By Shiming Duan,  Last Update: 8/13/2010 5:09 PM

 

VI.         Robust Control and Time-Domain Specification via Eigenvalue Assignment

            In Chapter 5, the method for eigenvalue assignment via the Lambert W function was introduced. In practice, systems are usually subject to various kinds of uncertainties, which may come from estimation errors, approximation errors or modeling errors. To ensure the performance of the design for such perturbed systems, it is important to consider the effects of these uncertainties. In this chapter, a new approach based on the Lambert W function is proposed to solve the robust control problem for systems of DDEs. Furthermore, a method based on the analysis of rightmost eigenvalues is proposed to help the design of feedback control systems achieve desired time-domain specification.

 

Stability Radius

 

To calculate the stability radius for a perturbed time delay system

one can follow the following procedure:

 

Step 1: Calculate

Step 2: Calculate  where  and  are the largest and smallest singular values of

Step 3: Find the maximum value of   over the interval , where 

 

Since , the maximum value find in Step 3 will be the supremum value of  

which gives the stability radius of the system with uncertainty.

 

Robust Feedback Control Design

 

Step 1: Compute the stability radius, , of the given system , where  denotes the largest singular value of

 

Step 2: Apply the method introduced in Chapter 5 to select feedback gains

 

Step 3: Compute a theoretical stability radius, r2, of the closed-loop system using the procedure introduced above

 

Step 4: If r1 > r2, then system may be destabilized by the uncertainty; go back to step 2 and select a more conservative gain (the gain that moves the rightmost poles farther to the left in the s-plane)

 

 

Design Based on Time-Domain Specifications

 

Using the Lambert W function approach, one can not only assign the real parts of the rightmost eigenvalues, but also their imaginary parts. It is not feasible to assign the entire eigenspectrum by using a finite number of control parameters. However, since rightmost poles usually dominate the transient responses of the systems of DDEs, design based on rightmost eigenvalues will often provide a good approximation.

 

For ideal second-order system of ODEs, the time-domain specifications can be calculated as

 

For systems of DDEs, although there are infinite number of eigenvalues, approximating these specifications from rightmost eigenvalues still provides useful information for design.

 

 

Example 6.1 Nonlinear Hydraulic System (Robust Feedback Control)

Consider the hydraulic water tank system with the nonlinear model

and the parameters have the same values as in Ex. 5.1. In this example, a feedback control for this nonlinear system will be designed to keep the level of water at x0=0.5m. Compared to the linear model used in Ex. 5.1, this nonlinear model has both nonlinear terms and delay terms. Here we will demonstrate how to apply the Lambert W function approach to solve a robust feedback control problem.

 

First, note that and linearize the system about the equilibrium point (x0, u0):

 

where  and  represents the approximation error from the Taylor expansion.  Assume that the depth of the tank is 1m and the level of the water surface is measurable. Therefore, with , the closed-loop system becomes

 

 

The approximation error for  () is shown in Fig.6.1. Thus, one can obtain the bounds  for the figure.

 

Fig.6.1 Uncertainty raised due to approximation error

 

First, one can easily obtain . Then, choose the gain K to place the rightmost eigenvalue. A list of gains and corresponding rightmost eigenvalues is listed in the following table. In Fig. 6.2, it can be seen that the stability radius increases as the eigenvalue moves left. Since , we can place the rightmost eigenvalue at -2 with a stability radius of  such that the system will remain stable despite the uncertainty.

 

Table 6.1 The gains K corresponding to each rightmost eigenvalues

Rightmost Eigenvalues

−0.0009

−0.0011

−0.0013

−0.0015

−0.0017

−0.00019

K

 

Fig.6.2 The rightmost eigenvalues vs. the stability radius of the system in Ex.6.1

 

 

 

 

Example 6.2 Pendulum System (Design based on Time-Domain Specification)

Consider again the eigenvalue assignment problem for the pendulum system in Ex. 5.2 .

where  and

In Ex. 5.2, we designed a full state feedback control and assigned the rightmost poles to be . The selection of the rightmost eigenvalues is surely not unique. In this example, we will find the relationship between the performance of the system and the location of rightmost eigenvalues.

 

Follow the same procedure introduced as in Ex.5.2 to place the right poles at  and  with different feedback gains. As shown in the Table 6.2, although the three pairs of poles considered have the same real parts, the transient responses can be quite different depending on the selection of the imaginary parts. Therefore, one can assign the rightmost eigenvalues based on the estimated performance first and validate the actual performance by simulation afterwards (see Fig.6.3). Although analysis based on rightmost eigenvalues only provides an approximation, the method gives useful guidelines for design.

 

Table 6.2  The rightmost eigenvalues with corresponding time-domain specifications

Rightmost Eigenvalues

K

1.5

2

2.5

1.80

2.24

2.69

0.555

0.447

0.371

(s)

0.999

0.805

0.669

(s)

4.6

4.6

4.6

(s)

2.094

1.571

1.257

(%)

12

21

28

 

Fig.6.3 Responses of the systems with feedback control corresponding to the rightmost eigenvalues in Table 6.2