{} f(x) \\ y &{} = &{} h(x) \\ \end{array} \right. \end{aligned}$$
(1)

where \(x \in {\mathbf R}^n\) is the state and \(y \in {\mathbf R}^p\), the output of the plant (1). We assume that the state x belongs to an open neighbourhood X of the origin of \({\mathbf R}^n\). We assume that \(f: X \rightarrow {\mathbf R}^n\) is a \(C^1\) vector field and also that \(f(0) = 0\). We also assume that the output mapping \(h: X \rightarrow {\mathbf R}^p\) is a \(C^1\) map, and also that \(h(0) = 0\). Let \(Y \mathop {=}\limits ^{\varDelta } h(X)\).

Definition 1

Consider a \(C^1\) dynamical system described by

$$\begin{aligned} \dot{z} = g(z, y), \ \ \ [ z \in {\mathbf R}^m ] \end{aligned}$$
(2)

where z is defined in a neighbourhood Z of the origin of \({\mathbf R}^m\) and \(g: Z \times Y \rightarrow {\mathbf R}^m\) is a \(C^1\) map with \(g(0, 0) = 0\). Consider also the map \(q: Z \rightarrow {\mathbf R}^n\) described by

$$\begin{aligned} w = q(z) \end{aligned}$$
(3)

Then the candidate system (2) is called a general asymptotic (respectively, general exponential) observer for the plant (1) corresponding to (3) if the following two requirements are satisfied:

  1. (O1)

    If \(w(0) = x(0)\), then \(w(t) \equiv x(t)\), for all \(t \ge 0\).

  2. (O2)

    There exists a neighbourhood V of the origin of \({\mathbf R}^n\) such that for all initial estimation error \(w(0) - x(0) \in V\), the estimation error \(e(t) = w(t) - x(t)\) tends to zero asymptotically (respectively, exponentially) as \(t \rightarrow \infty \).    \(\blacksquare \)

Remark 1

If a general exponential observer (2) satisfies the additional properties that \(m = n\) and q is a \(C^1\) diffeomorphism, then it is called a full-order general exponential observer. A full-order general exponential observer (2) with the additional property that \(q = \text{ id }_X\) is called an identity exponential observer, which is the same as the standard definition of local exponential observers for nonlinear systems.    \(\blacksquare \)

The estimation error e is defined by

$$\begin{aligned} e = q(z) - x \end{aligned}$$
(4)

Now, we consider the composite system

$$\begin{aligned} \left. \begin{array}{ccl} \dot{x} &{} = &{} f(x) \\ \dot{z} &{} = &{} g(z, h(x)) \\ \end{array} \right. \end{aligned}$$
(5)

The following lemma is straightforward.

Lemma 1

([20]) The following statements are equivalent.

  1. (a)

    The condition (O1) in Definition 1 holds for the composite system (1) and (2).

  2. (b)

    The submanifold defined via \(q(z) = x\) is invariant under the flow of the composite system (5).    \(\blacksquare \)

3 A Necessary Condition for General Asymptotic Observers for Continuous-Time Nonlinear Systems

In this section, we first prove a necessary condition for the plant (1) to have general asymptotic observers.

Theorem 1

([20]) A necessary condition for the existence of a general asymptotic observer for the plant (1) is that the plant (1) is asymptotically detectable, i.e. any state trajectory x(t) of the plant dynamics in (1) with small initial condition \(x_0\), satisfying \(h(x(t)) \equiv 0\) must be such that

$$\begin{aligned} x(t) \rightarrow 0 \ \ \ \text{ as } \ \ \ t \rightarrow \infty \end{aligned}$$
(6)

Proof

Let (2) be a general asymptotic observer for the plant (1). Then the conditions (O1) and (O2) in Definition 1 are satisfied. Now, let x(t) be any state trajectory of the plant dynamics in (1) with small initial condition \(x_0\) satisfying \(y(t) = h(x(t)) \equiv 0\). Then the observer dynamics (2) reduces to

$$\begin{aligned} \dot{z} = g(z, 0) \end{aligned}$$
(7)

Taking \(z_0 = 0\), it is immediate from (7) that \(z(t) = z(t; z_0) \equiv 0\).

Hence, \(w(t) = q(z(t)) \equiv 0\).

By condition (O2), we know that the estimation error trajectory \(e(t) = w(t) - x(t)\) tends to zero as \(t \rightarrow \infty \). Since \(w(t) \equiv 0\), it follows that \(x(t) \rightarrow 0\) as \(t \rightarrow \infty \).

This completes the proof.    \(\blacksquare \)

Using Theorem 1, we can prove the following result which says that there is no general asymptotic observer for the following plant

$$\begin{aligned} \left. \begin{array}{ccl} \left[ \begin{array}{c} \dot{x} \\ \dot{\lambda } \\ \end{array} \right] &{} = &{} \left[ \begin{array}{c} F(x, \lambda ) \\ 0 \\ \end{array} \right] \\ y &{} = &{} h(x) \\ \end{array} \right. \end{aligned}$$
(8)

if \(F(0, \lambda ) = 0\) (i.e. if the equilibrium \(x = 0\) of the dynamics \(\dot{x} = F(x, \lambda )\) does not change with the real parametric uncertainty \(\lambda \)).

Theorem 2

Suppose that the plant (8) satisfies the assumption

$$\begin{aligned} F(0, \lambda ) \equiv 0 \end{aligned}$$
(9)

Then there is no general asymptotic observer for the plant (8).

Proof

This is an immediate consequence of Theorem 1. We show that the plant (8) is not asymptotically detectable.

This is easily seen by taking \(x(0) = x_0 = 0\) and \(\lambda (0) = \lambda _0 \ne 0\).

Then we have

$$\begin{aligned} y(t) = h(x(t)) \equiv 0 \quad \text{ and } \quad x(t) \equiv 0 \end{aligned}$$
(10)

but \(\lambda (t) \equiv \lambda _0 \ne 0\). Hence, the plant (8) is not asymptotically detectable. From Theorem 1, we deduce that there is no general asymptotic observer for the plant (8).

This completes the proof.    \(\blacksquare \)

4 Necessary and Sufficient Conditions for General Exponential Observers for Continuous-Time Nonlinear Systems

In this section, we establish a basic theorem that completely characterizes the existence of general exponential observers of the form (2) for Lyapunov stable nonlinear plants of the form (1).

For this purpose, we define the system linearization pair for the nonlinear plant (1) as

$$\begin{aligned} A = \frac{\partial f}{\partial x}(0) \quad \text{ and } \quad h = \frac{\partial h}{\partial x}(0) \end{aligned}$$
(11)

We also define

$$\begin{aligned} E = \frac{\partial g}{\partial z}(0, 0) \quad \text{ and } \quad K = \frac{\partial g}{\partial y}(0, 0) \end{aligned}$$
(12)

Now, we state and prove the following result, which gives a complete characterization of the general exponential observers for Lyapunov stable nonlinear systems.

Theorem 3

Suppose that the plant dynamics in (1) is Lyapunov stable at \(x = 0\). Then the system (2) is a general exponential observer for the plant (1) with respect to (3) if and only if the following conditions are satisfied:

  1. (a)

    The submanifold defined via \(q(z) = x\) is invariant under the flow of the composite system (5).

  2. (b)

    The dynamics

    $$\begin{aligned} \dot{z} = g(z, 0) \end{aligned}$$
    (13)

    is locally exponentially stable at \(e = 0\).

Proof

Necessity. Suppose that the system (2) is a general exponential observer for the plant (1). Then the conditions (O1) and (O2) in Definition 1 are readily satisfied. By Lemma 1, the condition (O1) implies the condition (a). To see that the condition (b) also holds, we take \(x(0) = 0\). Then \(x(t) \equiv 0\) and \(y(t) = h(x(t)) \equiv 0\), for all \(t \ge 0\).

Thus, the Eq. (2) simplifies into

$$\begin{aligned} \dot{z}(t) = g(z(t), y(t)) = g(z(t), 0) \end{aligned}$$
(14)

By the condition (O2) in Definition 1, it is immediate that \(e(t) = q(z(t)) - x(t) = q(z(t)) \rightarrow 0\) exponentially as \(t \rightarrow \infty \) for all small initial conditions \(z_0\). Hence, we must have \(z(t) \rightarrow 0\) exponentially as \(t \rightarrow \infty \) for the solution trajectory z(t) of the dynamics (14). Hence, we conclude that the dynamics (14) is locally exponentially stable at \(z = 0\). Thus, we have established the necessity of the conditions (a) and (b).

Sufficiency. Suppose that the conditions (a) and (b) are satisfied by the plant (1) and the candidate observer (2). Since the condition (a) implies the condition (O1) of Definition 1 by Lemma 1, it suffices to show that condition (O2) in Definition 1 also holds.

By hypotheses, the equilibrium \(z = 0\) of the dynamics (13) is locally exponentially stable and the equilibrium \(x = 0\) of the plant dynamics in (1) is Lyapunov stable. Hence, E must be Hurwitz and A must have all eigenvalues with non-negative real parts.

We have two cases to consider.

Case I: A is Hurwitz.

By Hartman–Grobman theorem [3], it follows that the composite system (5) is locally topologically conjugate to the system

$$\begin{aligned} \left[ \begin{array}{c} \dot{x} \\ \dot{z} \\ \end{array} \right] = \left[ \begin{array}{cc} A &{} 0 \\ K C &{} E \\ \end{array} \right] \, \left[ \begin{array}{c} x \\ z \\ \end{array} \right] \end{aligned}$$
(15)

We note that

$$\begin{aligned} \text{ eig }\left[ \begin{array}{cc} A &{} 0 \\ K C &{} E \\ \end{array} \right] = \text{ eig }(A) \cup \text{ eig }(E) \end{aligned}$$
(16)

Since both A and E are Hurwitz matrices, it follows from (16) that the system matrix of (15) is Hurwitz. Hence, it is immediate that \(x(t) \rightarrow 0\) and \(z(t) \rightarrow 0\) exponentially as \(t \rightarrow \infty \). Hence, it follows trivially that \(e(t) = q(z(t)) - x(t) \rightarrow 0\) exponentially as \(t \rightarrow \infty \) for all small initial conditions x(0) and z(0).

Case II: A is not Hurwitz.

Without loss of generality, we can assume that the plant dynamics in (1) has the form

$$\begin{aligned} \left. \begin{array}{ccl} \dot{x}_1 &{} = &{} A_1 x_1 + \phi _1(x_1, x_2) \\ \dot{x}_2 &{} = &{} A_2 x_2 + \phi _2(x_1, x_2) \\ \end{array} \right. \end{aligned}$$
(17)

where \(x_1 \in {\mathbf R}^{n_1}, x_2 \in {\mathbf R}^{n_2}\) (with \(n_1 + n_2 = n\)), \(A_1\) is an \(n_1 \times n_1\) matrix having all eigenvalues with zero real part, \(A_2\) is an \(n_2 \times n_2\) Hurwitz matrix and \(\phi _1, \phi _2\) are \(C^1\) functions vanishing at \((x_1, x_2) = (0, 0)\) together with all their first-order partial derivatives.

Now, \(x = 0\) is a Lyapunov stable equilibrium of the dynamics in (1). Also, the equilibrium \(z = 0\) of the dynamics (13) is locally exponentially stable. Hence, by a total stability result [5], it follows that \((x, z) = (0, 0)\) is a Lyapunov stable equilibrium of the composite system (5) (by its triangular structure).

Also, by the center manifold theorem for flows [25], we know that the composite system (5) has a local center manifold at \((x, z) = (0, 0)\), the graph of a \(C^1\) map,

$$\begin{aligned} \left[ \begin{array}{c} x_2 \\ z \\ \end{array} \right] = \pi (x_1) = \left[ \begin{array}{c} \pi _1(x_1) \\ \pi _2(x_1) \\ \end{array} \right] \end{aligned}$$
(18)

Since \(q(z) = x\) is an invariant manifold for the composite system (5), it is immediate that along the center manifold, we have

$$\begin{aligned} q(\pi _2(x_1)) = \left[ \begin{array}{c} x_1 \\ \pi _1(x_1) \\ \end{array} \right] . \end{aligned}$$
(19)

By the principle of asymptotic phase in the center manifold theory [25], there exists a neighbourhood V of \((x, z) = (0, 0)\) such that for all \((x(0), z(0)) \in V\), we have

$$\begin{aligned} ||\left[ \begin{array}{c} x_2(t) - \pi _1(x_1(t)) \\ z(t) - \pi _2(x_1(t)) \\ \end{array} \right] ||\, \le \, M \exp (-a t) ||\left[ \begin{array}{c} x_2(0) - \pi _1(x_1(0)) \\ z(0) - \pi _2(x_1(0)) \\ \end{array} \right] || \end{aligned}$$
(20)

for some positive constants M and a.

Hence, it is immediate that

$$\begin{aligned} z(t) \rightarrow \pi _2(x_1(t)) \ \ \text{ exponentially } \text{ as } \ \ \ t \rightarrow \infty \end{aligned}$$
(21)

From (19) and (21), it follows that

$$\begin{aligned} q(z(t)) \rightarrow \left[ \begin{array}{c} x_1(t) \\ \pi _1(x_1(t)) \\ \end{array} \right] \ \ \text{ exponentially } \text{ as } \ \ \ t \rightarrow \infty \end{aligned}$$
(22)

From (20) and (22), it follows that

$$\begin{aligned} q(z(t)) \rightarrow x(t) = \left[ \begin{array}{c} x_1(t) \\ x_2(t) \\ \end{array} \right] \ \ \text{ exponentially } \text{ as } \ \ \ t \rightarrow \infty \end{aligned}$$
(23)

Thus, the condition (O2) also holds.

This completes the proof.    \(\blacksquare \)

Theorem 4

A necessary condition for the system (2) to be a general exponential observer for a Lyapunov stable plant (1) is that

$$\begin{aligned} \text{ dim }(z) \ge n_1 \end{aligned}$$
(24)

where \(n_1\) denotes the number of critical eigenvalues of the system matrix A.

Proof

If the system matrix A is Hurwitz, then \(n_1 = 0\) and there is nothing to prove.

If A is not Hurwitz, then by the center manifold theory, we have

$$\begin{aligned} q(\pi _2(x_1)) = \left[ \begin{array}{c} x_1 \\ \pi _1(x_1) \\ \end{array} \right] \end{aligned}$$
(25)

For every small \(x_1 \in {\mathbf R}^{n_1}\), we know that the vector

$$ \left[ \begin{array}{c} x_1 \\ \pi _1(x_1) \\ \end{array} \right] $$

has a pre-image \(\pi _2(x_1)\) under the mapping, q. Hence, it is immediate that the dimension of the domain of the mapping, q, cannot be lower than the dimension of the state \(x_1\).

This completes the proof.    \(\blacksquare \)

5 Definition of General Observers for Discrete-Time Nonlinear Systems

In this section, we consider a nonlinear plant of the form

$$\begin{aligned} \left. \begin{array}{ccl} x(k + 1) &{} = &{} f(x(k)), \\ y(k) &{} = &{} h(x(k)) \\ \end{array} \right. \end{aligned}$$
(26)

where \(x \in {\mathbf R}^n\) is the state and \(y \in {\mathbf R}^p\), the output of the plant (26). We assume that the state x belongs to an open neighbourhood X of the origin of \({\mathbf R}^n\). We assume that \(f: X \rightarrow {\mathbf R}^n\) is a \(C^1\) map and also that \(f(0) = 0\). We also assume that the output mapping \(h: X \rightarrow {\mathbf R}^p\) is a \(C^1\) map, and also that \(h(0) = 0\). Let \(Y \mathop {=}\limits ^{\varDelta } h(X)\).

Definition 2

Consider a discrete-time dynamical system described by

$$\begin{aligned} z(k + 1) = g(z(k), y(k)), \ \ \ [ z \in {\mathbf R}^m ] \end{aligned}$$
(27)

where z is defined in a neighbourhood Z of the origin of \({\mathbf R}^m\) and \(g: Z \times Y \rightarrow {\mathbf R}^m\) is a \(C^1\) map with \(g(0, 0) = 0\). Consider also the map \(q: Z \rightarrow {\mathbf R}^n\) described by

$$\begin{aligned} w = q(z) \end{aligned}$$
(28)

Then the candidate system (27) is called a general asymptotic (respectively, general exponential) observer for the plant (26) corresponding to (28) if the following two requirements are satisfied:

  1. (O1)

    If \(w(0) = x(0)\), then \(w(k) \equiv x(k)\), for all \(k \in {\mathbf Z}_+\), where \({\mathbf Z}_+\) denotes the set of all positive integers.

  2. (O2)

    There exists a neighbourhood V of the origin of \({\mathbf R}^n\) such that for all initial estimation error \(w(0) - x(0) \in V\), the estimation error \(e(k) = w(k) - x(k)\) tends to zero asymptotically (respectively, exponentially) as \(k \rightarrow \infty \).    \(\blacksquare \)

Remark 2

If a general exponential observer (27) satisfies the additional properties that \(m = n\) and q is a \(C^1\) diffeomorphism, then it is called a full-order general exponential observer. A full-order general exponential observer (27) with the additional property that \(q = \text{ id }_X\) is called an identity exponential observer, which is the same as the standard definition of local exponential observers for nonlinear systems.    \(\blacksquare \)

The estimation error e is defined by

$$\begin{aligned} e = q(z) - x \end{aligned}$$
(29)

Now, we consider the composite system

$$\begin{aligned} \left. \begin{array}{ccl} x(k + 1) &{} = &{} f(x(k)) \\ z(k + 1) &{} = &{} g(z(k), h(x(k))) \\ \end{array} \right. \end{aligned}$$
(30)

The following lemma is straightforward.

Lemma 2

([19]) The following statements are equivalent.

  1. (a)

    The condition (O1) in Definition 1 holds for the composite system (26) and (27).

  2. (b)

    The submanifold defined via \(q(z) = x\) is invariant under the flow of the composite system (30).    \(\blacksquare \)

6 A Necessary Condition for General Asymptotic Observers for Discrete-Time Nonlinear Systems

In this section, we first prove a necessary condition for the plant (26) to have general asymptotic observers.

Theorem 5

([19]) A necessary condition for the existence of a general asymptotic observer for the discrete-time plant (26) is that the plant (26) is asymptotically detectable, i.e. any state trajectory x(k) of the plant dynamics in (26) with small initial condition \(x_0\), satisfying \(h(x(k)) \equiv 0\) must be such that

$$\begin{aligned} x(k) \rightarrow 0 \ \ \ \text{ as } \ \ \ k \rightarrow \infty \end{aligned}$$
(31)

Proof

Let (27) be a general asymptotic observer for the plant (26). Then the conditions (O1) and (O2) in Definition 2 are satisfied. Now, let x(k) be any state trajectory of the plant dynamics in (26) with small initial condition \(x_0\) satisfying \(y(k) = h(x(k)) \equiv 0\). Then the observer dynamics (27) reduces to

$$\begin{aligned} z(k + 1) = g(z(k), 0) \end{aligned}$$
(32)

Taking \(z_0 = 0\), it is immediate from (32) that \(z(k) = z(k; z_0) \equiv 0\).

Hence, \(w(k) = q(z(k)) \equiv 0\).

By condition (O2), we know that the error trajectory \(e(k) = w(k) - x(k)\) tends to zero as \(k \rightarrow \infty \). Since \(w(k) \equiv 0\), it follows that \(x(k) \rightarrow 0\) as \(k \rightarrow \infty \).

This completes the proof.    \(\blacksquare \)

Using Theorem 5, we can prove the following result which says that there is no general asymptotic observer for the following plant

$$\begin{aligned} \left. \begin{array}{ccl} \left[ \begin{array}{c} x(k + 1) \\ \lambda (k + 1) \\ \end{array} \right] &{} = &{} \left[ \begin{array}{c} F(x(k), \lambda (k)) \\ \lambda (k) \\ \end{array} \right] \\ y(k) &{} = &{} h(x(k)) \\ \end{array} \right. \end{aligned}$$
(33)

if \(F(0, \lambda ) \!=\! 0\) (i.e. if the equilibrium \(x = 0\) of the dynamics \(x(k + 1) \!=\! F(x(k), \lambda (k))\) does not change with the real parametric uncertainty \(\lambda \)).

Theorem 6

Suppose that the plant (33) satisfies the assumption

$$\begin{aligned} F(0, \lambda ) \equiv 0 \end{aligned}$$
(34)

Then there is no general asymptotic observer for the plant (33).

Proof

This is an immediate consequence of Theorem 5. We show that the plant (33) is not asymptotically detectable.

This is easily seen by taking \(x(0) = x_0 = 0\) and \(\lambda (0) = \lambda _0 \ne 0\).

Then we have

$$\begin{aligned} y(k) = h(x(k)) \equiv 0 \quad \text{ and } \quad x(k) \equiv 0 \end{aligned}$$
(35)

but \(\lambda (k) \equiv \lambda _0 \ne 0\). Hence, the plant (33) is not asymptotically detectable. From Theorem 5, we deduce that there is no general asymptotic observer for the plant (33).

This completes the proof.    \(\blacksquare \)

7 Necessary and Sufficient Conditions for General Exponential Observers for Discrete-Time Nonlinear Systems

In this section, we establish a basic theorem that completely characterizes the existence of general exponential observers of the form (27) for Lyapunov stable nonlinear plants of the form (26).

For this purpose, we define the system linearization pair for the nonlinear plant (26) as

$$\begin{aligned} A = \frac{\partial f}{\partial x}(0) \quad \text{ and } \quad h = \frac{\partial h}{\partial x}(0) \end{aligned}$$
(36)

We also define

$$\begin{aligned} E = \frac{\partial g}{\partial z}(0, 0) \quad \text{ and } \quad K = \frac{\partial g}{\partial y}(0, 0) \end{aligned}$$
(37)

Now, we state and prove the following result, which gives a complete characterization of the general exponential observers for Lyapunov stable nonlinear systems.

Theorem 7

Suppose that the plant dynamics in (26) is Lyapunov stable at \(x = 0\). Then the system (27) is a general exponential observer for the plant (26) with respect to (28) if and only if the following conditions are satisfied:

  1. (a)

    The submanifold defined via \(q(z) = x\) is invariant under the flow of the composite system (30).

  2. (b)

    The dynamics

    $$\begin{aligned} z(k + 1) = g(z(k), 0) \end{aligned}$$
    (38)

    is locally exponentially stable at \(e = 0\).

Proof

Necessity. Suppose that the system (27) is a general exponential observer for the plant (26). Then the conditions (O1) and (O2) in Definition 2 are readily satisfied. By Lemma 2, the condition (O1) implies the condition (a). To see that the condition (b) also holds, we take \(x(0) = 0\). Then \(x(k) \equiv 0\) and \(y(k) = h(x(k)) \equiv 0\), for all \(k \in {\mathbf Z}_+\).

Thus, the Eq. (27) simplifies into

$$\begin{aligned} z(k + 1) = g(z(k), y(k)) = g(z(k), 0) \end{aligned}$$
(39)

By the condition (O2) in Definition 2, it is immediate that \(e(k) = q(z(k)) - x(k) = q(z(k)) \rightarrow 0\) exponentially as \(k \rightarrow \infty \) for all small initial conditions \(z_0\). Hence, we must have \(z(k) \rightarrow 0\) exponentially as \(k \rightarrow \infty \) for the solution trajectory z(k) of the dynamics (39). Hence, we conclude that the dynamics (39) is locally exponentially stable at \(z = 0\). Thus, we have established the necessity of the conditions (a) and (b).

Sufficiency. Suppose that the conditions (a) and (b) are satisfied by the plant (26) and the candidate observer (27). Since the condition (a) implies the condition (O1) of Definition 2 by Lemma 2, it suffices to show that condition (O2) in Definition 2 also holds.

By hypotheses, the equilibrium \(z = 0\) of the dynamics (38) is locally exponentially stable and the equilibrium \(x = 0\) of the plant dynamics in (26) is Lyapunov stable. Hence, E must be convergent and A must have all eigenvalues \(\zeta \) with \(|\zeta | \le 1\).

We have two cases to consider.

Case I: A is convergent.

By Hartman–Grobman theorem for maps, it follows that the composite system (30) is locally topologically conjugate to the system

$$\begin{aligned} \left[ \begin{array}{c} x(k + 1) \\ z(k + 1) \\ \end{array} \right] = \left[ \begin{array}{ccc} A &{} 0 \\ K C &{} E \\ \end{array} \right] \, \left[ \begin{array}{c} x(k) \\ z(k) \\ \end{array} \right] \end{aligned}$$
(40)

We note that

$$\begin{aligned} \text{ eig }\left[ \begin{array}{cc} A &{} 0 \\ K C &{} E \\ \end{array} \right] = \text{ eig }(A) \cup \text{ eig }(E) \end{aligned}$$
(41)

Since both A and E are convergent matrices, it follows from (41) that the system matrix of (40) is convergent. Hence, it is immediate that \(x(k) \rightarrow 0\) and \(z(k) \rightarrow 0\) exponentially as \(k \rightarrow \infty \). Hence, it follows trivially that \(e(k) = q(z(k)) - x(k) \rightarrow 0\) exponentially as \(k \rightarrow \infty \) for all small initial conditions x(0) and z(0).

Case II: A is not convergent.

Without loss of generality, we can assume that the plant dynamics in (26) has the form

$$\begin{aligned} \left. \begin{array}{ccl} x_1(k + 1) &{} = &{} A_1 x_1(k) + \phi _1(x_1(k), x_2(k)) \\ x_2(k + 1) &{} = &{} A_2 x_2(k) + \phi _2(x_1(k), x_2(k)) \\ \end{array} \right. \end{aligned}$$
(42)

where \(x_1 \in {\mathbf R}^{n_1}, x_2 \in {\mathbf R}^{n_2}\) (with \(n_1 + n_2 = n\)), \(A_1\) is an \(n_1 \times n_1\) matrix having all eigenvalues with unit modulus, \(A_2\) is an \(n_2 \times n_2\) convergent matrix and \(\phi _1, \phi _2\) are \(C^1\) functions vanishing at \((x_1, x_2) = (0, 0)\) together with all their first-order partial derivatives.

Now, \(x = 0\) is a Lyapunov stable equilibrium of the dynamics in (26). Also, the equilibrium \(z = 0\) of the dynamics (38) is locally exponentially stable. Hence, by a total stability result [5], it follows that \((x, z) = (0, 0)\) is a Lyapunov stable equilibrium of the composite system (30) (by its triangular structure).

Also, by the center manifold theorem for maps, we know that the composite system (30) has a local center manifold at \((x, z) = (0, 0)\), the graph of a \(C^1\) map,

$$\begin{aligned} \left[ \begin{array}{c} x_2 \\ z \\ \end{array} \right] = \pi (x_1) = \left[ \begin{array}{c} \pi _1(x_1) \\ \pi _2(x_1) \\ \end{array} \right] \end{aligned}$$
(43)

Since \(q(z) = x\) is an invariant manifold for the composite system (30), it is immediate that along the center manifold, we have

$$\begin{aligned} q(\pi _2(x_1)) = \left[ \begin{array}{c} x_1 \\ \pi _1(x_1) \\ \end{array} \right] . \end{aligned}$$
(44)

By the principle of asymptotic phase in the center manifold theory [25], there exists a neighbourhood V of \((x, z) = (0, 0)\) such that for all \((x(0), z(0) \in V\), we have

$$\begin{aligned} ||\left[ \begin{array}{c} x_2(k) - \pi _1(x_1(k)) \\ z(k) - \pi _2(x_1(k)) \\ \end{array} \right] ||\, \le \, M a^k ||\left[ \begin{array}{c} x_2(0) - \pi _1(x_1(0)) \\ z(0) - \pi _2(x_1(0)) \\ \end{array} \right] || \end{aligned}$$
(45)

for some positive constant M and \( 0 < a < 1\).

Hence, it is immediate that

$$\begin{aligned} z(k) \rightarrow \pi _2(x_1(k)) \ \ \text{ exponentially } \text{ as } \ \ \ k \rightarrow \infty \end{aligned}$$
(46)

From (44) and (46), it follows that

$$\begin{aligned} q(z(k)) \rightarrow \left[ \begin{array}{c} x_1(k) \\ \pi _1(x_1(k)) \\ \end{array} \right] \ \ \text{ exponentially } \text{ as } \ \ \ k \rightarrow \infty \end{aligned}$$
(47)

From (45) and (47), it follows that

$$\begin{aligned} q(z(k)) \rightarrow x(k) = \left[ \begin{array}{c} x_1(k) \\ x_2(k) \\ \end{array} \right] \ \ \text{ exponentially } \text{ as } \ \ \ k \rightarrow \infty \end{aligned}$$
(48)

Thus, the condition (O2) also holds.

This completes the proof.    \(\blacksquare \)

Theorem 8

A necessary condition for the system (27) to be a general exponential observer for a Lyapunov stable plant (26) is that

$$\begin{aligned} \text{ dim }(z) \ge n_1 \end{aligned}$$
(49)

where \(n_1\) denotes the number of critical eigenvalues of the system matrix A.

Proof

If the system matrix A is convergent, then \(n_1 = 0\) and there is nothing to prove.

If A is not convergent, then by the center manifold theory, we have

$$\begin{aligned} q(\pi _2(x_1)) = \left[ \begin{array}{c} x_1 \\ \pi _1(x_1) \\ \end{array} \right] \end{aligned}$$
(50)

For every small \(x_1 \in {\mathbf R}^{n_1}\), we know that the vector

$$ \left[ \begin{array}{c} x_1 \\ \pi _1(x_1) \\ \end{array} \right] $$

has a pre-image \(\pi _2(x_1)\) under the mapping, q. Hence, it is immediate that the dimension of the domain of the mapping, q, cannot be lower than the dimension of the state \(x_1\). This completes the proof.    \(\blacksquare \)

8 Conclusions

This work has studied the problem of constructing general asymptotic and exponential observers for both continuous-time and discrete-time nonlinear systems. First, asymptotic detectability has been shown to be a necessary condition for the construction of general asymptotic observers for nonlinear systems. Next, necessary and sufficient conditions have been derived for the construction of general exponential observers for nonlinear systems. It has been established that the dimension of a general exponential observer cannot be lower than the number of critical eigenvalues of the system linearization matrix of the plant dynamics of a given nonlinear plant.