Mathematical Problems in Engineering

Volume 2008 (2008), Article ID 341720, 21 pages

http://dx.doi.org/10.1155/2008/341720

## Parameterized Regulator Synthesis for Bimodal Linear Systems Based on Bilinear Matrix Inequalities

Department of Mechanical and Industrial Engineering, University of Toronto, Toronto, ON, Canada M5S 3G8

Received 3 July 2007; Revised 18 January 2008; Accepted 7 March 2008

Academic Editor: Tamas Kalmar-Nagy

Copyright © 2008 Zhizheng Wu and Foued Ben Amara. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

A regulator design method is presented for switched bimodal linear systems, where it is desired to reject known disturbance signals and/or track known reference inputs. The switching in the bimodal system is defined by a switching surface. The regulator design approach consists of three steps. The first step is based on constructing a switched observer-based state feedback central controller for the switched linear system. The second step involves augmenting the switched central controller with additional dynamics to construct a parameterized set of switched controllers. In the third step, two sufficient regulation conditions are derived for the resulting switched closed loop system. The regulation conditions present guidelines for the selection of the additional dynamics used to parameterize the switched controllers to yield the desired regulator. A regulator synthesis approach is proposed based on solving properly formulated bilinear matrix inequalities. Finally, a numerical example is presented to illustrate the performance of the proposed regulator.

#### 1. Introduction

Recently, switched control systems have attracted much attention in the control community since they present challenging problems of practical importance. Significant progress has been made in this area, most notably in the stability analysis for such systems [1–7]. Numerous other results have been published and are presented in [8–10] and the references therein. In practice, in addition to stability requirements, there is a need to find controllers that would achieve regulation against known reference or disturbance signals. For example, in hard disk drives, maintaining a constant small distance between the read/write head and the disk surface is an important target that would allow greater data storage densities on the hard disk. However, the close proximity of the read/write head to the disk surface leads to intermittent contact between the two and results in a switched system regulation problem [11, 12]. Solutions to the output regulation problem for the case of linear systems have been presented in [13–15] and for the case of nonlinear systems in [16, 17]. However, these results cannot be applied to switched systems since the vector fields of switched systems are discontinuous and nonsmooth. The work of Devasia et al. [18] and Sakurama and Sugie [19] are closely related to this problem. Devasia et al. [18] studied the exact output tracking problem for linear switched systems and presented necessary and sufficient conditions for the existence of a regulator for exact output tracking with parameter jumps. The derived solution was based on the assumption that the switching times are known a priori. Therefore, the necessary and sufficient conditions were derived only for the elimination of switching induced output transients and cannot be applied to systems where switching is signal driven or where the switching times cannot be set a priori. Sakurama and Sugie [19] recently discussed the trajectory tracking problem for bimodal switched systems. The reference trajectory for the switched system is assumed known and the switching times are assumed to be measurable. An error variable and an error system are introduced, based on which the tracking controller was designed using a Lyapunov-like function. However, conditions for the existence of the desired regulator are sometimes very difficult to check.

In this paper, a controller design approach is proposed for single-input single-output switched bimodal systems, where it is desired to reject known disturbance signals and track known reference inputs simultaneously. Switching in the bimodal system is defined by a switching surface. A regulator synthesis method based on solving bilinear matrix inequalities (BMIs) is presented. The proposed regulator design approach consists of three steps. In the first step, a switched observer-based state-feedback central controller is constructed for the switched linear system. The second step involves augmenting the switched central controller with additional dynamics to construct a parameterized set of switched controllers. Stability analysis of the resulting switched closed loop system is then presented. In the third step, two sufficient regulation conditions are derived for the switched closed loop system. The first sufficient condition is derived based on the input-output stability property of the switched closed loop system. The second sufficient condition for regulation is derived by transforming the forced switched closed loop system into an unforced impulsive switched system. As such, the regulation problem is transformed into a stability analysis problem for the impulsive switched system. Based on the parameterized controller structure and the derived regulation conditions, proper BMIs are formulated and a regulator synthesis method is proposed. The main advantage of the proposed regulator synthesis approach is that it offers a numerical procedure that can practically be implemented and used to develop the desired switched regulator.

The rest of the paper is organized as follows. In Section 2, the general regulation problem for switched bimodal linear systems is presented. In Section 3, the construction of a parameterized set of switched controllers for the switched system is discussed and the stability properties of the resulting closed loop switched system are analyzed. Regulation conditions for the switched system are presented in Section 4 and the regulator synthesis method for switched systems is proposed in Section 5. The controller design method is illustrated in Section 6 using a numerical example, followed by the conclusion in Section 7.

In the following, and denote the maximum and minimum eigenvalues of a matrix, respectively. The symbol represents the identity matrix. For a given matrix , denotes the vector with the same entries as those of .

#### 2. Regulation Problem for Switched Bimodal Systems

This paper considers switched bimodal systems subject to external inputs representing disturbance and/or reference signals. Both the system dynamics as well as the external input signal are assumed to switch according to a switching law defined by a switching surface. These types of switched systems are motivated by practical applications, such as the flying height regulation problem for the read/write head in hard disk drives [11, 12].

Consider the switched system given by the following state space representation: where is the state vector, is the control input, is the measurement signal to be fed to the controller, is the performance variable to be regulated and is assumed to be measurable, is the state vector of the exogenous systems generating the signal , have simple eigenvalues on the imaginary axis, is the index of the system under consideration at time , and is a constant satisfying . The switching between the systems and is performed according to the value of the performance variable , and is determined based on the location of with respect to a switching surface given by The switching surface is not fixed but changes with time given that the term is in general a time-varying term. The switching between the two modes takes place as follows. If and becomes strictly greater than , then the mode switches to ; and if and becomes less than or equal to , then the mode switches to . In the following, it is assumed that for any given , the system must operate in only one of the two modes corresponding to . This assumption is motivated by physical considerations in some applications of interest, such as the example presented at the end of the paper or the system treated in [11, 12]. For the switched system (2.1), it is desired to construct an output feedback controller to regulate the performance variable of the switched system against the external input signal . Given the switching nature of the plant, the output feedback controller is also chosen to be a switching feedback controller , , with a state space representation given by where , and where the switching among controllers is to obey the same rule given in (2.1) for switching between the two plant models. Therefore, the resulting closed loop system is given by The output feedback regulation problem for the switched system (2.1) can then be stated as follows.

*Output-Feedback Regulation Problem*

Given the switched system (2.1), find a switched output feedback
controller of the form (2.3)
such that the resulting closed loop system satisfies the following conditions.

(C1)*Internal stability.* With ,
the equilibrium point of the unforced switched closed loop system is
exponentially stable.(C2)*Output regulation.* For
all and ,
the response of the closed loop system involving the
switched system (2.1) and the switched controller (2.3) satisfies In the following
section, a
framework within which regulation conditions will be derived is presented.

#### 3. Parameterization of a Set of Controllers

The controller design approach presented in this paper relies on the construction of a parameterized set of output feedback controllers for the switched system (2.1). In this section, the construction of such a set is first discussed, followed by an analysis of the stability properties of the resulting switched closed loop system. The construction of a parameterized set of switched controllers involves two steps. The first step consists of designing a central controller in the form of an observer-based state-feedback controller. The second step involves augmenting the central controller with additional dynamics to construct a parameterized set of controllers.

##### 3.1. Observer-Based State-Feedback Controller

Consider the following observer-based state-feedback controller for the switched system (2.1) where is the estimate of the plant state vector and is an estimate of the plant output . The mode is determined according to the rule given in (2.1). Moreover, it is assumed that there are no impulsive changes in the controller states at the switching times.

##### 3.2. Parameterized Output-Feedback Controller

The construction of a parameterized set of controllers for the switched system is based on considering, for each , a linear fractional transformation involving a fixed system and a proper system as shown in Figure 1. The proposed controllers are similar to the parameterized stabilizing controllers for linear systems [20, 21], but where no stability assumptions are placed on the system . The state space representation of the system is given by and the system is given by where and . In particular, throughout the rest of the paper, the system is such that the matrices , , and also change according to .

*Remark 3.1. *The state space representation
of the system given in (3.2) differs from that traditionally used
in the construction of parameterized sets of stabilizing controllers for linear
systems in that the state equation in (3.2) contains the additional term .
The presence of this term makes it possible to derive the sufficient conditions
for regulation presented in Theorem 4.2.

By combining the systems and in (3.2) and (3.3), the state-space representation of the regulator is then given by (2.3), where Let denote the state estimation error. It follows that the resulting closed loop system involving the plant (2.1) and the regulator given by (2.3) and (3.4) can be written as follows: Let denote the state vector of the resulting closed loop system with , and let The resulting parameterized switched closed loop system dynamics can then be expressed as In the following, both the internal stability as well as the input-output stability properties of the system (3.7) are analyzed.

##### 3.3. Stability of the Parameterized Switched Closed Loop System

In this section, it is desired to study the stability properties of the system (3.7). The stability analysis involves two steps, namely, internal stability analysis and input-output stability analysis. In the first step, the internal stability of the closed loop system is studied by considering the system (3.7) in the absence of the signal , and studying the stability properties of the origin for the resulting unforced switched system. The second step builds on the internal stability results and presents input-output stability results for the system (3.7).

###### 3.3.1. Internal Stability

Consider first the system (3.7) in the absence of the signal . The state equation for the resulting system is given byNote that, there are no impulsive changes in the state variables of the above switched system at the switching times. The internal stability of the system (3.8) is then given by the following well known result.

Lemma 3.2 (see [22]). *If there
exists a matrix and a constant such that the following inequalities hold ** then the origin is an exponentially stable
equilibrium point for the switched system (3.8) with arbitrary switching, and **where
. *

*Remark 3.3. *The matrix inequality (3.9) can be solved using numerical
algorithms. Here, we search for a common Lyapunov function for the switched
systems (3.8) to guarantee the stability of the
origin for the system (3.8)
under arbitrary switching, since the switching surface is not a fixed function of the state vector and changes with .

###### 3.3.2. Input-Output Stability

In this section, we will consider the input-output stability properties of the system (3.7) with the external input signal . Let . Then we have the following stability result.

Lemma 3.4 (see [9]). *Assume
the origin for the switched system (3.8) is an exponentially stable
equilibrium point. Then the state vector in (3.7) is bounded, and the states will
ultimately evolve inside the bounded set given by ** where . **The input-output stability of
the closed loop system follows immediately from the above result.*

#### 4. Regulation Conditions for the Switched System

The purpose of introducing the parameterized controllers is to find, for each , appropriate , , and in (3.3) to solve the output-feedback regulation problem for the switched system (3.7). The solution to the regulation problem for the switched system (3.7) is presented in two steps. In the first step, regulation conditions for each of the individual closed loop systems , , are presented. In the second step, two sufficient conditions for regulation in the switched closed loop systems are derived. To derive the sufficient condition for regulation, a coordinate transformation is defined first, allowing the forced switched closed loop system to be transformed into an unforced impulsive switched system. Hence, the regulation problem is transformed into a stability analysis problem for the origin of the resulting impulsive switched system. Conditions for achieving asymptotic stability in the new impulsive switched system are then presented, which is equivalent to achieving regulation in the original switched closed loop system.

##### 4.1. Regulation Conditions for and

Let and in (2.4). For each of the systems , , and using a controller of the form (2.3), regulation conditions are given by the following lemma.

Lemma 4.1 (see [15]). *For each ,
consider the system in (2.1) and a controller in (2.3), and assume that is a stability matrix. Then, for each , in (2.4) achieves regulation if and only if
there exists a matrix that solves the linear matrix equations: **
Similarly,
using the parameterized feedback regulator given in (2.3) and (3.4), regulation conditions for the
switched closed loop system (3.7)
can be derived and are presented in the following theorem.*

Theorem 4.2. * Assume
that for each ,
the closed loop system given in (3.7) is exponentially stable. Then each of
the systems and achieves regulation only if,
for each ,
there exists a pair of matrices which satisfy the following equations: ** Furthermore,
if the linear matrix equations in (4.2) admit a solution ,
and the matrices and in (3.3) are taken to be of the form ** then,
for each , in (3.7) achieves regulation. *

*Proof. *Equation
(4.1) can be rewritten as Partition as .
Then (4.1) is the same as Letting ,
(4.2) follows immediately.

To prove the second part of the
theorem, note that by assumption, for each , in (3.7) is a stability matrix. Moreover,
since is related to by a similarity transformation, then is also a stability matrix. Now suppose and satisfy (4.2) and that and satisfy (4.3). To show that the closed loop system
achieves regulation, let .
Using the expression for in (3.4) and substituting in the expression for in (4.3) yields It follows that: Given that from (4.3), then
based on (4.2) we have Therefore, (4.5) is satisfied, which implies (4.1) is satisfied and regulation can be
achieved.

The results presented above represent regulation conditions for each of the closed loop systems in (3.7), individually. In the following sections, regulation conditions for the system (3.7) subject to switching are discussed.

##### 4.2. Equivalent Impulsive Switched Closed Loop System Model

Based on Lemma 4.1, and using a properly defined coordinate transformation, the forced switched closed loop system (3.7) can be transformed into an unforced impulsive switched system. To introduce the appropriate coordinate transformation, note that the original regulation condition (4.1) can be rewritten as whereConsider now the following coordinate transformation Using (4.9), the forced closed loop switched system (3.7) can be transformed into an unforced switched system given by It should be noted from (4.11) that the coordinate transformation varies depending on the value of . Consequently, the states in the new system (4.12) undergo impulsive changes at the switching times. Let be the set of switching times , . Therefore, for a given switching time , we have that for where . Hence, the impulsive switched closed loop system is given by In the new coordinate system, the switching surface given in (2.2) is fixed and is expressed as follows: Using the coordinate transformation in (4.11), the original output-feedback regulation problem for the switched system (3.7) is transformed into an asymptotic stability analysis problem for the origin of the unforced impulsive switched system (4.14).

##### 4.3. Regulation Conditions for the Switched System

First, let denote the distance between the switching surface and the origin. Then we have . Define Let , where , and define the set as follows:Using the definition for in (4.11), and the fact that the original state vector will ultimately evolve inside a bounded set given by (3.11), it follows that there exists a finite time at which will enter the set and continue to evolve in thereafter. A sufficient condition for regulation is then given as follows.

Theorem 4.3. *Assume that the switched closed
loop system given in (3.7) is internally stable under arbitrary
switching, and that for each , given in (3.7) achieves output regulation. If ** then the origin is an asymptotically
stable equilibrium point for the impulsive switched system
(4.14), implying that the switched system
(3.7) achieves regulation. *

*Proof. *If ,
then the switching surface given by (4.15) does not intersect the bounded set .
The trajectory of the state vector will enter the set at and continue to evolve inside thereafter. Consequently, there will be no
more switching. Since the system that is active in the halfspace containing is such that is a stability matrix, and since ,
the trajectory of the state vector asymptotically converges to the origin,
implying that regulation in the original switched closed loop system (3.7) is achieved.

If condition (4.18) is not satisfied, that is, , then the switching surface must intersect with the bounded set . To present regulation conditions in this case, define the matrices , , and . Sufficient conditions for regulation are presented in the following theorem.

Theorem 4.4. * Assume the switched closed loop
system given in (3.7) is internally stable under arbitrary
switching and that, for each , given in (3.7) achieves output regulation. Moreover,
assume that
, ,
and that the direction
of vector is not parallel to that of and . If the following conditions are
satisfied, ** or ** then the origin is an asymptotically
stable equilibrium point for the impulsive switched system
(4.14),
implying that the switched system (3.7) achieves regulation. *

*Proof. *The following proof is presented for the case of .
The case of is treated using the same ideas. Let region 1
and region 2 denote the halfspaces where the systems and are active, respectively (see
Figures 2
and 3).
Hence, the origin is in region 1 where the system is active. Define the hypersurfaces and .
Then for all points in , and for all points in , .
Since, by assumption, ,
the switching surface intersects with the bounded set .
Let be the intersection of the switching surface with .
By assumption, the direction of vector is not parallel to that of and .
Therefore, the switching surface must intersect with either of the hypersurfaces or .
Define the intersection of with the hypersurfaces and as and ,
respectively. For each ,
let denote the distance between the origin and the
set .
Then, can be obtained by solving the problem of
minimizing subject
to the constraint . Since, by assumption, ,
the solution to the
above problem is , . If , then the sets , ,
are outside the bounded set .
In this situation, the set does not intersect with the sets or .
If and , then the set will be located
in one of the two halfspaces defined by the hypersurface and where
all the points satisfy . Similarly, if and , then the set will be located
in one of the two halfspaces defined by the hypersurface and where
all the points satisfy . By deriving conditions on ,
and using the stability properties of and ,
it is possible to conclude about the asymptotic stability of the origin of the
impulsive switched closed loop system (4.14).
In the following, two cases are considered in
detail.*Case 1 ( and (see Figure 2)). * In this case,
all the state trajectories leaving the set and entering region 1 have .
Therefore, if the state trajectory leaves the set and enters region 1, then it will not hit it
again. Two possible cases can be considered here depending on whether the state
trajectory enters the bounded set from region 1 or region 2 at time .
In this case, the state trajectory enters the set from region 1, then the state trajectory
cannot hit the switching surface since for all states on the switching surface within
the set .
Therefore, the state trajectory is confined to evolve in
region 1 inside the set .
Since is a stable system, the state trajectory will
converge to the origin asymptotically, which implies that regulation for the
switched system (3.7) is achieved. Consider
now the case where the state trajectory enters the set from region 2. Since the state
trajectory is confined to evolve inside the set and since the system is asymptotically stable, the state trajectory
must hit the switching surface and switching must take place. Following
switching, and based on the dynamics of the impulsive switched closed system as
presented in (4.14), the state vector immediately
following switching will be in region 1. Since for all states on the switching surface within
the set for the system ,
the state trajectory cannot cross the switching surface again and will continue
to evolve in region 1 thereafter. The state trajectory will approach the origin
asymptotically, which implies that
regulation will be achieved.*Case 2 ( and (see Figure 3)). *In this case,
all the state trajectories leaving the set and entering region 2 have .
Assume the state trajectory enters the set from region 2 at the time .
Given that the system is asymptotically stable, and that the
trajectories are confined to evolve inside ,
the state trajectory must cross the switching surface to approach the origin.
However, this is not possible since, with respect to the system , for all .
Therefore, it is not possible to have the state trajectory enter the set from region 2 at time .
Consider now the case, where the state trajectory enters the set from region 1. If the state trajectory crosses
the switching surface to enter into region 2, then that will result in a
contradiction similar to that discussed for the case, where the state
trajectory enters the set from region 2. Therefore, once the state
trajectory enters the set from region 1, it will continue to evolve in
region 1 forever and will never cross the switching surface to enter into region
2 for .
Since is a stable system, then the state trajectory
will converge to the origin asymptotically, which implies that regulation for
the switched system (3.7) is achieved.

Therefore, based on the above analysis, if the conditions for
Case 1 given by (4.19) or the
conditions for Case 2 given by (4.20) are satisfied, then regulation in the
switched system (3.7) can be achieved.

#### 5. Regulator Synthesis for the Switched System

Based on the regulation conditions for the switched system proposed in the previous section, a regulator synthesis approach is presented in this section. The proposed synthesis approach is based on solving a set of properly formulated bilinear matrix inequalities (BMIs). The main idea behind the regulator synthesis approach is as follows. Consider an output-feedback controller as given in (2.3) and (3.4), where and . Since is the solution to the Sylvester equation (4.2), is only parameterized in the unknown matrices , , and . Assume is such that the resulting closed system satisfies (3.9). Then, based on Lemma 3.2 and Theorem 4.2, the switched closed loop system given in (3.7) is internally stable under arbitrary switching and, for each , given in (3.7) also achieves output regulation. In this case, if any of (4.18), (4.19), and (4.20) is satisfied, then based on Theorems 4.3 and 4.4, regulation in the switched system (3.7) is achieved. The set of conditions in Lemma 3.2 and Theorems 4.3 and 4.4 yield bilinear matrix inequalities in the unknown parameters , , and . In the following, a solution procedure for the formulated BMIs is proposed to determine the unknown parameters , , and in .

The sufficient regulation condition (4.18) given in Theorem 4.3 is equivalent toUsing and letting , condition (4.18) is equivalent to If (5.2) cannot be satisfied, the switching surface intersects the set and conditions (4.19) and (4.20) in Theorem 4.4 need to be verified. For the switched closed loop system (3.7), the parameters and can be estimated using the following matrix inequalities [22]: If the matrices , , , and in are unknown, then the above equations define a bilinear matrix inequality. Combining (5.2) and (5.3), the synthesis procedure for the controller can be realized by solving the following BMIs: where the unknown parameters are , , , , , and . Once , , are determined based on the above BMIs, the matrices and can be calculated using (4.3) as given in Theorem 4.2.

In this paper, a regulator synthesis algorithm, referred to as the - iteration algorithm, will be used to find a parameterized regulator that satisfies either one of the regulation conditions (4.18), (4.19), or (4.20). The - algorithm iteratively solves for the Lyapunov matrix and the parameters in the controller given in (2.3). The basic idea of the - iteration is that a BMI can be converted into an LMI when some of the parameters in the BMI are fixed. The approach for solving BMI problems is to alternate between two optimization problems subject to LMIs, which are related to the matrix in (5.4) and (5.5) and the parameters in the controller in (2.3), respectively. In the algorithm, the input data is represented by , , , , , , , , , and , whereas the unknown variables to be determined are , , , , , , , and . The algorithm is summarized below, where , , , , , and denote the solutions , , , , , and obtained at the th iteration of the algorithm.

(1)*Calculating *: Determine by solving the Sylvester equation (4.2) and let with .
Let be as in (4.10). Determine using (4.16),
then calculate
.(2)*Initializing *: Initialize the
controller parameters , ,
and to make the switched closed loop system (3.7) internally stable, which can be
realized by solving the following two LMIs separately for the unknown matrices , , , ,
and with preset constants and , (3)*Initializing *:
Based on the initial controller
parameters , ,
and from step 2, initialize the maximum decay rate by solving the following optimization problem:(4)-*Step*: At the th iteration, ,
given the matrices , ,
and and the scalar ,
solve the following optimization problem for and :(5)-*Step*: At
the th iteration, ,
given the matrix and from step 4, solve the following optimization problem
for , , ,
and :(6)*Verification:* Verify the
constraints (5.6), (4.19), and (4.20), and if any of the three conditions is
satisfied, compute and using (4.3), then stop the
algorithm. If none of (5.6),
(4.19), and (4.20) is satisfied, then go to step 4.

The iterative loop is repeatedly executed until a solution is found, or there is no major reduction in relative to the previous iteration. The algorithm converges to a local solution of , since at each th iteration of the algorithm, we have and . Therefore, if no solution can satisfy the regulation conditions (5.6), (4.19), or (4.20) by the iterative procedure described above, then the initial parameters , , and can be adjusted by changing the decay rates and in (5.7), and restarting the iterative procedure again. If the above algorithm yields a solution, then, according to Theorems 4.3 or 4.4, the switched closed loop system will achieve regulation.

#### 6. Numerical Example

In this section, the regulator synthesis method proposed in this paper will be used to design a controller that cancels the contact vibrations in a mechanical system. Figure 4 shows the diagram of such a system consisting of a mass , a contact surface , and their respective coordinates. The mass is attached to a spring with stiffness and a damper with damping coefficient . The mass moves only in the vertical direction, whereas the contact surface underneath it moves to the left. The mass may enter into intermittent contact with the surface , resulting in contact vibrations. When the mass enters into contact with the surface , the contact characteristics are represented by a spring with stiffness and a damper with damping coefficient . The force represents the external force used to control the mass , whereas represents a disturbance force. This model can be found in many applications, such as the interface between the read/write head and the disk surface in hard disk drive systems. In the following, and with respect to the system shown in Figure 4, the control objective is for the mass to follow the displacement of the contact surface , while maintaining a desired constant separation in the vertical direction. Let be the deviation of the mass from its equilibrium position. Then the equations of motion of the mass can be written as where is the contact force expressed asand where is the displacement of the contact surface. The wavy contact surface profile is expressed as a linear combination of sinusoidal functions: with amplitudes , frequencies , phases , , and constant offset . Therefore, the distance between the mass and contact surface is . The control signal is defined as for the noncontact situation and , when contact takes place. Let denote the desired distance between the mass and contact surface . The output , to be fed to the controller, is defined asThe performance variable is defined to be the difference between the actual distance and the desired distance ,Therefore, the system will switch between the contact and noncontact modes according to the value of the performance variable . If , the system will operate in the noncontact mode, and if , the system will operate in the contact mode. Let and , then the switched system model is given by where Consider a model for the read/write head and disk surface interface in hard disk drive systems similar to that in [11, 12]. The surface profile is given by , and the force . Therefore we have . Let , , , , and be 200 mg, N/m, 0.05 N/m/Sec, N/m, and 0.3 N/m/Sec, respectively. In the following, a regulator for the switched system is designed using the synthesis procedure described in the previous section. Simulation results will illustrate the performance of the proposed regulator in maintaining the desired system output despite the presence of switching.

In order to find a feasible solution for the BMIs, we introduce a coordinate transformation as , where . Rewriting (6.6) in the state space form (2.1) results in It is desired to design a regulator that can reject the disturbance in the switched system. Based on Theorem 4.2, solving the Sylvester equation (4.2) yields Based on (4.16), we obtain . Since m and , and with , we have in (5.2). Using the - algorithm proposed above, we obtain Then, based on (4.3), we have Since , condition (5.6) is not satisfied. Then conditions (4.19) and (4.20) will be verified. First, we obtain . Hence, after a long enough time, the trajectory will enter into the bounded set with and continue to evolve thereafter. Given that and that it follows that regulation condition (4.20) in Theorem 4.4 is satisfied. Therefore, regulation can be achieved in the switched closed loop system using the designed controller.

The simulation results of the response of the closed loop system under switching are illustrated in Figures (5-6). It can be seen that if the performance variable is smaller than micrometers, the mass enters into contact with the contact surface and the model of the system switches. It can also be seen that the disturbance changes at the switching times. But even in the presence of switching, the switched system performance variable still converges to zero, which means the mass asymptotically follows the contact surface at the desired separation 30 micrometers.

#### 7. Conclusion

The problem of regulation in switched bimodal systems against known disturbance or reference signals is discussed. A regulator design approach based on the parameterization of a set of controllers that can achieve regulation for the switched system is presented. The forced switched closed loop system involving the parameterized controller is transformed into an unforced impulsive switched system. Consequently, the original regulation problem is transformed into a stability analysis problem for the origin of the impulsive switched system. Sufficient conditions for regulation and a regulator synthesis method based on solving a set of bilinear matrix inequalities are presented. A simulation example is used to illustrate the effectiveness of the proposed regulator.

#### Acknowledgments

The authors would like to thank the associate editor and the anonymous reviewers for their comments. This research was supported by the Natural Sciences and Engineering Research Council of Canada and the provincial government of Ontario.

#### References

- D. Cheng, L. Guo, Y. Lin, and Y. Wang, “Stabilization of switched linear systems,”
*IEEE Transactions on Automatic Control*, vol. 50, no. 5, pp. 661–666, 2005. View at Publisher · View at Google Scholar · View at MathSciNet - V. D. Blondel and J. N. Tsitsiklis, “Complexity of stability and controllability of elementary hybrid systems,”
*Automatica*, vol. 35, no. 3, pp. 479–489, 1999. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - T. Ooba and Y. Funahashi, “On a common quadratic Lyapunov function for widely distant systems,”
*IEEE Transactions on Automatic Control*, vol. 42, no. 12, pp. 1697–1699, 1997. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - R. Shorten, K. S. Narendra, and O. Mason, “A result on common quadratic Lyapunov functions,”
*IEEE Transactions on Automatic Control*, vol. 48, no. 1, pp. 110–113, 2003. View at Publisher · View at Google Scholar · View at MathSciNet - M. S. Branicky, “Multiple Lyapunov functions and other analysis tools for switched and hybrid systems,”
*IEEE Transactions on Automatic Control*, vol. 43, no. 4, pp. 475–482, 1998. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - M. Johansson and A. Rantzer, “Computation of piecewise quadratic Lyapunov functions for hybrid systems,”
*IEEE Transactions on Automatic Control*, vol. 43, no. 4, pp. 555–559, 1998. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - J. M. Gonçalves, A. Megretski, and M. A. Dahleh, “Global analysis of piecewise linear systems using impact maps and surface Lyapunov functions,”
*IEEE Transactions on Automatic Control*, vol. 48, no. 12, pp. 2089–2106, 2003. View at Publisher · View at Google Scholar · View at MathSciNet - D. Liberzon,
*Switching in Systems and Control*, Systems & Control: Foundations & Applications, Birkhäuser, Boston, Mass, USA, 2003. View at Zentralblatt MATH · View at MathSciNet - Z. Sun and S. S. Ge,
*Switched Linear Systems. Control and Design*, Springer, London, UK, 2005. View at Zentralblatt MATH - E. K. Boukas,
*Stochastic Switching Systems: Analysis and Design*, Control Engineering, Birkhäuser, Boston, Mass, USA, 2006. View at MathSciNet - K. Linda, K. Ono, and M. Yamane, “Dynamic characteristics and design consideration of a tripad slider in the near-contact regime,”
*ASME Journal of Tribology*, vol. 124, no. 3, pp. 600–606, 2002. View at Google Scholar - Z. Wu and F. Ben Amara, “Adaptive regulation of the flying height for 2-DOF tripad slider in hard disk drives,” in
*Proceedings of the IEEE International Conference on Mechatronics and Automation*, vol. 1, pp. 160–165, Niagara Falls, Ontario, Canada, July-August 2005. - E. J. Davison, “The robust control of a servomechanism problem for linear time-invariant multivariable systems,”
*IEEE Transactions on Automatic Control*, vol. 21, no. 1, pp. 25–34, 1976. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - B. A. Francis and W. M. Wonham, “The internal model principle of control theory,”
*Automatica*, vol. 12, no. 5, pp. 457–465, 1976. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - B. A. Francis, “The linear multivariable regulator problem,”
*SIAM Journal on Control and Optimization*, vol. 15, no. 3, pp. 486–505, 1977. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - A. Isidori and C. I. Byrnes, “Output regulation of nonlinear systems,”
*IEEE Transactions on Automatic Control*, vol. 35, no. 2, pp. 131–140, 1990. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - C. I. Byrnes, F. D. Priscoli, A. Isidori, and W. Kang, “Structurally stable output regulation of nonlinear systems,”
*Automatica*, vol. 33, no. 3, pp. 369–385, 1997. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - S. Devasia, B. Paden, and C. Rossi, “Exact-output tracking theory for systems with parameter jumps,”
*International Journal of Control*, vol. 67, no. 1, pp. 117–131, 1997. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - K. Sakurama and T. Sugie, “Trajectory tracking control of bimodal piecewise affine systems,”
*International Journal of Control*, vol. 78, no. 16, pp. 1314–1326, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - K. Zhou, J. C. Doyle, and K. Glover,
*Robust and Optimal Control*, Prentice Hall, Upper Saddle River, NJ, USA, 1996. View at Zentralblatt MATH - F. Ben Amara, P. T. Kabamba, and A. G. Ulsoy, “Adaptive sinusoidal disturbance rejection in linear discrete-time systems—part I: theory,”
*Journal of Dynamic Systems, Measurement, and Control*, vol. 121, no. 4, pp. 648–654, 1999. View at Publisher · View at Google Scholar - S. Boyd, L. El Ghaoui, E. Feron, and V. Balakrishnan,
*Linear Matrix Inequalities in System and Control Theory*, vol. 15 of*SIAM Studies in Applied Mathematics*, Society for Industrial and Applied Mathematics, Philadelphia, Pa, USA, 1994. View at Zentralblatt MATH · View at MathSciNet