**3. Sums of squares polynomials and state-dependent linear matrix inequalities**

This chapter introduces some definitions and results on SOS polynomials. We also introduce that SDLMIs can be solved by the SOS decomposition.

We begin with the definitions of monomials, polynomials, and sums of squares polynomials.

**Definition 7 (monomials).** Let and . A monomial of *z*, *mα*(*z*), is a function given by

$$m\_{\alpha}(\mathbf{z}) = \prod\_{\ell=1}^{n} \mathbf{z}\_{\ell}^{a\_{\ell}}$$

**Definition 8 (polynomials).** Consider monomials of *z*, , where = (1, 2, …, )∈ℤ+ , and *ci* ∈ ℝ for *i* = 1,…,*m*. A polynomial of *z*, *f*(*z*), is a function given in the form of

$$f(\mathbf{z}) = \sum\_{i=1}^{m} c\_i \, m\_{\alpha\_i}(\mathbf{z}).$$

The degree of polynomial *f*(*z*), *d*, is given by

$$d = \max\_{i} |a\_{i}|.$$

Let ℛ*<sup>n</sup>* denote the set of polynomials of *n* variables. Then, we show the definition of the sums of squares polynomials.

**Definition 9 (sums of squares polynomials, SOSs).** Let *z* = (*z*1,…,*zn*). A sum of squares polynomial *σn*(*z*) is a function given in the form of

$$\sigma\_n(\mathbf{z}) = \sum\_{l=1}^m f\_l(\mathbf{z})^2, \quad f\_l(\mathbf{z}) \in \mathcal{R}\_n, \qquad l = 1, \dots, m.$$

The decomposition of given polynomials into SOSs is called as the SOS decomposition. Regarding the SOS decomposition, the following result is shown.

**Theorem ([2, 3])**. Consider the polynomial of *z* of degree 2*d*, *f*(*z*). The polynomial *f*(*z*) is an SOS polynomial if and only if there exist a column vector *X*(*z*) whose elements are monomials of *z* of degree no greater than *d* and a positive semidefinite matrix *Q* such that

$$f(\mathbf{z}) = X(\mathbf{z})^T Q X(\mathbf{z})$$

holds.

**Definition 5 (positive semidefiniteness).** A function *h:* ℝ*<sup>n</sup>* → ℝ is said to be positive semide-

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

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

**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

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

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

**3. Sums of squares polynomials and state-dependent linear matrix**

This chapter introduces some definitions and results on SOS polynomials. We also introduce

We begin with the definitions of monomials, polynomials, and sums of squares polynomials.

finite if *h*(*x*) ≥ 0 for any *x* ∈ ℝ*<sup>n</sup>*.

322 Nonlinear Systems - Design, Analysis, Estimation and Control

sublevel set

is bounded.

*V*(*x*) satisfies that

**inequalities**

positive definite (respectively, positive semidefinite).

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

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

functions as solutions to the SOS conditions.

that SDLMIs can be solved by the SOS decomposition.

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

We show a simple example of SOSs.

**Example 1.** Consider a polynomial *f*(*z*) given by

$$f(\mathbf{z}) = \mathbf{z}^2 + 2\mathbf{z} + 2\mathbf{z}$$

where *z* ∈ ℝ. Apparently, this polynomial is expressed as the sum of squares polynomial

$$f(\mathbf{z}) = \mathbf{z}^2 + 2\mathbf{z} + \mathbf{2} = (\mathbf{z} + 1)^2 + 1\mathbf{z}$$

Regarding Theorem 2, the polynomial is also expressed as

$$f(\mathbf{z}) = \begin{bmatrix} \mathbf{z} & \mathbf{1} \end{bmatrix} \begin{bmatrix} \mathbf{1} & \mathbf{1} \\ \mathbf{1} & \mathbf{2} \end{bmatrix} \begin{bmatrix} \mathbf{z} \\ \mathbf{1} \end{bmatrix} . \tag{2}$$

and the matrix in the right-hand side of (2) is positive definite.

The SOS decomposition can be solved by some numerical solvers, such as YALMIP [18] and SOSTOOLS [19]. When some coefficients of polynomials are decision variables in an SOS decomposition, by using the numerical solvers, we can find the feasible solutions such that the SOS decomposition holds. Therefore, we can adapt the SOS decomposition to the design of feedback laws in control problems.

With the relation to the stability theory presented in Section 2, the sufficient condition of the stability is given as the SOS conditions.

**Theorem 3.** [2] Consider system (1). If there exist a positive definite function and an SOS polynomial , such that

$$\begin{aligned} \phi(\mathbf{x}) - \epsilon(\mathbf{x}) &> 0, \\ \frac{\partial \phi}{\partial \mathbf{x}}(\mathbf{x}) f(\mathbf{x}) + \epsilon(\mathbf{x}) &< 0, \qquad \forall \; \mathbf{x} \neq \mathbf{0} \end{aligned}$$

then the equilibrium *x* = 0 is asymptotically stable.

Theorem 3 shows a direct application of the SOSs to the analysis of the stability. This implies that the SOS decomposition can be applied to the synthesis of the stabilizing feedback laws. This chapter develops a method to design dynamic output feedback laws based on the SDLMI approach [8]. The SDLMI is defined as the optimization problem:

$$\begin{aligned} \text{minimize } & \sum\_{i=1}^{m} a\_i \, c\_i\\ \text{subject to } & F\_0(\mathbf{z}) + \sum\_{i=1}^{m} c\_i \, F\_i(\mathbf{z}) \ge 0, \qquad \text{for } \forall \, \mathbf{z} \in \mathbb{R}^n, \end{aligned}$$

where *ai* ∈ ℝ are the fixed coefficients, *ci* are the decision variables, the matrix functions *Fi* : are state-dependent symmetric matrices. The constraint should be satisfied for any *z* ∈ ℝ*<sup>n</sup>*. This differs from standard LMIs and is the derivation of the word, state-dependent.

A relation of the SDLMIs and the SOS decompositions is shown as follows.

**Theorem 4.** ([8]) Let *d* > 0 and *F*: a symmetric polynomial matrix the elements of which are polynomials of *z* with degree 2*d*. Moreover, consider a vector *v* ∈ . If *vT F*(*z*)*v* is a sum of squares polynomial, then *F*(*z*) ≥ 0 holds for any *z* ∈ ℝ*<sup>q</sup>* .

Theorem 4 states that if we find that the polynomial *vT F*(*z*)*v* is decomposed into an SOS with respect to (*z,v*), it implies the positive definiteness of *F*(*z*) for any *z* ∈ ℝ*<sup>n</sup>*. We can derive stability conditions in terms of SDLMIs. This leads to the design of feedback laws for the stabilization based on the combination of the SDLMIs and the SOS decomposition. We develop the synthesis of dynamic output feedback laws based on Theorem 4 in the following sections.
