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The sign stability concept in ecological systems is introduced into the analysis and synthesis of switched linear system to explore new control design technique. The necessary and sufficient condition for sign stability of a switched linear system under arbitrary switching is achieved via the notion of complete isogenous sign stable set (CISSS). A new approach for the stabilization of switched system is presented. Although the controllers are devised for each subsystem, respectively, the switched system is sign stabilized by the constitution of CISSS. The provided method has natural robustness and more design freedoms than the familiar Lyapunov function method, which bears relative conservativeness as the requirement of solving LMIs. The presented technique is validated by an example of flight control within a large-scale flight envelop. Simulation results indicate that the proposed method can stabilize the flight attitude under large variations of system parameters and external perturbations.

The sign stability (or qualitative stability) concept is first proposed in bionomics and utilized to analyze the interactions of different species in a large-scale ecosystem which lacks exact model but presents high robustness under various perturbations [

For a given matrix, the signs

Based on theoretical researches, the sign stability technique is applied in control engineering, especially the robust controllers design for aerospace flight control system. A three-axis attitude stabilization controller for an axisymmetric satellite is provided in [

However, the existing applications of sign stability only consider linear time-invariant systems which cannot approximate the flight dynamics within the full flight envelope. One of the alternative solutions is the switched linear system which is a good approximation of complicated system characteristics. Flight-control-oriented analysis and synthesis of switched systems are studied in the prior literatures and show large potential in engineering practice [

Compared to the prior references, the significant contribution of this work is the conclusion on sign stability analysis of switched linear system, which is presented by a necessary and sufficient condition. The system model treated with sign stability approach is extended from LTI system to a hybrid system. The mix of continuous linear systems and discrete signals make switched system suitable to describe a wide range of engineering systems such as power systems, automotive engine control, flight control, and networked control systems. As a result, the application areas of sign stability approach are enlarged remarkably.

Another important contribution is that a new stabilization technique of switched system is presented. As for the asymptotic stability of switched system under arbitrary switching, the common Lyapunov function is a necessary and sufficient condition and a usual method. In [

With this consideration, the paper is organized as follows. In Section

In this section, preliminary knowledge about the sign stability is reviewed, and then the definition of sign stability for switched linear system is proposed to formulate the main problem of this paper.

A switched linear autonomous system is usually described as

For the

The

The necessary and sufficient condition for the sign stability of matrix is provided in referenced researches [

The following are necessary conditions for sign stability of matrix

For all

For all

It is worth announcing that the color test, given by Jeffries in [

Compared with the definition of sign stability for matrix, sign stability of switched linear system is defined as below.

The switched linear system (

From Definition

In a sign stable switched system, there may be several subsystems of which the matrices have the same sign-pattern. In other words, the switched system derived from a given sign stable sign-pattern set is stable even if more than one matrices are built according to the same sign-pattern.

The subset of an arbitrary sign stable switched system (or sign-pattern set) is also sign stable. The conclusion is obtained under arbitrary switching laws.

With the concept given in Definition

The following statements are equivalent:

the switched linear system

where

the linear time-variant system

From Definition

The following are necessary conditions for sign stability of switched linear system (

For all

For all

For (1), noting that each subsystem of switched system (

For (2), with no loss of generality, it is supposed that there are 2 subsystems in switched system (

It is easily understood that

Due to condition (3) of Lemma

(i) For the situation that

(ii) For the situation that

That is,

Considering that all the elements permit arbitrary sign-preserving variations, let

From condition (3) of Lemma

That is,

To ensure (

Synthesizing situations (i) and (ii), condition (2) is proved.

It can be concluded by Theorem

A sign-pattern set is called an isogenous set if the elements in the same position of all the sign-patterns satisfy one of the following terms: (i) they have the same sign (including 0); (ii) they are all 0; (iii) the elements in the opposite position are all 0.

The sign-pattern set of a sign stable switched system is an isogenous set.

We also say that two sign-patterns are isogenous with each other if they conform to the condition of Definition

Considering an arbitrary isogenous sign-pattern set, we define the joining operation of all the sign-patterns to develop a special sign-pattern named the original sign-pattern of the set. The joining operation should keep the following rules.

If elements in a certain position of all the sign-patterns have the same nonzero sign or 0, then the corresponding element of the original sign-pattern is the same nonzero sign.

If elements in a certain position of all the sign-patterns are all 0, then the corresponding element of the original sign-pattern is 0.

If elements in a certain position of all the sign-patterns have different signs, then the corresponding element of the original sign-pattern is written as

The original sign-pattern obtained from a known isogenous set may be a new sign-pattern or an existent one which is already in the set. The original sign-pattern of set (

Furthermore, it is obvious that the new original sign-pattern and the known isogenous set can be joined to build a new set which is still isogenous. Actually, new sign-patterns can be derived to extend the isogenous set by replacing some nonzero elements of the original sign-pattern by 0. To demonstrate the aforementioned operation, the set which is developed from (

Of course, we can continue to add new isogenous sign-patterns of (

A sign-pattern set is named a complete isogenous sign stable set (CISSS) if it contains a sign stable original sign-pattern and all of the derivative isogenous sign-patterns that are sign stable.

Definition

It also should be emphasized that the selection of the uncertain element

In what follows, the main result of this work is proposed.

The necessary and sufficient condition for sign stability of switched linear system (

(1) The necessity can be easily proved by Corollary

(2) If the sign-pattern set

According to Definition

With Theorem

Two simple examples are given to illuminate the constitution of CISSS.

For

It has been summarized that the original sign-pattern is needed to have as few “0” elements as possible. Hence, all the diagonal elements are chosen to be “−,” and the off-diagonal elements are selected to possess as many pairs of opposite signs as possible. For

Consider the

Utilizing the criterion in [

In fact, in the above process of finding CISSS, the two pairs of “0 and

The controller design problem for the highly maneuverable technology (HiMAT) vehicle is presented to demonstrate the main results in this paper. As declared in [

Partial flight envelope and operating points.

The linearization model for each operating point is regarded as a subsystem of the switched system. The switched system is then given by

The system matrices and control matrices for several operating points are shown to illustrate the sign-patterns:

Furthermore, the sign-pattern set of the open-loop system matrices is given by the following:

As explained in Section

Although the complete form of the CISSS derived from (

It can be seen that the closed-loop system matrices

The simulation across different regions of the flight envelope is taken to validate the proposed technique. The flight trajectory travels through vicinities of 7 operating points (

Switching law of the flight trajectory.

To validate the stability within the full flight envelope, gust disturbance is considered in the form of perturbations acted on the angle of attack, which is a group of pulse signals with a period of 10 seconds, a width of 1 second, and a magnitude of 3 degrees. The initial values of the angle of attack, pitch rate, and pitch angle are set to 3 degrees, 3 degrees per second, and 5 degrees, respectively. To illustrate the robustness of the proposed approach, multiplicative perturbations on the closed-loop subsystem matrices are introduced. Each element in the matrices is multiplied by an arbitrary positive value, which will preserve all the signs of the products. The perturbation matrices are given by the following:

The resulted closed-loop subsystem matrices are obtained by multiplying each element of the original matrix with the corresponding element in the perturbation matrix. Thus, it leads to 3 closed-loop switched systems with different multiplicative perturbations. The time histories of the system states and control surface deflections are depicted in Figure

Time histories of system states and control inputs under perturbations: (a) angle of attack, (b) pitch rate, (c) pitch angle, (d) elevator, (e) elevon, (f) canard, and (g) symmetric aileron.

As depicted in Figure

A new concept of sign stable switched linear system is established to develop novel techniques for flight control. The main result is provided in the form of necessary and sufficient condition for the sign stability of switched system under arbitrary switching. Hence, the application areas of sign stability approach are enlarged remarkably. A new robust stabilization approach for switched linear system is proposed via the notion of CISSS. The sign stabilization controllers are devised for each subsystem, respectively; therefore the design process possesses more freedoms. Compared with the traditional Lyapunov function method, the proposed technique has natural robustness and decreased conservativeness. The aforementioned points are verified by a stabilization control problem of aircraft with a large-scale flight envelope.

The authors declare that there is no conflict of interests regarding the publication of this paper.

This work is supported by the National Natural Science Foundation of China (61273083).