**2. Preliminary: stability of nonlinear systems**

numerical methods for the analysis and synthesis of nonlinear polynomial systems. Applications to control problems are feedback design [7, 8], motion planning [9], modeling, and control of fuzzy systems [10] to mention a few. Applications of the SOS approach to nonpolynomial

The SOS approach has been the basis of numerical methods for the analysis and the synthesis of nonlinear systems. Although the Lyapunov-based approach offers the methods for the analysis and the synthesis, the construction of Lyapunov functions is often a difficult task. The SOS approach provides a technique to find Lyapunov functions by formulating the Lyapunov inequality conditions into the SOS conditions. The stability of nonlinear systems is analyzed by a direct application of SOS decompositions to the Lyapunov stability analysis. However, applications of the SOS approach to Lyapunov-based feedback design are much complicated because decision variables do not enter the Lyapunov inequalities conditions linearly. So far, two main approaches have been proposed. One is a method in [8], which formulates the design conditions into state-dependent linear matrix inequalities (SDLMIs) conditions. The SDLMIs are solved by the SOS decompositions. The other method is based on an iterative algorithm shown in reference [7], which also considers the enlargement of the regions of attraction of the

In the actual control problems, we often cannot measure all the values of the state variables of control systems. This fact leads to the necessity of the design of output feedback laws. The design of output feedback laws is more complicated task than that of state feedback laws because the stability conditions of the closed-loop systems become complex. As far as the authors know, so far, a few output feedback design methods have been proposed, for example, [[7], Section 3.5] and [13–15]. The further developments of design methods for output feedback

It is well known that we often can design dynamic feedback laws even when the design of static output feedback laws is difficult. This leads to the motivation of developing a design method based on the SOS approach for the design of dynamic output feedback laws. In reference [7], an iterative method for the design of dynamic output feedback laws has been shown. However, we need to give control Lyapunov functions (CLFs) to start the iteration in the method, and this might be a difficult task especially for complex or high-dimensional systems. The state-dependent LMI approach can be an alternative approach because it does not need to give any CLF. However, a concrete method for dynamic output feedback laws has

We provide the design methods of dynamic output feedback laws for the stabilization based on the SDLMI approach. This method is based on the design method of state feedback laws based on the SDLMI approach [8]. The proposed method employs a two-step algorithm. We first design a virtual state feedback law for a given system using the method of reference [8]. Then, we design a dynamic output feedback by using an SDLMI again based on the virtual state feedback law. The use of the virtual state feedback inherits the design approach of output feedback laws in reference [16], which indicates the general design approach of output feedback laws not necessarily for the SOS approach. We also show some numerical examples to demonstrate the effectiveness of the proposed method to the actual control problems.

systems are found in reference [11, 12].

320 Nonlinear Systems - Design, Analysis, Estimation and Control

closed-loop systems.

laws have been desired.

not been shown in this direction yet.

This section provides the stability theory of nonlinear systems. We present the definitions of stability, and then, we introduce the Lyapunov stability theory. The Lyapunov stability theory forms the basis for the analysis and synthesis of the stability of dynamical systems. The theory states that the existence of a kind of functions implies the stability.

This section considers the stability of an autonomous nonlinear system

$$
\dot{\mathbf{x}} = f(\mathbf{x}), \quad \mathbf{x}(\mathbf{t}\_0) = \mathbf{x}\_0 \tag{1}
$$

where *x*∈ℝ*<sup>n</sup>* is the state, *f*:ℝ*<sup>n</sup>* → ℝ*<sup>n</sup>* is the vector fields, and *x*<sup>0</sup> ∈ ℝ*<sup>n</sup>* is the initial value of the state. In the following, we assume that the origin *x* = 0 is the equilibrium of system (1), that is, *f*(0) = 0, and we consider the stability of the origin.

To begin with, we show the definitions of the stability.

**Definition 1 (stability).** The equilibrium *x* = 0 is said to be Lyapunov stable if for any > 0, there exists *δ* = *δ*( ) > 0, such that for any *‖x*0‖ < *δ*, the solution *x*(*t*) of (1) satisfies that

$$\|\ x(t) \| < \epsilon, \quad \forall \ t \in [t\_0, \infty).$$

**Definition 2 (asymptotic stability).** The equilibrium *x* = 0 is said to be asymptotically stable if it is stable and there exists *δ* > 0, such that for any ‖*x*0‖ < *δ*, the solution of (1) satisfies that

$$\mathbf{x}(t) \to \mathbf{0} \text{ as } t \to \infty.$$

**Definition 3 (global asymptotic stability**). The equilibrium *x* = 0 is said to be globally asymptotically stable if it is stable and for any *x*<sup>0</sup> ∈ ℝ*<sup>n</sup>*, the solution *x*(*t*) of (1) satisfies that

$$\mathbf{x}(t) \to \mathbf{0} \text{ as } t \to \infty.$$

To introduce the Lyapunov stability theory, we provide the definitions of the properties of functions.

**Definition 4 (positive definiteness).** A function *h:* ℝ*<sup>n</sup>* → ℝ is said to be positive definite if *h*(*x*) > 0 for any *x* ≠ 0 and *h*(0) = 0.

**Definition 5 (positive semidefiniteness).** A function *h:* ℝ*<sup>n</sup>* → ℝ is said to be positive semidefinite if *h*(*x*) ≥ 0 for any *x* ∈ ℝ*<sup>n</sup>*.

We say that a function *h*(*x*) is negative definite (negative semidefinite) if the function −*h*(*x*) is positive definite (respectively, positive semidefinite).

**Definition 6 (properness).** A function *h:* ℝ*<sup>n</sup>* → ℝ is said to be proper if for any *K* ∈ ℝ, the sublevel set

$$\{x \in \mathbb{R}^n \mid h(x) \le \kappa\}$$

is bounded.

The Lyapunov stability theory is stated as follows [17].

**Theorem 1.** Let *U* be an open subset of ℝ*<sup>n</sup>* which contains the origin. Suppose that a function *V*:*U* → ℝ is continuously differentiable, positive definite, and proper. The equilibrium of system (1), *x* = 0, is stable if and only if the function *V*(*x*) satisfies that

$$\frac{dV}{dt}(\chi) = \frac{\partial V}{\partial \chi}(\chi) f(\chi) \le 0, \ \forall \ x \in \mathbb{R}^n.$$

Moreover, the equilibrium of system (1), *x* = 0, is asymptotically stable if and only if the function *V*(*x*) satisfies that

$$\frac{dV}{dt}(\mathbf{x}) = \frac{\partial V}{\partial \mathbf{x}}(\mathbf{x}) f(\mathbf{x}) < 0, \ \forall \ \mathbf{x} \neq \mathbf{0}.$$

When *U* = ℝ*<sup>n</sup>*, the global asymptotic stability holds.

The Lyapunov theory is used to investigate the stability of nonlinear systems. However, to investigate the stability of each system by Lyapunov theory, we need to find a Lyapunov function for it. However, to find the Lyapunov functions is often a difficult task. Further, when we try to design stabilizing feedback laws based on the Lyapunov theory, we also need to find the Lyapunov function candidates for the closed-loop systems. Therefore, we require a method to find Lyapunov functions for each nonlinear system. The SOS approach provides Lyapunov functions as solutions to the SOS conditions.
