A method for the design of robust controllers ensuring the quadratic stability for switching systems

Abstract

In recent years, it has been remarkable to see the increasing number of studies related to the theory and application of fractional-order controllers, especially PI^λD^μ controllers, in many areas of science and engineering. Research activities are focused on developing new analysis and design methods to ensure robustness in new or classical control problems. In this paper, we investigate switching systems. A frequency-domain design method is developed for switching systems for both integer- or fractional-order controllers, taking into account specifications regarding performance and robustness and ensuring the quadratic stability of the controlled system. Some examples are given to show the applicability and effectiveness of the proposed tuning method.

Keywords

Switching system quadratic stability common Lyapunov method fractional-order controllers PID

1. Introduction

In recent years, the study of switching systems has received growing attention. Switching systems are a class of hybrid dynamical systems consisting of a family of continuous (or discrete) time subsystems and a rule that orchestrates the switching between them (see e.g. Liberzon, 2003; Sun and Ge, 2005b). The widespread application of such systems is motivated by increasing performance requirements, and by the fact that high performance control systems can be realized by switching between relatively simple linear time invariant (LTI) systems. However, the potential gain of switching systems is offset by the fact that the switching action introduces behaviour in the overall system that is not present in any of the composite subsystems. A survey of the main problems in the stability and design of switching systems was reported in Liberzon and Morse (1999), Sun and Ge (2005a) and Xiang and Xiao (2011). There are many methodologies and approaches developed in the switching systems theory in order to mitigate some problems encountered in practice (refer to Dong, 2011, and the references therein for a current review). However, despite much effort, relatively little work can be found in the literature from a robustness point of view.

The widespread use of PID controllers has motivated many researchers to look for better design methods or alternative controllers for many years. As reported, for example in Aström and Hägglund (2006), more than 90% of the control loops are PID-type. The generalization of the PID controller to noninteger order, namely PI^λD^μ, was proposed by Podlubny (1999), whose transfer function, in general form, is given by

{PI}^{λ} D^{μ} (s) = K_{p} + K_{i} s^{λ} + K_{d} s^{μ}

(1)

where λ and μ are the noninteger orders of the integrator and differentiator terms respectively, and λ, μ ∈ ℝ⁺. Intuitively, with noninteger-order controllers, there is more flexibility in adjusting the gain and phase characteristics than when using integer-order ones; in other words, fractional-order controllers take advantage of allowing more adjustable time and frequency responses of the system to be attained. The better performance of this type of controller in comparison with the integer ones has been demonstrated in many works (refer for example to the pioneering study Podlubny, 1999). It is important to realize that there is a very wide range of control problems and consequently also a need for a wide range of design techniques for PI^λD^μ controllers (a review can be found in Monje et al. (2010), Chen et al. (2009) and Valério (2010)). There are already many tuning methods available for fractional-order controllers applied to integer-order systems (see e.g. Monje et al., 2004; Silva and Machado, 2006; Feliu-Batlle et al., 2007; Vinagre et al., 2007; Bohannan, 2008; Monje et al., 2008; Feliu-Batlle et al., 2009; Ladaci et al., 2009; Li et al., 2010; Padula and Visioli, 2011; Luo and Chen, 2012), fractional-order systems (see e.g. Luo and Chen, 2009; Luo et al., 2010, 2011; Özbay et al., 2012) or both integer-order systems and fractional-order systems (refer to e.g. Hajiloo et al., 2012). Nevertheless, its application to switching systems has not been paid much attention. We expect it to open a new perspective in the field, especially since there are more parameters to be tuned and, consequently, more robust performances can be attained.

Given this context, the objective of this paper is to develop a frequency-domain method to design robust controllers, of both integer- or noninteger-order, for switching systems, given a set of specifications and ensuring the quadratic stability of the controlled system. A preliminary version of this paper can be found in HosseinNia et al. (2012a).

The remainder of this paper is organized as follows. Section 2 provides some fundamentals for the quadratic stability of switching systems, as well as the statement of the problem under consideration. Section 3 addresses the tuning method for the design of robust controllers for switching systems. Three examples are given in Section 4 to show the effectiveness of the developed method. Finally, Section 5 draws the conclusions of this paper.

2. Preliminaries

In this section, some conditions of the quadratic stability of switching systems will be recalled, specifically the frequency domain method introduced in Kunze et al. (2008), which will be useful to present the design method in Section 3. The problem for the class of systems under consideration will also be stated.

2.1. Stability of switching systems

Consider a switching system as

\overset{\cdot}{x} = Ax, A \in co {A_{1}, \dots, A_{L}}

(2)

where co denotes the convex combination and A_i ∈ ℝ^n×n, i = 1, … , L, is the switching subsystem. System (2) can alternatively be written as (Pardalos and Rosen, 1987)

\overset{\cdot}{x} = Ax, A = \sum_{i = 1}^{L} λ_{i} A_{i}, \forall λ_{i} \geq 0, \sum_{i = 1}^{L} λ_{i} = 1

(3)

A characteristic polynomial of order n of system (2) can be given by

c (s) = s^{n} + c_{n - 1} s^{(n - 1)} + \dots + c_{1} s + c_{0}

(4)

where n is an integer number indicating the order of the system and

A = [- c_{n - 1} - c_{n - 2} \dots - c_{1} - c_{0} 1 0 ··· 0 00 1 ··· 0 0 : ⋮ ⋮: 0 0 ··· 0 1]

With definitions above, some important stability conditions of switching systems are offered next:

Quadratic stability (Boyd et al., 1994). A system described by (3) is quadratically stable if and only if there exists a matrix P = P^T > 0, P ∈ ℝ^n×n, such that

A_{i}^{T} P + {PA}_{i} < 0, \forall i = 1, \dots, L

Quadratic stability in frequency domain (Kunze et al., 2008). Consider c₁(s) and c₂(s), two stable polynomials of order n corresponding to the subsystems $\overset{\cdot}{x} = A_{1} x$ and $\overset{\cdot}{x} = A_{2} x$ , respectively. Then the following statements are equivalent:

c₁(s)/c₂(s) and c₂(s)/c₁(s) are strictly positive real (SPR);

$| \arg (c_{1} (j ω)) - \arg (c_{2} (j ω)) | < π 2$ , ∀ω;

A₁ and A₂ are quadratically stable, which means that ∃P = P^T > 0 ∈ ℝ^n×n such that $A_{1}^{T} P + {PA}_{1} < 0$ and $A_{2}^{T} P + {PA}_{2} < 0$ .

2.2. Problem statement

It is well known that a switching system can be potentially destabilized by an appropriate choice of switching signal, even if the switching is between a number of Hurwitz-stable closed-loops systems. Even in the case where the switching is between systems with identical closed-loop characteristic polynomials, it is sometimes possible to destabilize the switching system by means of switching (Leith et al., 2003). Likewise, the concept of robustness with respect to parameter variations is well defined for LTI systems. However, this issue is somewhat more difficult to quantify for switched linear systems. In particular, robustness may be defined with respect to a number of design parameters, including not only the parameters of the closed-loop system matrices, but also the switching signal.

Let us illustrate the importance of designing a robust controller for switching systems by means of a particular example. Consider a switching system given by the following second-order transfer function:

H_{i} (s) = 2 (τ_{i} s + 1)^{2}, i = 1, 2

(5)

with τ₁ = 1 and τ₂ = 0.1. One can state that only the time constant τ of the system changes. As can be observed in Figure 1, both subsystems have the same phase margin of 90°. Applying quadratic stability conditions in the frequency domain to the closed-loop system, we have

| \arg ((3 - ω^{2}) + 2 j ω) - \arg ((3 - 0.01 ω^{2}) + 0.2 j ω) | le π 2, \forall ω

(6)

Figure 2 depicts condition (6) graphically. It can be seen that quadratic stability is not guaranteed. As a result, a method for designing robust and quadratically stable controllers for such a class of switching systems is required.

Figure 1.

Bode plot of the controlled subsystems H₁ and H₂.

Figure 2.

Phase difference between the two characteristic polynomials of the closed-loop subsystems H₁ and H₂.

3. Design method

As commented in the introduction, the objective of this paper is to design controllers for switching systems so that the system fulfills different specifications regarding performance and robustness, while ensuring its quadratic stability. A scheme of the control approach is shown in Figure 3 for a general system G_i(s), i = 1, 2, … , L.

Figure 3.

Scheme of the controlled system.

Specifications related to phase margin, gain crossover frequency and output disturbance rejection are going to be considered in this design method. Indeed, other kinds of specifications can be met, depending on the particular requirements of the application. It should be noticed that, apart from these design specifications, which can change with the application, the stability conditions have to be also fulfilled. Actually, if the number of subsystems which constitute the system to be controlled is L, there are L − 1 stability conditions to be fulfilled. Therefore, denoting the number of specifications as N, a controller with L + N − 1 parameters is required in order to fulfill all given specifications and the stability conditions.

Consider G_n as a subsystem with the worst conditions with regards to each specification and assume that the phase margin and gain crossover frequency of a subsystem G_n are denoted by φ_{m
_n} and ω_{cp
_n}, respectively, where c_i, i = 1, 2, … , L, are the characteristic polynomials of each closed-loop subsystem and K(jω) is the controller to be tuned. Thus, the design problem is formulated as follows:

Frequency domain specifications:

Phase margin:

\arg (K (j ω_{{cp}_{n}}) G_{n} (j ω_{{cp}_{n}})) + π > φ_{m_{n}}

(7)

Gain crossover frequency:

| K (j ω_{{cp}_{n}}) G_{n} (j ω_{{cp}_{n}}) |_{dB} = 0 dB

(8)

Output disturbance rejection:

| S (j ω) = 11 + G_{n} (j ω) K (j ω) |_{dB} \leq M, \forall ω \leq ω_{s}

(9)

where M is the desired value of the sensitivity function S for frequencies less than ω_s.

Stability conditions:

| \arg (c_{1} (j ω)) - \arg (c_{2} (j ω)) | < π 2, \forall ω \geq 0 : | \arg (c_{L - 1} (j ω)) - \arg (c_{L} (j ω)) | < π 2, \forall ω \geq 0

(10)

It is important to remark that (7)–(9) refer to the worst conditions concerning phase margin, crossover frequency and sensitivity for a subsystem G_n among all subsystems. The same can be done for any other specification, such as high frequency noise rejection, steady-state error cancellation, etc. (see e.g. Monje et al., 2008, for more tuning specifications). The set of conditions in (10) ensures the quadratic stability of the switching system. In the case of fractional-order systems or time-delayed systems, an approximation of fractional-order derivatives or delay can be used to apply these specifications.

As an example, in the case of N = 2, a list of types of switching systems and their possible controllers is given in Table 1 (any other type of controller can be tuned using the same idea). As can be stated, the use of fractional-order controllers may have the advantage of allowing more specifications or subsystems to be fulfilled or controlled, respectively, and consequently, more robust performances to be attained.

Table 1.

Type of switching system and possible controllers when N = 2

L	Type of controller	Transfer function
2	PID	$K_{p} + K_{i} s + K_{d} s$
2	Fractional PI (FPI)	$K_{p} + K_{i} s^{λ}$
2	Fractional PD (FPD)	K_p + K_ds^μ
3	PID with noise filter (NPID)	$K_{p} + K_{i} s + K_{d} s 1 + s / NN$
4	Fractional PID (FPID)	$K_{p} + K_{i} s^{λ} + K_{d} s^{μ}$

To determine the controller parameters, the set of nonlinear equations (7)–(10) has to be solved. To do so, the optimization toolbox of Matlab can be used to work out the best solution with minimal error. More precisely, the function FMINCON is able to find the constrained minimum of a function of several variables. It solves problems of the form min_xf(x) subject to C(x) ≤ 0, C_eq(x) = 0 and x_m ≤ x ≤ x_M, where f(s) is the function to minimize; C(x) and C_eq(x) represent the nonlinear inequalities and equalities respectively (nonlinear constraints); x is the minimum we are looking for; and x_m and x_M define a set of lower and upper bounds on the design variables, x.

In this particular case, the specification (7) will be taken as the main function to minimize, and the rest of specifications (i.e. (8)–(10)) will be taken as constraints for the minimization, all of them subjected to the optimization parameters defined within the function FMINCON. The success of this optimization process depends mainly on the initial conditions considered for the parameters of the controller.

4. Examples

This section gives some examples of application of the proposed method for designing robust and quadratically stable controllers for switching systems. Specifically, three cases will be considered next for different numbers of design specifications and systems with different numbers of subsystems: the velocity control of a car, the control of a switching system with L = 4 given two design specifications, and the velocity control of a servomotor given three design specifications.

Example 1

Velocity control of a vehicle with first-order dynamics given two design specifications.

In HosseinNia et al. (2011, 2012b) we proposed a hybrid model of a vehicle taking into account its different dynamics when accelerating and braking as follows:

G_{1} (s) \approx 4.39 s + 0.1746

(11)

G_{2} (s) \approx 4.45 s + 0.445

(12)

where G₁ and G₂ refer to the throttle and brake dynamics respectively, and the input and output of the system are the reference velocity and the actual velocity of the car.

From the viewpoint of the comfort of the car's occupants, phase margin and crossover frequency has to be chosen around 80 ° and 0.8 rad/s, respectively, in order to obtain a smooth closed-loop response with an overshoot close to 0. Therefore, given two specifications (N = 2) and two subsystems (L = 2), controllers with three parameters are required for this application. In particular, two different three-parameter controllers are designed: a fractional PI (FPI) controller and a traditional PID controller, both of the forms given in Table 1. Solving the set of equations (7)–(10) for the previous specifications, the parameters of both controllers are:

FPI: K_p
₁ = 0.15, K_i
₁ = 0.07 and α = 0.71;

PID: K_p
₂ = 0.1, K_i
₂ = 0.11 and K_d = 0.223.

Figure 4 shows the frequency response of the controlled car when applying the FPI and PID controllers. As can be seen, the design specifications are fulfilled for both subsystems for both designed controllers: the phase margin obtained with the FPI is even higher than 80°. An important issue that should be noted is that the system controlled with the PID controller has constant magnitude at high frequencies, which may cause the system to be sensitive to high frequency noises and consequently lead to instability. The phase difference between the two characteristic polynomials of the closed-loop controlled subsystems for both cases is shown in Figure 5. It is observed that the maximum phase differences are 27.35° and 10.57° when using the FPI and PID, respectively, so the controlled system is quadratically stable in both cases.

Figure 5.

Phase difference between the two characteristic polynomials of the closed-loop system in Example 1 when applying (a) FPI (b) PID.

Figure 4.

Bode plots of the controlled system in Example 1 when applying (a) FPI (b) PID.

To show the system performance in time domain, Figure 6 depicts a manoeuvre which simulates the car's acceleration to 20 km/h and, after that, the braking to 0 km/h (stopping completely) for different switching, including a comparison of its velocity for the FPI and PID cases. As observed in this figure, the car has an adequate performance for both the throttle and the brake actions when applying the FPI controller (dash-dotted black line), achieving the reference velocity in a suitable time and without overshoot in both cases. Although both controllers fulfilled the specifications, the response when using the PID (dashed red line) has a considerabley high value of overshoot. As a result, it can be said that the occupants' comfort is guaranteed when applying the proposed FPI controller.

Figure 6.

Time response of the controlled system with both FPI and PID during random switching.

Example 2

Control of a switching system with L = 4 given two design specifications.

Now consider a switching system given by

G_{1} (s) = 1.55 s + 1

(13)

G_{2} (s) = 1.23 s + 1

(14)

G_{3} (s) = 1.12 s + 1

(15)

G_{4} (s) = 1 s + 1

(16)

The aim is to design a robust controller so that the controlled system is quadratically stable during a defined switching (from subsystem 1 to subsystem 2, subsystem 2 to subsystem 3, subsystem 3 to subsystem 4 and vice versa) and has zero steady-state error and a fast response with a settling time of less than 2 s. Equivalently, the design specifications are chosen as a phase margin of 80° at 5 rad/s. Therefore, a five-parameter controller is required; in other words, a fractional-order PID (FPID) of the form given in Table 1 will be tuned next. In addition, the three stability conditions to be fulfilled are

| \arg (c_{1} (j ω)) - \arg (c_{2} (j ω)) | < π 2, \forall ω \geq 0

(17)

| \arg (c_{2} (j ω)) - \arg (c_{3} (j ω)) | < π 2, \forall ω \geq 0

(18)

| \arg (c_{3} (j ω)) - \arg (c_{4} (j ω)) | < π 2, \forall ω \geq 0

(19)

Then, solving the set of equations (7), (8) and (17)–(19), the FPID parameters are K_p = 10.20, K_i = 30.18, K_d = 2.80, λ = 0.83 and μ = 0.47.

Figures 7 and 8 show the Bode plot and the phase difference, respectively, between each pair of subsystems. It is obvious that the controlled system fulfills the design specifications (ω_cp > 5 rad/s and φ_m > 80°) and is quadratically stable. As can be seen, the worst case corresponds to the first subsystem, but is still within the margin of the specifications.

Figure 7.

Bode plot of the controlled system in Example 2 when applying FPID.

Figure 8.

Phase difference between the each pair of characteristic polynomials of the closed-loop system in Example 2: (17) solid line; (18) dashed line; and (19) dash-dotted line.

In order to show the performance of the system by applying the designed controller, the closed-loop response is simulated for constant and variable references, as shown in Figures 9 and 10. It can be observed that all the subsystems fulfill the settling time less than 2 s.

Figure 9.

Step response of the system in Example 2 for constant reference.

Figure 10.

Step response of the system in Example 2 for variable reference. Each color is related to the subsystem which is activated, which shows the controller maintains its stability during the switching.

Example 3

Velocity control of a servomotor given three design specifications.

Let us now consider the velocity of a servomotor described by

G_{1} (s) = 0.5562 s + 1

(20)

G_{2} (s) = 0.55100 s + 1

(21)

where dynamics (20) correspond to the normal servomotor dynamics, and dynamics (21) may be caused when the brake is activated (Monje, 2006). In this example, we will design a controller using specifications of phase margin, gain crossover frequency and output disturbance rejection as follows: φ_m = 80° at ω_cp = 0.6 rad/s and a desired value of the sensitivity function of M = −20 dB for frequencies less than 0.1 rad/s. Among all four-parameter controllers, a PID with noise filter (NPID) of the form in Table 1 is chosen. Then, solving the set of four equations, the parameters of the controller are K_p = 81.35, K_i = 35.195, K_d = 95.71 and N = 19.86.

The fulfillment of the design specifications and stability is proved by Figures 11 –13, which represent the Bode plot, the phase difference of closed-loop polynomials and the sensitivity function, respectively. Finally, Figure 14 shows the time response of the controlled system for variable reference. It can be observed that its performance is adequate, even during switching.

Figure 11.

Bode plot of the controlled system in Example 3 when applying NPID.

Figure 12.

Phase difference of characteristic polynomials of the closed-loop system in Example 3.

Figure 13.

Sensitivity function S in Example 3.

Figure 14.

: Step response of the system in Example 3 for variable reference.

5. Conclusion

This paper has addressed the design of robust controllers for switching systems in frequency domain considering specifications regarding performance and robustness and ensuring the quadratic stability of the controlled system. Three examples were given to show the effectiveness of the proposed tuning method for different kinds of switching systems and for different numbers of specifications to be fulfilled. In particular, different integer- and fractional-order controllers were designed to fulfill a set of desired specifications ensuring the quadratic stability of the switching systems.

In future work, we will focus on the validation of the proposed method in a real application.

Footnotes

Funding

This work was supported by the Spanish Ministry of Science and Innovation under the project DPI2009-13438-C03.

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