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This paper proposes a method for determining the stabilizing parameter regions for general delay control systems based on randomized sampling. A delay control system is converted into a unified state-space form. The numerical stability condition is developed and checked for sample points in the parameter space. These points are separated into stable and unstable regions by the decision function obtained from some learning method. The proposed method is very general and applied to a much wider range of systems than the existing methods in the literature. The proposed method is illustrated with examples.

Finding stabilizing regions for control systems in parameter space becomes important in recent years. Stabilizing parameter regions will be instructive for controller tuning with greatest robustness or controller optimization with regard to other specific indexes. Most papers in the literature discuss about the stabilizing parameter regions for proportional-integral-derivative (PID) controllers. Wang et al. [

In this paper, we design a general algorithm for determining stabilizing parameter regions for delay control systems based on randomized sampling. Each unknown parameter is assumed to follow the uniform distribution in a given range and a certain number of independent and identically distributed (i.i.d.) random sample points are generated in the parameter space based on the randomized algorithms [

The rest of this paper is organized as follows. Section 2 presents the idea of proposed method. Section 3 develops the stability criterion. Determining stabilizing parameter regions is discussed in Section 4. Section 5 gives simulation examples and Section 6 concludes the paper.

We consider a unity feedback control system as shown in

We consider the model in [

with a PI controller

where and are unknown parameters. With the method in [

The randomized algorithms have been applied to design robust controllers [

spective range. Then a certain number of i.i.d. random points are sampled in the parameter space. Repeated samples are omitted. According to the randomized algorithms [

where we set a priori as the accuracy parameter and as the confidence level. Both and are usually taken small values, say less than 0.1. We choose and for our example. It could be calculated from (1) that and then we choose. Throughout this paper, is used for all simulation cases.

Next, we check whether each of these points could stabilize the system by some stability criterion. The characteristic equation of the closed-loop system is

We can simply calculate the closed-loop poles for stability testing. If a point of could stabilize the system, it is labeled as “stable”. Otherwise, if a point could not stabilize the system, it is labeled as “unstable”. However, calculating the closed-loop poles is not possible for systems with time delays. In this case, we present a Linear Matrix Inequality (LMI) stability criterion which will be discussed in next section.

Lastly, the points in the parameter space are divided into stable and unstable regions by the decision function obtained from some learning method, such as the Neural Networks and the Support Vector Machines (SVM) [

As stated in previous section, it is impossible to calculate the closed-loop poles for systems with time delays. Therefore, in this section, we present an effective algorithm for stability testing which can be applied to a much wider range of systems. Given a delay system with PI or PID controller, we first convert it into a unified statespace form, which is a generalization of the method in [

Consider a plant:

with a PI controller:

Let

so that

The vector can be viewed as a new state variable of the system, whose dynamics is governed by

where

Let and. Equation (4) can be rewritten as

or

Substituting (5) into (3) yields

where

and

When (2) is with a PID controller

the conversion could not be proceeded. This is because depends on since

. Then the control signal cannot be expressed only by state vectors as (4) or (5). In such a case, we could use a practical D controller:

where is chosen by users to limit derivative gain on higher frequencies. Then, the practical PID controller falls in a format of general dynamic controller, which is handled in Section 3.3 below.

Consider a plant:

with a PID controller:

Let and. We have

and

Denoting, we have

where

Combining (7) and the definition of yields

and

Denoting, , ,

, , and

, we have

Suppose that is invertible. Let

,

, and, where

Then (7) is equivalent to

with

i.e.,

which is also in the form of (6) with and.

Remark 1. The systems (2) and (7) only contain one time delay. However, it would not be difficult to make conversion for systems with multiple time delays, which is omitted here for brevity.

The previous two cases only tackle delay systems with PI or PID controller whose parameters appear in a linear form. In practical control systems, the controllers may be of higher orders and the parameters of controllers may also appear in a nonlinear form, such as the lead-lag compensators [

Consider a plant (9)

under the following dynamic controller:

whose minimal state-space realization can be expressed by

Let and. Denoting

, we have

and

Combining the above expressions gives (10)

and

i.e.,

where

and

Remark 2. The system (6) is a special case of (11).

Theorem 1. The system (11) is asymptotically stable if there exist symmetric positive definite matrices , and, such that

where (13) holds,

and

Here and in the sequel, a block induced by symmetry is denoted by an ellipsis *.

Proof. Define the Lyapunov functional as

The derivative of is

It follows from Jensen’s inequality [

Then we have (14).

Let

and

One sees

By Schur complement, (12) guarantees

Therefore, the system (11) is asymptotically stable.

Each point in the parameter space corresponds to a sample of the parameter vector p, which is denoted by,. We check whether each of these points could stabilize the system by the developed LMI stability criterion. If a point could stabilize the system, it is labeled as “stable”. Otherwise, if could not stabilize the system, it is labeled as “unstable”.

The points in the parameter space can be separated into stable and unstable regions by the decision function obtained from some learning method. In this paper, we choose SVM as the learning method due to its superior performance in a wide range of applications. Support Vector Machines (SVM), which was first introduced by Vapnik [

In this paper, SVM is employed to solve a binary classification problem. Given the data set with, where is a point in the parameter space and (stable) or −1 (unstable) is the label of the point, SVM is to solve the following problem:

where is the Lagrange multiplier, is the penalty parameter which can be set by users and is a mapping from to a higher dimensional space.

There have already been many SVM tool kits that can be used to solve the classification problems. LIBSVM [

In this section, four examples are presented to illustrate the effectiveness of the proposed method.

Example 1. The analytical method in [

with a P controller. This control system is converted to the form in (11) with

Let. Performing our method with the LibSVM arguments “-t” = 2 and “-c” = 100, the stabilizing parameter region is obtained and shown in

Example 2. The graphical method in [

with a P controller. This control system is converted to the form in (11) with

Let. Performing our method with “-t” = 2 and “-c” = 1000, the stabilizing parameter region is obtained and shown in

Example 3. Consider the plant (15) with under the controller

Note that b appears in a nonlinear fashion, which is different from parameters of PID controllers. We can rewrite (16) as

This control system is converted to the form in (11) with

Let. Performing our method with “-t” = 2 and “-c” = 1000, the stabilizing parameter region is obtained and shown in

Example 4. The proposed method also works well with a high-dimensional parameter space. Consider the plant:

with a controller:

This control system is converted to the form in (11) with

Let. Performing our method with “-t” = 2 and “-c” = 1000, the stabilizing parameter region is obtained and shown in

The above examples have well illustrated the effectiveness of the proposed method which can be applied to a much wider range of systems than the existing methods in the literature.

This paper proposes a new and general method for determining the stabilizing parameter regions for delay control systems. We first take a certain number of random sample points in the parameter space. Next, we represent a delay control system in a unified state-space form. Then the numerical stability condition is developed and checked for sample points in the parameter space. These points are divided into two classes according to whether they can stabilize the system. The stabilizing parameter regions could be well defined by the decision function obtained from some learning method. The effectiveness of the proposed method is well illustrated with examples. The proposed method does not have essential constraints and has a wide range of applications. Note that our method could be applied to a higher-dimensional parameter space, though the stabilizing parameter regions are difficult to be shown by graphics.

It should be pointed out that the presented LMI stability criterion is only sufficient since it is based on Lyapunov theory. A sufficient and necessary stability criterion and the additional potential values of the proposed method are to be investigated in future works.