Abstract
We consider the stabilization of nonlinear polynomial systems and the design of dynamic output feedback laws based on the sums of squares (SOSs) decompositions. To design the dynamic output feedback laws, we show the design conditions in terms of the state-dependent linear matrix inequalities (SDLMIs). Because the feasible solutions of the SDLMIs are found by the SOS decomposition, we can obtain the dynamic output feedback laws by using numerical solvers. We show numerical examples of the design of dynamic output feedback laws.
Keywords
- sums of squares polynomials
- output feedback stabilization
- Lyapunov methods
- state-dependent LMIs
1. Introduction
In the last few decades, control design methods based on numerical methods have appeared in the control literature. Major progress in the 1980s was the emergence of numerical methods based on linear matrix inequalities (LMIs) [1]. The methods provide the numerical solutions to linear control problems in the formulation of the semidefinite programming. The LMI approach provides the design methods of feedback laws for the asymptotic stabilization, H-infinity control, and robust control. For the nonlinear control problems, the sums of squares (SOS) approach is introduced as a generalization of the LMI approach to nonlinear systems [2–6]. A feature of the sums of squares polynomials is negative semidefiniteness, and this is suitable for the stability analysis of nonlinear systems based on the Lyapunov theory. The studies [2, 3] have shown that the sums of squares decomposition can be solved in the formulation of the semidefinite programming. The result leads to the development of 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 systems are found in reference [11, 12].
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 closed-loop systems.
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 laws have been desired.
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 not been shown in this direction yet.
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.
2. Preliminary: stability of nonlinear systems
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
where
To begin with, we show the definitions of the stability.
To introduce the Lyapunov stability theory, we provide the definitions of the properties of functions.
We say that a function
is bounded.
The Lyapunov stability theory is stated as follows [17].
Moreover, the equilibrium of system (1),
When
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.
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.
The degree of polynomial
Let
The decomposition of given polynomials into SOSs is called as the SOS decomposition. Regarding the SOS decomposition, the following result is shown.
holds.
We show a simple example of SOSs.
where
Regarding Theorem 2, the polynomial is also expressed as
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.
then the equilibrium
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:
where
A relation of the SDLMIs and the SOS decompositions is shown as follows.
Theorem 4 states that if we find that the polynomial
4. Problem setting: stabilization through dynamic output feedback
This chapter considers the stabilization problem via dynamic output feedback laws and the synthesis of the stabilizing feedback laws. This section states the problem setting.
The approach presented here is based on the SDLMI approach, which derives the sufficient conditions of the existence of stabilizing feedback laws as the SDLMI conditions. We can obtain stabilizing feedback control laws and Lyapunov functions by solving the SDLMI conditions using numerical solvers.
Consider a nonlinear system given as
where
where
where
We have the closed-loop system of (4) with the dynamic output feedback law (5), given by
We consider the stabilization of the closed-loop system (6). To this end, we give a method to design the matrix functions
by choosing
where the matrices
5. Design of dynamic output feedback laws through SOSs
This section provides a design method of dynamic feedback laws (5) for the output stabilization of system (4). We show stability conditions of the closed-loop system of (6) as SDLMI conditions. We can obtain the stabilizing laws by solving the SDLMI conditions via SOS decomposition using numerical solvers.
The main idea of the proposed method is as follows. Instead of the dynamic feedback law (5), assume that there exists a static state feedback law
where
becomes the Lyapunov function of the closed-loop system (6) with some positive definite matrix
In the following, if the matrix
As discussed above, we design a stabilizing state feedback law as the first step. The state feedback law also can be designed by using SDLMIs. We introduce the following result shown in reference [8].
are SOS polynomials, where
Then, the origin of (4) is asymptotically stabilized by a state feedback given by
For the design of the output feedback laws, we show the following theorem as the main result, which gives a design condition of the feedback law (5) in terms of state-dependent matrix inequalities.
are SOS polynomials, where
is an SOS polynomial where
where the matrices
is the Lyapunov function of the closed-loop system of (4) with the state feedback law
Then, to consider a dynamic output feedback law in the form of (5), we consider a function given by
where the function
Then, the time derivative of function (14) along the trajectory of the closed-loop system (6) is given as
where
and
Therefore, the time derivative of the function
Then, condition (13) of the theorem and Theorem 4 imply that
From (17) and (18), we can conclude that
When we design the dynamic output feedback law (5) according to Theorem 6, we first solve the SOS decomposition of condition (12) to find the matrix
6. Numerical examples of dynamic output feedback stabilization
6.1. Numerical example 1
This section shows some numerical examples of the dynamic output feedback stabilization by the proposed method shown in Section 5.
We show the first example of the stabilization. Consider a system given by
where
We consider the output feedback stabilization of system (19) using the dynamic feedback law (5). We consider a low-dimensional dynamic feedback, and we assume that
Then, by using
Figure 1 shows the time responses of the state variables
Then, we also obtain a dynamic output feedback control law in the case where the elements of
We also obtain the values of
The obtained feedback control law also stabilizes system (19). Figure 2 shows the time responses of the state
6.2. Numerical example 2
We consider the following example, which models a circuit with negative-resistance oscillator, taken from reference [17] and modified. Consider a system given by
where
Following the design procedure in the previous section, we design the dynamic feedback control law with
Then, we solve the SOS decomposition (13) to find the matrices
Figure 3 shows the time responses of the states
7. Conclusion
We considered the design of dynamic output feedback laws via the SOS decomposition. For the design of the feedback laws, we derived the design conditions as the state-dependent matrix inequalities. According to the derived conditions, we can design the stabilizing feedback laws as the feasible solutions to the SDLMIs by using the numerical solvers. Future works include to derive less conservative conditions and to develop design methods of dynamic output feedback laws for advanced control, such as H-infinity control.
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