A general method for model-order reduction of switched linear dynamical systems is presented. The proposed technique uses convex generalized gramian which is a convex combination of the generalized gramians. It is shown that different classical reduction methods can be developed into the generalized gramian framework for model reduction of linear systems and further for the reduction of switched systems by construction of the convex generalized gramian. Balanced reduction within specified frequency bound is taken as an example which is developed within this framework. In order to avoid numerical instability and also to increase the numerical efficiency, convex generalized gramian-based Petrov-Galerkin projection is constructed instead of the similarity transform approach for reduction. It is proven that the method preserves the stability of the original switched system at least for stabilizing switching signal and it is also less conservative than the method which is based on the common generalized gramian. Some discussions on the coefficient of the vertices of the convex variables are presented. The performance of the proposed method is illustrated by numerical examples.

The highly complicated models are the response to the ever-increasing need for accurate mathematical modeling of physical as well as artificial processes for simulation and control. This problem demands efficient automatic computational tools to replace such complex models by an approximate simpler models, which are capable of capturing dynamical behavior and preserving essential properties of the complex one, either the complexity appears as high-order describing dynamical system or complex nonlinear structure. Due to this fact model reduction methods have become increasingly popular over the last two decades [

Most of the studies related to model reduction presented so far have been devoted to linear case and just few methods have been proposed for nonlinear cases which are not strong comparing to linear reduction methods.

On the other hand, most of the methods that are proposed so far for control and analysis in hybrid and switched systems theory are suffering from high computational burden when dealing with large-scale dynamical systems. Because of the weakness of standard model reduction techniques in dealing directly with hybrid structure without sacrificing essential features and also pressing needs for efficient analysis and control of large-scale dynamical hybrid and switched systems, it is necessary to study model reduction of hybrid and switched systems in particular. This fact has motivated the researchers in hybrid systems to study model reduction [

The model reduction problem for switched systems of Markovian type was studied in [

In [

In this paper we propose convex generalized gramian-based framework for model reduction of switched system. This general framework can be categorized as gramian-based model reduction methods. Balanced model reduction is one of the most common gramian-based model reduction schemes. It was presented in [

To apply balanced reduction, first the system is represented in a basis where the states which are difficult to reach are simultaneously difficult to observe. This is achieved by simultaneously diagonalizing the reachability and the observability gramians, which are solutions to the reachability and the observability Lyapunov equations. Then, the reduced model is obtained by truncating the states which have this property. Balanced model reduction method is modified and developed from different viewpoints [

In this paper we first show that the generalized method in [

The paper is organized as follows. In the next section we review balanced reduction method and balanced reduction technique based on the generalized gramian. Section

The notation used in this paper is as follows.

Balanced truncation is a well-known method for model reduction of dynamical systems; see, for example, [

The reduced model can be easily obtained by truncating the states which are associated with the set of the least Hankel singular values. Applying the method to stable, minimal

In this section we present a general framework to build generalized gramian version of gramian-based methods. Then we present generalized balanced reduction within frequency bound within this framework following by some words about numerical implementation of the algorithm based on projection.

Suppose that

If

This lemma leads to the following proposition that makes the relation between Lyapunov equations and Lyapunov inequalities evident.

Suppose

It can be proven easily by subtracting (

Proposition

It is possible to propose generalized version of other gramian-based reduction methods in this framework. The only step that we need to take is to derive associated Lyapunov equations and their Lyapunov inequalities. In the following we propose generalized version of balanced reduction within frequency bound.

Over the past two decades, a great deal of attention has been devoted to balanced model reduction and it has been developed and improved from different viewpoints. Frequency weighted balanced reduction method is one of the devised gramian-based techniques based on ordinary balanced truncation [

Note that since

Balanced transformation can be ill-conditioned numerically when dealing with the systems with some nearly uncontrollable modes or some nearly unobservable modes. Difficulties associated with computation of the required balanced transformation in [

One of the most important subclasses of hybrid systems are linear switched systems. Linear switched system is a dynamical system specified by the following equations:

In this section we build a framework for model reduction of switched system described by (

Petrov-Galerkin projection for a dynamical system

The reduced-order model using this projection is:

In our framework we construct the aforementioned projection based on the convex generalized gramian which is defined as follows.

Convex controllability (observability) generalized gramian for the dynamical system (

One easy way to develop generalized gramian framework to model reduction of switched linear system is to apply the method locally on each subsystem independently, in other words, to reduce each subsystem by generalized gramian reduction method independently. Independent reduction of subsystems poses an extra-computational burden for construction of the independent projection matrices for each subsystem. Therefore it is preferable to construct single projection which is capable of reduction of all subsystems in one shot. Due to this fact, we introduce convex generalized gramian. Building the projection based on the convex generalized gramian enables us to reduce all subsystem in one shot and reduces the extra computational burden which the methods based on independent reduction of subsystems like the one in [

At this point it is possible to develop different gramian-based reduction methods into this framework for reduction of switched system finding generalized controllability/observability gramian for each subsystem, constructing convex controllability/observability generalized gramian. The next step can be simultaneous diagonalization of the convex generalized gramian and balancing and reduction of all subsystems based on Hankel singular values of the convex generalized gramian. In order to avoid numerical bad conditioning and also to increase the efficiency we use Schur or square root algorithm instead of balancing and directly Petrov-Galerkin projection matrices can be computed. This procedure is less conservative and provides more accurate results.

In the method that we proposed in [

The matrix pencil

In general this is the convex hull of the family of matrices which is defined as

The procedure is almost the same as what we mentioned before. The only deference is that we restrict one of the convex gramians to satisfy

In order to clarify the method we extend generalized balanced reduction within frequency bound that is presented in previous section, for model reduction of switched linear system.

First, we need to find the generalized controllability gramian

For example in the case of bimodal systems,

The convex controllability gramian within

In (

If stability preservation is of concern we have to choose

If we plug

One of issues in model reduction is preservation of the stability which needs to be studied. We need to recall two stability results in Theorems

Switched bimodal dynamical system (

Switched dynamical system (

The proofs for these theorems are by construction, in other words in the proofs the switching signal for which the switched system is stable are constructed based on

Consider

If

In the proposed method, we have

On the other hand,

Hence

This proposition along with the Theorems

In the particular scenarios the stability of the original switched system is guaranteed to be preserved under arbitrary switching signal. This is shown in Proposition

The Convex Generalized Gramian framework is stability preserving under arbitrary switching signal if

We have

Similarly

Assume that (

Hence

In stability theory for switched system it is well-known sufficient condition for quadratic stability [

In the case that (

We have

Some research has been focused on conditions for finding

Let

In Proposition

For all

Therefore the sufficient condition for the stability of

Our framework is said to be feasible if (

In this section we have applied the proposed method for reduction of two bimodal switched linear systems. The first example is of order 5 and the second one is of order 25.

Consider a single-input-single output switched linear of the form(

In order to reduce the switched system first we construct convex gramians over the frequency domain

Generalized Hankel Singular Values (

Randomly generated switching signal.

Step response of original switched linear system (solid line) and the reduced-order model (dotted).

Figure

Consider bimodal switched linear system of order 25. The original system is SISO and it is reduced to 14, 17, 18, and 19 using the proposed reduction method over

The generalized Hankel singular values are shown in Figure

Generalized Hankel Singular Values (

The step responses of the original and reduced-order switched systems associated to the switching signal of Figure

Switching signal.

Step response of original switched linear system (solid line) and the reduced-order model which is of order 19 (dotted).

Step response of original switched linear system (solid line) and the reduced-order model which is of order 18 (dotted).

Step response of original switched linear system (solid line) and the reduced-order model which is of order 17 (dotted).

Step response of original switched linear system (solid line) and the reduced-order model which is of order 14 (dotted).

In this example also we represent the infinity norm of transfer function of the original subsystems and the reduced counterpart which is of order 19 in Figures

The infinity norm of original of transfer matrix of first subsystem (solid line) and its reduced-order counterpart of order 19 (dotted) over frequency domain

The infinity norm of original of transfer matrix of second subsystem (solid line) and its reduced-order counterpart of order 19 (dotted) over frequency domain

A general framework for model order reduction of switched linear dynamical systems has been presented. In this paper we have reformulated the frequency domain balanced reduction method into this scheme but generally various gramian-based reduction methods can be reformulated in the proposed generalized method easily and can be applied for reduction of switched system. The stability issue has been studied in the paper. The method provides single projectors for all subsystems which enable us to reduce all of the subsystems in one step. It is less conservative than the previous method based on common generalized gramian. The method is dependent to selection of parameters. This opens a window toward further modifications in optimization framework.