Output Regulation in Discrete-Time Switched Bimodal Systems Based on Youla Parameterized Stabilizing Controllers

Abstract

Motivated by a class of contact vibration control problems in mechanical systems, this paper considers a regulation problem for discrete-time switched bimodal linear systems where it is desired to achieve output regulation against partially known deterministic and unknown random exogenous signals. First, a set of observer-based Youla parameterized stabilizing controllers is constructed, based on which the regulation conditions for the switched system against the deterministic signals along with an 𝐻₂ performance constraint against the unknown random signals are derived. Then a corresponding regulator synthesis algorithm is developed based on solving properly formulated linear matrix inequalities. The proposed regulator is successfully evaluated on an experimental setup involving a switched bimodal mechanical system subject to contact vibrations, hence, demonstrating the effectiveness of the proposed regulation approach.

1. Introduction

Recently, switched control systems have been attracting much attention in the control community since they present problems that are not only academically challenging, but also of practical importance. In the past decades, significant progress has been reported in the literature [1–4] on switched systems and impulsive systems. However, most of the results focus on the stability analysis for such systems. In practice, in addition to stability requirements, there is a need to find controllers that would achieve regulation against reference or disturbance signals. For example, in hard or optical disk drives, maintaining a constant small distance between the read/write head and the disk surface is an important target for the design of ultrahigh storage density drives [5–7]. In this case, the read/write head enters into intermittent contact with the disk surface, which results in a switched system regulation problem [5, 6]. Existing results in the literature on the exact output regulation problem for switched systems have been derived using different approaches and under different assumptions. The case of continuous-time switched systems with preknown switching times and where the model for the exogenous inputs is either completely or partially known is treated in [8]. Regulation in continuous-time piecewise affine (PWA) bimodal systems with completely known reference trajectories and where switching is defined according to a switching surface has been treated in [9] using a generalized error signal. General continuous-time PWA systems have been treated in [10] for the case of perfectly known reference trajectory. A different approach to the regulation problem in continuous-time bimodal linear systems against known exogenous signals was presented by Wu and Ben Amara in [11–13]. The design of the desired regulator was performed within a set of Youla (or 𝑄-) parameterized regulators for the system. The synthesis algorithms are formulated in sets of bilinear matrix inequalities [11] or linear matrix inequalities [12, 13], respectively.

In this paper, a Youla parameterized controller design approach is proposed for discrete-time switched bimodal linear systems. The goal is to design a controller to achieve regulation against partially known deterministic disturbance or reference signals in the presence of unknown random disturbances. The problem treated in this paper is motivated by the flying height regulation problem in hard or optical disk drives [5–7], where the read/write head is supposed to track, at a small constant distance, the disk surface profile in the presence of known and unknown disturbances and vibrations. The close proximity of the read/write head to the disk surface results in intermittent contact between the two and a bimodal system behaviour. The regulation problem is considered by constructing a set of observer-based parameterized stabilizing controllers that satisfy a sufficient regulation condition for the switched system. A corresponding regulator synthesis algorithm is developed based on solving properly formulated linear matrix inequalities.

The rest of the paper is organized as follows. In Section 2, the general regulation problem for the discrete-time switched bimodal system with deterministic input signals and unknown random signals is presented. In Section 3, a Youla parameterized set of stabilizing switched controllers for the switched system is constructed. In Section 4, regulation conditions for the switched system are presented. In Section 5, a regulator synthesis algorithm is proposed based on solving properly formulated linear matrix inequalities. In Section 6, experimental results are presented to illustrate the effectiveness of the proposed regulators, followed by the conclusion in Section 7.

2. The Regulation Problem for Switched Bimodal Systems

Consider the discrete-time switched bimodal system given by the following state space representation: Σ_𝑟∶𝑥(𝑘+1)=𝐴_𝑟𝑥(𝑘)+𝐵_𝑟𝑢(𝑘)+𝐷^𝑥_𝑟𝑑_𝑟(𝑘)+𝐹^𝑥_𝑟𝑑_𝑤(𝑘),𝑥(0)=𝑥₀,𝑦(𝑘)=𝐶^𝑦_𝑟𝑥(𝑘)+𝐷^𝑦_𝑟𝑑_𝑟(𝑘)+𝐹^𝑦_𝑟𝑑_𝑤(𝑘),𝑒(𝑘)=𝐶^𝑒_𝑟𝑥(𝑘)+𝐷^𝑒_𝑟𝑑_𝑟(𝑘)+𝐹^𝑒_𝑟𝑑_𝑤(𝑘),(𝑘)𝑟=1if𝑒=𝐶^𝑒_𝑟𝑥(𝑘)+𝐷^𝑒_𝑟𝑑_𝑟(𝑘)+𝐹^𝑒_𝑟𝑑_𝑤(𝑘)(𝑘)≤𝛿,2if𝑒=𝐶^𝑒_𝑟𝑥(𝑘)+𝐷^𝑒_𝑟𝑑_𝑟(𝑘)+𝐹^𝑒_𝑟𝑑_𝑤(𝑘)>𝛿, (2.1) where 𝑥∈𝐑^𝑛 is the state vector, 𝑢∈𝐑 is the control input, 𝑦∈𝐑 is the measurement signal to be fed to the controller, 𝑒∈𝐑 is the performance variable to be regulated and is assumed to be measurable, 𝑟∈{1,2} is the index of the system Σ_𝑟 under consideration at time 𝑘, and 𝛿 is a constant satisfying |𝛿|>0. The external signal 𝑑_𝑤∈𝐑^𝑙 represents unknown random signals. The external signal 𝑑_𝑟∈𝐑^ℎ, representing deterministic disturbance and/or reference signals, is also assumed to switch according to the rule given in (2.1). The signal 𝑑_𝑟 is given by: 𝑑_𝑟=𝑑¹_𝑟⋯𝑑^ℎ_𝑟^𝑇, (2.2) where each 𝑑^𝑗_𝑟 is of the following form: 𝑑^𝑗_𝑟(𝑘)=^𝑘𝑗_𝑖=1𝑐_𝑖^𝑟,𝑗cos𝜔_𝑖^𝑟,𝑗𝑘+𝜙_𝑖^𝑟,𝑗+𝑐_𝑘^𝑟,𝑗_𝑗+1,𝑟∈{1,2},𝑗=1,…,ℎ, (2.3) with known frequencies 𝜔_𝑖^𝑟,𝑗, and unknown amplitudes 𝑐_𝑖^𝑟,𝑗 and phases 𝜙_𝑖^𝑟,𝑗, 𝑖=1,…,𝑘_𝑗; 𝑗=1,…,ℎ. It should be noted that switching is dependent on a time varying switching surface defined by 𝑒(𝑘)=𝐶^𝑒_𝑟𝑥(𝑘)+𝐷^𝑒_𝑟𝑑_𝑟(𝑘)+𝐹^𝑒_𝑟𝑑_𝑤(𝑘), and furthermore the exogenous input 𝑑_𝑟 switches at the same time switching in the plant model takes place. Moreover, 𝑑_𝑟 is discontinuous during switching. The control objective with respect to the switched system (2.1) is to design an output feedback controller to regulate the performance variable 𝑒 of the switched system against the external input signals 𝑑_𝑟 and 𝑑_𝑤. More specifically, it is desired to achieve exact output regulation against the deterministic exogenous input 𝑑_𝑟 and to minimize the effects of the random input 𝑑_𝑤 to the performance variable 𝑒, simultaneously.

One of the motivations for the regulation problem presented for the system (2.1) above is the mechanical system shown in Figure 1. In this system, the mass 𝑀 is supposed to maintain a constant separation with the contact surface 𝑆 while the latter keeps moving laterally. The system exhibits a bimodal behaviour depending on whether contact between the mass 𝑀 and the contact surface 𝑆 takes place. A detailed model for such system is presented in [5, 11], which can be found in many practical applications, such as the flying height regulation problem for the read/write head in hard or optical disk drives.

Figure 1 

Example of a mechanical system with switched dynamics.

3. Construction of a Set of Parameterized Output Feedback Controllers

In this paper, the feedback controllers are obtained by considering Youla (𝑄-) parameterized output feedback controllers for the switched system (2.1). The construction of such set of controllers proceeds along the same lines as in [12, 13]. Each controller is expressed as a linear fractional transformation involving a fixed system 𝐽_𝑟 and a proper stable parameter 𝑄_𝑟 that could be chosen as desired (see Figure 2). The state space representation of the systems 𝐽_𝑟 and 𝑄_𝑟 are given by the following: 𝐽_𝑟∶𝑥(𝑘+1)=𝐴_𝑟+𝐿_𝑟𝐶^𝑦_𝑟+𝐵_𝑟𝐾_𝑟𝑥(𝑘)−𝐿_𝑟𝑦(𝑘)+𝐵_𝑟𝑦_𝑄(𝑘),𝑥(0)=𝑥₀,𝑢(𝑘)=𝐾_𝑟𝑥(𝑘)+𝑦_𝑄(𝑘),𝑦(𝑘)−𝑦(𝑘)=𝑦(𝑘)−𝐶^𝑦_𝑟𝑥(𝑘).𝑄_𝑟∶𝑥_𝑄(𝑘+1)=𝐴_𝑄𝑥_𝑄(𝑘)+𝐵_𝑄𝑦(𝑘)−𝑦(𝑘),𝑥_𝑄(0)=𝑥⁰_𝑄,𝑦_𝑄(𝑘)=𝐶_𝑄𝑟𝑥_𝑄(𝑘), (3.1) where 𝑥(𝑘) is an estimate of the plant state vector 𝑥(𝑘), 𝑦(𝑘)=𝐶^𝑦_𝑟𝑥(𝑘) is an estimate of the plant output 𝑦(𝑘), 𝐾_𝑟 and 𝐿_𝑟, 𝑟∈{1,2}, are state feedback and observer gains, respectively, 𝑥_𝑄∈𝑅^𝑛𝑄, 𝐴_𝑄 is a fixed stability matrix, 𝐵_𝑄 is a fixed matrix, and matrix 𝐶_𝑄𝑟 changes with 𝑟∈{1,2}. In this paper, the transfer function of the 𝑄_𝑟 parameter is considered as the form 𝑄_𝑟(𝑧)=𝐹(𝑧)∑^𝑛𝑞_𝑞=1𝜃^𝑞_𝑟𝑧^1−𝑞, where 𝐹(𝑧) is a stable function and 𝜃^𝑞_𝑟∈𝑅. Let 𝜒=[𝑥^𝑇𝑥^𝑇_𝑄𝑥^𝑇]^𝑇_1×𝑁, 𝑁=2𝑛+𝑛_𝑄, 𝑥(𝑘)=𝑥(𝑘)−𝑥(𝑘), then the resulting closed loop system is given by the following state space representation: ^𝑐𝑙_𝑟∶𝜒(𝑘+1)=𝐴_𝑟𝜒(𝑘)+𝐸^𝑑_𝑟𝑑_𝑟(𝑘)+𝐸^𝑤_𝑟𝑑_𝑤(𝑘),𝑒(𝑘)=𝐶_𝑟^𝑒𝑥𝜒(𝑘)+𝐷^𝑒_𝑟𝑑_𝑟(𝑘)+𝐹^𝑒_𝑟𝑑_𝑤(𝑘),(𝑘)𝑟=1if𝑒=𝐶_𝑟^𝑒𝑥𝜒(𝑘)+𝐷^𝑒_𝑟𝑑_𝑟(𝑘)+𝐹^𝑒_𝑟𝑑_𝑤(𝑘)(𝑘)≤𝛿,2if𝑒=𝐶_𝑟^𝑒𝑥𝜒(𝑘)+𝐷^𝑒_𝑟𝑑_𝑟(𝑘)+𝐹^𝑒_𝑟𝑑_𝑤(𝑘)>𝛿, (3.2) where 𝐴_𝑟=𝐴_𝑟+𝐵_𝑟𝐾_𝑟𝐵_𝑟𝐶_𝑄𝑟𝐵_𝑟𝐾_𝑟0𝐴_𝑄−𝐵_𝑄𝐶^𝑦_𝑟00𝐴_𝑟+𝐿_𝑟𝐶^𝑦_𝑟,𝐸^𝑑_𝑟=𝐷^𝑥_𝑟^𝑇(𝐵_𝑄𝐷^𝑦_𝑟)^𝑇−𝐷^𝑥_𝑟+𝐿_𝑟𝐷^𝑦_𝑟^𝑇𝑇,𝐸^𝑤_𝑟=𝐹^𝑥_𝑟^𝑇(𝐵_𝑄𝐹^𝑦_𝑟)^𝑇−𝐹^𝑥_𝑟+𝐿_𝑟𝐹^𝑦_𝑟^𝑇𝑇,𝐶_𝑟^𝑒𝑥=𝐶^𝑒_𝑟00. (3.3)

Figure 2 

Closed loop system with a 𝑄-parameterized controller.

To examine the internal stability of the closed loop system, consider the system (2.1) in the absence of the signals 𝑑_𝑟 and 𝑑_𝑤. The state equation for the resulting system is given by the following: 𝑥()𝑥𝑘+1_𝑄()𝑥()=𝐴𝑘+1𝑘+1^𝑟+𝐵_𝑟𝐾_𝑟𝐵_𝑟𝐶_𝑄𝑟𝐵_𝑟𝐾_𝑟0𝐴_𝑄−𝐵_𝑄𝐶^𝑦_𝑟00𝐴_𝑟+𝐿_𝑟𝐶^𝑦_𝑟𝑥(𝑘)𝑥_𝑄(𝑘)𝑥(𝑘). (3.4) Based on (3.4), define the following three subsystems: 𝑆1∶𝑥(𝑘+1)=𝐴_𝑟+𝐵_𝑟𝐾_𝑟𝑥(𝑘),𝑆2∶𝑥(𝑘+1)=𝐴_𝑟+𝐿_𝑟𝐶^𝑦_𝑟𝑥(𝑘),𝑆3∶𝑥_𝑄(𝑘+1)=𝐴_𝑄𝑥_𝑄(𝑘). (3.5)(3.6)(3.7)

Lemma 3.1. If the switched systems (3.5) and (3.6) are each asymptotically stable under arbitrary switching and 𝐴_𝑄 is a stability matrix, then the switched system (3.4) is also asymptotically stable under arbitrary switching.

Proof. If the switched system (3.6) is asymptotically stable under arbitrary switching signal, then we have [2] ‖𝑥(𝑘)‖=𝜙(𝑘,0)𝑥(0)≤𝑐̃‖𝑥(0)‖𝑎^𝑘, (3.8) for some 𝑐̃>0 and 𝑎∈(0,1) where 𝜙(𝑘,0) is the state transition matrix of the switched system (3.6). Therefore, 𝑥→0 as 𝑘→∞ for any 𝑥(0). Similarly, if the system (3.7) is asymptotically stable, then we have 𝑥_𝑄(𝑘)=𝜙_𝑄(𝑘,0)𝑥_𝑄(0)≤𝑐_𝑄𝑥_𝑄(0)𝑎^𝑘_𝑄, (3.9) for some 𝑐_𝑄>0 and 𝑎_𝑄∈(0,1), where 𝜙_𝑄(𝑘,0) is the state transition matrix of the system (3.7). The expression of the state variable 𝑥_𝑄 in (3.4) is given by: 𝑥_𝑄(𝑘)=𝜙_𝑄(𝑘,0)𝑥_𝑄(0)−^𝑘−1_𝑗=0𝜙_𝑄𝑘,𝑗𝐵_𝑄𝐶^𝑦_𝑟𝑥𝑗. (3.10) It follows 𝑥_𝑄(𝑘)≤𝜙_𝑄(𝑘,0)𝑥_𝑄(0)+^𝑘−1_𝑗=0𝜙_𝑄𝑘,𝑗𝐵_𝑄𝐶^𝑦_𝑟𝑥𝑗≤𝑐_𝑄𝑥_𝑄(0)𝑎^𝑘_𝑄+𝑐_𝑄𝑐̃‖𝑥(0)‖𝐵_𝑄𝐶^𝑦_𝑟max^𝑘−1_𝑗=0𝑎_𝑄^𝑘−𝑗𝑎^𝑗. (3.11) Hence, 𝑥_𝑄(𝑘)≤𝑐_𝑄𝑥_𝑄(0)𝑎^𝑘_𝑄+𝑐_𝑄𝑐𝑎̃_𝑄‖𝑥(0)‖𝐵_𝑄𝐶^𝑦_𝑟max𝑎_𝑄−𝑎𝑎^𝑘_𝑄−𝑎^𝑘,if𝑎_𝑄≠𝑎𝑥_𝑄(𝑘)≤𝑐_𝑄𝑥_𝑄(0)𝑎^𝑘_𝑄+𝑐_𝑄𝑐̃‖𝑥(0)‖𝐵_𝑄𝐶^𝑦_𝑟max𝑘𝑎^𝑘_𝑄,if𝑎_𝑄=𝑎, (3.12) which yields 𝑥_𝑄(𝑘)≤𝑐_𝑄𝑥_𝑄(0)+2𝑐_𝑄𝑐̃‖𝑥(0)‖𝐵_𝑄𝐶^𝑦_𝑟max𝑎_𝑄−𝑎𝑎^𝑘_𝑚,𝑎_𝑄≠𝑎, (3.13) where ‖𝐵_𝑄𝐶^𝑦_𝑟‖_max is the maximum value of ‖𝐵_𝑄𝐶^𝑦_𝑟‖ and 𝑎_𝑚=max(𝑎_𝑄,𝑎). Therefore, we have 𝑥_𝑄→0 as 𝑘→∞ for any 𝑥_𝑄(0). Similarly, we have that 𝑥→0 as 𝑘→∞ for any 𝑥(0) in (3.4).

The internal stability of the switched closed loop system (3.4) is then given by the following result.

Lemma 3.2. The origin is an asymptotically stable equilibrium point for the switched system (3.4) under arbitrary switching if 𝐴_𝑄 is a stability matrix and there exist matrices 𝑃_𝐾=𝑃^𝑇_𝐾>0, 𝑃_𝐿=𝑃^𝑇_𝐿>0, and matrices 𝐾_𝑟, 𝐿_𝑟, 𝑟∈{1,2}, such that 𝐴_𝑟+𝐵_𝑟𝐾_𝑟^𝑇𝑃_𝐾𝐴_𝑟+𝐵_𝑟𝐾_𝑟−𝑃_𝐾𝐴<0,_𝑟+𝐿_𝑟𝐶^𝑦_𝑟^𝑇𝑃_𝐿𝐴_𝑟+𝐿_𝑟𝐶^𝑦_𝑟−𝑃_𝐿<0. (3.14)

Proof. Condition (3.14) implies that each of the switched systems S1 and S2 admits a common Lyapunov function. Hence, each of those two systems is asymptotically stable under arbitrary switching. Since 𝐴_𝑄 is a stability matrix, system S3 is also asymptotically stable. Based on Lemma 3.1, it follows immediately that the switched system (3.4) is asymptotically stable under arbitrary switching.

Remark 3.3. Based on Lemma 3.1 and Lemma 3.2, a Youla parameterized stabilizing controller can be designed by separately considering the subsystems S1, S2, and S3. If feasible parameters 𝐾_𝑟 and 𝐿_𝑟 can be found to make the switched systems S1 and S2 asymptotically stable, then a stable 𝑄 parameter of the form 𝑄(𝑧)=((𝑏₁𝑧^𝑠−1+⋯+𝑏_𝑠)/(𝑧^𝑠+𝑎₁𝑧^𝑠−1+⋯+𝑎_𝑠))∑^𝑛𝑞_𝑖=1𝜃_𝑖𝑧^1−𝑖 could make the overall switched system (3.4) asymptotically stable. The state-space realization of 𝑄 can be given as: 𝐴_𝑄=0⋯00−𝑎_𝑠0001⋯00−𝑎_𝑠−1000⋮⋱⋮⋮⋮⋮⋮⋮0⋯10−𝑎₂0000⋯01−𝑎₁0000⋯0010000⋯000⋱000⋯000010_{𝑛𝑄×𝑛𝑄},𝐵_𝑄=𝑏_𝑠𝑏_𝑠−1⋮𝑏₂𝑏₁0⋮0_𝑛𝑄×1,𝐶_𝑄=0_{1×(𝑠−1)}𝜃^𝑇, (3.15) where 𝜃=[𝜃₁,…,𝜃_𝑛𝑞]^𝑇 and 𝑛_𝑄=𝑠+𝑛_𝑞−1. It follows that 𝐴_𝑄 is a fixed stability matrix, 𝐵_𝑄 is a fixed matrix, and the matrix 𝐶_𝑄 changes with the parameter vectors 𝜃. A set of stabilizing controllers for switched system (3.4) is hence parameterized by the parameter vector 𝜃∈𝑅^𝑛𝑞. The desired controller could be searched among 𝜃∈𝑅^𝑛𝑞 according to the regulation conditions to be discussed in Sections 4 and 5.

4. Regulation Conditions for the Switched Bimodal Systems

This section presents regulation conditions for the switched closed loop system (3.2) against the external input signals 𝑑_𝑟 and 𝑑_𝑤. Due to the presence of the unknown random signal 𝑑_𝑤, it is not possible to achieve exact output regulation. Consequently, the goal behind the regulator design is to achieve exact output regulation against the deterministic exogenous input 𝑑_𝑟 and minimize the 𝐻₂ norm of the closed loop system relating the input 𝑑_𝑤 to the performance variable 𝑒. It is proposed in this section to achieve these goals by exploiting the flexibility in the selection of the 𝑄 parameter in the parameterized set of stabilizing controllers for the switched system. It is well known that the exact regulation conditions for each of the two systems Σ₁^cl and Σ₂^cl against 𝑑_𝑟 can be presented in the form of interpolation conditions [13, 14] and can be written in the form of linear equations in the unknown parameter vectors 𝜃_𝑟=[𝜃¹_𝑟,…,𝜃^𝑛𝑞_𝑟]^𝑇, 𝑟∈{1,2}, as follows: 𝐴_𝜃𝑟𝜃_𝑟+𝐵_𝜃𝑟=0, (4.1) where 𝐴_𝜃𝑟 and 𝐵_𝜃𝑟 are properly formulated matrices. However, using the resulting vector 𝜃_𝑟, 𝑟∈{1,2}, the controllers do not guarantee achieving regulation in the switched closed loop system. In the following, additional conditions are placed on the controllers satisfying (4.1) so that switching in the closed loop system would stop after a finite amount of time and exact output regulation would be achieved in the switched closed loop system (3.2) against the deterministic input 𝑑_𝑟. Furthermore, in order to minimize the effects of the unknown random disturbance 𝑑_𝑤 on the performance variable 𝑒, an 𝐻₂ performance constraint is added in the design of the regulator for the closed loop system. Consider the closed loop system (3.2), and let 𝑇_𝑤 denote the transfer function from 𝑑_𝑤 to 𝑒 for ∑₁^cl when 𝛿>0 or for ∑₂^cl when 𝛿<0 and ‖𝑇_𝑤‖₂ denote the 𝐻₂ norm of 𝑇_𝑤, then we have the following regulation condition for the switched system (3.2).

Theorem 4.1. Consider the closed loop system (3.2) subject to bounded inputs 𝑑_𝑟 and 𝑑_𝑤. Let 𝜅=max_{𝑟∈{1,2},𝑘≥0}(‖𝑑_𝑟(𝑘)‖²+‖𝑑_𝑤(𝑘)‖²)≠0, and 𝛼>0 a preset constant and 0<𝜀≪1. If for 𝑟∈{1,2}, there exist matrices 𝑃>0, 𝑋>0, 𝑆>0, and positive scalars 𝜇 and 𝛽≤|𝛿| such that the following matrix inequalities are satisfied: −𝑃+𝛼𝑃0𝐴_𝑟^𝑇𝑃()𝜇0−1−𝜀𝜅𝐼𝐸^𝑑_𝑟𝐸^𝑤_𝑟^𝑇𝑃𝑃𝐴_𝑟𝑃𝐸^𝑑_𝑟𝐸^𝑤_𝑟𝐶−𝑃<0,𝛼𝑃0_𝑟^𝑒𝑥𝑇0𝛽−𝜇𝜅𝐼𝐷^𝑒_𝑟^𝑇𝐹^𝑒_𝑟^𝑇𝐶_𝑟^𝑒𝑥𝐷^𝑒_𝑟𝐹^𝑒_𝑟𝐴𝛽𝐼>0,_𝜃𝑟𝜃_𝑟+𝐵_𝜃𝑟𝐴=0,𝑟=1𝑖𝑓𝛿>0,2𝑖𝑓𝛿<0,−𝑋𝑋_𝑟0𝐴^𝑇_𝑟𝑋−𝑋𝐶_𝑟^𝑒𝑥𝑇0𝐶_𝑟^𝑒𝑥−𝐼<0,𝑟=1𝑖𝑓𝛿>0,2𝑖𝑓𝛿<0,𝑋0𝑋𝐸^𝑤_𝑟0𝐼𝐹^𝑒_𝑟𝐸^𝑤_𝑟^𝑇𝑋𝐹^𝑒_𝑟^𝑇𝑆(𝑆)>0,𝑟=1𝑖𝑓𝛿>0,2𝑖𝑓𝛿<0,𝑇𝑟<𝛾², (4.2)(4.3)(4.4)(4.5)(4.6)(4.7) then the state vector 𝜒(𝑘) and the performance variable 𝑒(𝑘) of the switched closed loop system (3.2) are ultimately bounded. Moreover, the switched closed loop system achieves exact output regulation against the deterministic input 𝑑_𝑟 and satisfies ‖𝑇_𝑤‖₂<𝛾 with respect to the unknown random input 𝑑_𝑤.

Proof. Consider the state space representation (3.2) and the quadratic function 𝑉(𝜒)=𝜒^𝑇𝑃𝜒, based on the Schur complement formula on (4.2), for any nonzero vector ^{𝑑𝜒(𝑘)}_𝑟𝑑^(𝑘)_𝑤(𝑘)∈𝑅^2𝑁+ℎ+1, we have 𝑑𝜒(𝑘)_𝑟𝑑(𝑘)_𝑤(𝑘)^𝑇𝐴_𝑟^𝑇𝑃𝐴_𝑟−𝑃+𝛼𝑃𝐴_𝑟^𝑇𝑃𝐸^𝑑_𝑟𝐸^𝑤_𝑟𝐸^𝑑_𝑟𝐸^𝑤_𝑟^𝑇𝑃𝐴_𝑟𝐸^𝑑_𝑟𝐸^𝑤_𝑟^𝑇𝑃𝐸^𝑑_𝑟𝐸^𝑤_𝑟−(1−𝜀)𝜇𝜅𝐼𝜒(𝑘)𝑑_𝑟(𝑘)𝑑_𝑤(𝑘)<0. (4.8) It follows from the above inequality that for the switched system (3.2): 𝑉𝜒()𝜒(𝑘)<()𝜇𝑘+1−𝑉1−𝜀𝜅𝑑_𝑟(𝑘)𝑑_𝑤(𝑘)^𝑇𝑑_𝑟(𝑘)𝑑_𝑤(𝑘)−𝛼𝑉𝜒(𝑘). (4.9) Hence, 𝑉(𝜒(𝑘+1))−𝑉(𝜒(𝑘))<0 holds whenever 𝜒(𝑘)≥()()𝜇𝛼𝑉1−𝜀𝜇≥1−𝜀𝜅𝑑_𝑟(𝑘)²+𝑑_𝑤(𝑘)². (4.10) Consequently, 𝑉(𝜒(𝑘)) cannot ultimately exceed the value (1−𝜀)𝜇/(𝛼), and we have lim_𝑘→∞𝑉(𝜒(𝑘))≤(1−𝜀)𝜇/(𝛼).
Therefore, there exists a finite time 𝑘_𝑏 such that 𝑉𝜒(𝑘)≤𝜇𝛼,∀𝑘≥𝑘_𝑏. (4.11) Using the Schur complement formula, inequality (4.3) is equivalent to: 0𝛼𝑃0𝛽−𝜇𝜅𝐼−1𝛽𝐶_𝑟^𝑒𝑥𝑇𝐷^𝑒_𝑟^𝑇𝐹^𝑒_𝑟^𝑇𝐶_𝑟^𝑒𝑥𝐷^𝑒_𝑟𝐹^𝑒_𝑟>0. (4.12) Multiplying from the left by 𝑑𝜒(𝑘)_𝑟𝑑(𝑘)_𝑤(𝑘)𝑇 and from the right by 𝑑𝜒(𝑘)_𝑟𝑑(𝑘)_𝑤(𝑘) yields 𝑒²(𝑘)<𝛽𝛼𝑉𝜒(𝑘)+𝛽−𝜇𝜅𝑑_𝑟𝑑(𝑘)_𝑤(𝑘)^𝑇𝑑_𝑟(𝑘)𝑑_𝑤(𝑘)<𝛽𝛼𝑉𝜒(𝑘)+𝛽−𝜇. (4.13) Then, based on (4.11), we have 𝑒²(𝑘)<𝛽², ∀𝑘≥𝑘_𝑏. Hence, 𝑒(𝑘) is ultimately bounded and we have |𝑒(𝑘)|<𝛽,∀𝑘≥𝑘_𝑏. (4.14) If 𝛽≤|𝛿| is satisfied, then after a long enough time 𝑘_𝑏, the output 𝑒(𝑘) will always be such that |𝑒(𝑘)|<𝛽≤|𝛿| and there will be no more switching. For the case of 𝛿>0, the system ∑₁^cl is active for 𝑘≥𝑘_𝑏. Given that the system ∑₁^cl is asymptotically stable and that the parameter vector 𝜃₁ satisfies the regulation condition in (4.1), then exact output regulation for the switched system is achieved against the deterministic input 𝑑_𝑟. Furthermore, based on (4.5)–(4.7), it is easy to obtain 𝑇𝑟((𝐹^𝑒_𝑟)^𝑇𝐹^𝑒_𝑟+(𝐸^𝑤_𝑟)^𝑇𝑋𝐸^𝑤_𝑟)<𝛾², where 𝐴^𝑇_𝑟𝑋𝐴_𝑟−𝑋+(𝐶_𝑟^𝑒𝑥)^𝑇𝐶_𝑟^𝑒𝑥<0 and 𝑋>0, which implies ‖𝑇_𝑤‖₂<𝛾 [15]. The analysis is similar for the case of 𝛿<0.

Remark 4.2. The main reason for considering common Lyapunov functions as opposed to multiple Lyapunov functions in (3.14), (4.2) and (4.3) is as follows.The existence of a common Lyapunov function for each of the switched systems S1 and S2 in (3.14) implies the existence of a common Lyapunov function in (4.2) and (4.3) for the system (3.4). The less conservative regulation conditions (4.2) and (4.3) can be formulated using multiple Lyapunov functions as in the approach presented in [13]. However, as outlined in Section 5, the use of a common Lyapunov function allows for the regulation conditions to be transformed into linear matrix inequalities (LMIs) in the regulator synthesis algorithm using an appropriate congruence transformation so that a global optimal solution can be found for the controller. The use of multiple Lyapunov functions would have made such transformation very difficult. Moreover, the synthesis algorithm normally would in that case be formulated using bilinear matrix inequalities, which would typically yield a conservative local solution for the controller.

Remark 4.3. In Theorem 4.1, the 𝐻₂ performance constraint applies only to the closed loop system that is active when regulation against the deterministic exogenous input is achieved, that is, the closed loop system operating in mode 1 if 𝛿>0 and in mode 2 if 𝛿<0. This is due to the fact that if conditions (4.2) and (4.3) are satisfied, then switching in the closed loop system stops after a finite amount of time. Consequently, it is enough to apply the additional 𝐻₂ performance constraint only to the closed loop system that is active when switching stops after a finite amount of time. Optimal performance for switched systems defined based on other performance measures, as presented for example in [16, 17], can also be considered. However, that will introduce conservativeness in the final solution since continuous switching is assumed to take place in those approaches, which is not the case for the systems considered in this paper.

5. Regulator Synthesis

The synthesis method aims at finding the proper 𝐾_𝑟, 𝐿_𝑟, and 𝐶_𝑄𝑟 such that conditions (4.2)–(4.7) are satisfied. The main idea behind the regulator synthesis is to first design the gains 𝐾_𝑟 and 𝐿_𝑟 based on (3.14). Since 𝐴_𝑄 and 𝐵_𝑄 in (4.2) and (4.5) are fixed, then it is only necessary to find the matrices 𝐶_𝑄𝑟 such that the regulation conditions (4.2)–(4.7) are satisfied. Partition 𝑃 in (4.2) as [𝑃₁]_𝑛×𝑛𝑃₂𝑃^𝑇₂𝑃₃ and define Ω₁=𝑃₁⁻¹, Ω₂=𝑃₁⁻¹𝑃₂,Ω₃=𝑃₃−𝑃^𝑇₂𝑃₁⁻¹𝑃₂,Ω=^Ω10_{−Ω𝑇2𝐼} and 𝑀_𝑟Ω₁,Ω₂,Ω₃,𝐶_𝑄𝑟=𝐴_𝑟+𝐵_𝑟𝐾_𝑟Ω₁Ω₂𝐴_𝑄−𝐵_𝑄𝐶_𝑟0𝐴_𝑟+𝐿_𝑟𝐶_𝑟−𝐴_𝑟+𝐵_𝑟𝐾_𝑟Ω₂+𝐵_𝑟𝐶_𝑄𝑟𝐵_𝑟𝐾_𝑟0Ω₃𝐴_𝑄−𝐵_𝑄𝐶_𝑟0𝐴_𝑟+𝐿_𝑟𝐶_𝑟. (5.1) Multiplying (4.2) from the left side by diag(Ω,𝐼,Ω) and from the right side by diag(Ω,𝐼,Ω)^𝑇 yields ()Ω𝛼−1¹00Ω₃0𝑀^𝑇_𝑟Ω₁,Ω₂,Ω₃,𝐶_𝑄𝑟()𝜇0−1−𝜀𝜅𝐼𝐸^𝑑_𝑟𝐸^𝑤_𝑟^𝑇Ω𝐼0_𝑇2Ω₃𝑀_𝑟Ω₁,Ω₂,Ω₃,𝐶_𝑄𝑟𝐼Ω₂0Ω₃𝐸^𝑑_𝑟𝐸^𝑤_𝑟−Ω₁00Ω₃<0. (5.2) Multiplying (4.3) from the left side by diag(Ω,𝐼,𝐼) and from the right side by diag(Ω^𝑇,𝐼,𝐼) yields 𝛼Ω₁00Ω₃0Ω₁0−Ω^𝑇₂𝐼𝐶_𝑟^𝑒𝑥𝑇0𝛽−𝜇𝜅𝐼𝐷^𝑒_𝑟^𝑇𝐹^𝑒_𝑟^𝑇𝐶_𝑟^𝑒𝑥Ω₁−Ω₂𝐷0𝐼^𝑒𝑟𝐹^𝑒_𝑟𝛽𝐼>0. (5.3) Similarly, we partition 𝑋 in (4.5) as [𝑋₁]_𝑛×𝑛𝑋₂𝑋^𝑇₂𝑋₃ and define Ψ₁=𝑋₁⁻¹, Ψ₂=𝑋₁⁻¹𝑋₂, Ψ₃=𝑋₃−𝑋^𝑇₂𝑋₁⁻¹𝑋₂ and Ψ=^Ψ10_{−Ψ𝑇2𝐼}. Multiplying (4.5) from the left side by diag(Ψ,Ψ,𝐼) and from the right side by diag(Ψ^𝑇,Ψ^𝑇,𝐼) yields the following: −Ψ₁𝟎𝟎Ψ₃𝑀_𝑟Ψ₁,Ψ₂,Ψ₃,𝐶_𝑄𝑟𝟎𝑀^𝑇_𝑟Ψ₁,Ψ₂,Ψ₃,𝐶_𝑄𝑟−Ψ₁𝟎𝟎Ψ₃Ψ₁𝟎−Ψ^𝑇₂𝐼𝐶_𝑟^𝑒𝑥𝑇𝟎𝐶_𝑟^𝑒𝑥Ψ₁−Ψ₂𝟎𝐼−𝐼<0,𝑟=1if𝛿>0,2if𝛿<0. (5.4) Multiplying (4.6) from the left side by diag(Ψ,𝐼,𝐼) and from the right side by diag(Ψ^𝑇,𝐼,𝐼) yields Ψ₁𝟎𝟎Ψ₃𝟎𝐼Ψ₂0Ψ₃𝐸^𝑤_𝑟𝟎𝐼𝐹^𝑒_𝑟𝐸^𝑤_𝑟^𝑇Ψ𝐼0_𝑇2Ψ₃𝐹^𝑒_𝑟^𝑇𝑆>0, (5.5) In the following, a regulator synthesis algorithm is presented to obtain 𝐶_𝑄𝑟 matrices that simultaneously minimize the 𝐻₂ performance 𝛾 and satisfy the regulation conditions in Theorem 4.1. The main steps in the proposed regulator synthesis procedure are summarized below.(1)Design of an internally stabilizing controller: design the controller parameters 𝐾_𝑟 and 𝐿_𝑟 to make the switched closed loop system (3.2) internally stable by solving (3.14) for the unknown matrices 𝑃_𝐾=𝑃^𝑇_𝐾>0, 𝑃_𝐿=𝑃^𝑇_𝐿>0 and matrices 𝐾_𝑟, 𝐿_𝑟, 𝑟∈{1,2}. (2)Find an upper bound 𝛼_𝑚 on the value of 𝛼 in (5.2) with unknown Ω₁>0, Ω₂, Ω₃>0, and 𝐶_𝑄𝑟, which is a generalized eigenvalue problem. Select a value of 𝛼 in the interval 𝛼∈(0,𝛼_𝑚). (3)Substitute |𝛿| for 𝛽 in (5.3). (4)Minimize 𝑇𝑟(𝑆) subject to (4.4), (5.2), (5.3), (5.4) and (5.5), in the unknown 𝜇, Ω₁, Ω₂, Ω₃, Ψ₁, Ψ₂, Ψ₃, 𝑆, and 𝐶_𝑄𝑟 with known 𝛼.

Note that (5.2) and (5.3) are linear in the unknown parameters 𝛽, 𝜇, Ω₁, Ω₂, Ω₃ and 𝐶_𝑄𝑟 only if 𝛼 is prefixed. The value of 𝛼 can be selected using a linear search in the interval 𝛼∈(0,𝛼_𝑚) in order to solve the optimization problem in Step 4. In the following, an experimental example is presented to show the effectiveness of the proposed regulator synthesis method.

6. Experimental Evaluation

The experimental setup represents a mechanical system subject to contact vibrations and is motivated by the flying height regulation problem in hard or optical disk drives [5–7]. The experimental setup consists mainly of a flexible beam subject to intermittent contact with a fictitious contact surface. The system has two modes of operation, one corresponding to the case of noncontact between the beam tip and the contact surface, and the second corresponds to the case when contact takes place. The control goal in the experiment is to maintain a constant separation between the tip of the flexible beam and the contact surface in the presence of external disturbances and switching between the noncontact mode and the contact mode of the system. Images of the experimental setup are shown in Figure 3. A flexible beam (the suspension beam in a hard disk drive) is actuated using a multilayer piezoelectric actuator (PZT). A Laser Doppler Vibrometer (LDV) provides real-time measurements of the position and velocity of the beam tip. A small-size permanent magnet is attached to the tip of the beam. In the experimental setup, contact between the beam tip and a contact surface is simulated by generating a vertical contact force using an electromagnetic actuator underneath and applying the generated force to the beam tip. The forces representing deterministic disturbances and random disturbances are also produced using the electromagnetic actuator. The sampling period in the experimental setup is of 1 KHz. The two subsystem models corresponding to the contact mode and the noncontact mode are obtained using system identification method. The regulator is designed so that the tip of the suspension beam tracks a contact surface 𝑠_𝑐=(5sin(240𝜋𝑡)−60)×10⁻⁶ m at a desired separation height of 60×10⁻⁶ m. The output 𝑦 and the performance variable 𝑒 are identical and are defined to be the difference between the actual separation and the desired separation height. The state space representation as in (2.1) is given as 𝐴₁=0−0.87351−0.5645,𝐴₂=,𝐵0−0.93131−0.2459₁=1.80450.9125,𝐵₂=,𝐷2.16011.0036_𝑥1=0.202100.20100,𝐹^𝑥₁=,𝐷0.20210.2010_𝑥2=0.226700.21870,𝐹^𝑥₂=,𝐶0.22670.2187_𝑒1=𝐶^𝑦₁=01,𝐶^𝑒₂=𝐶^𝑦₂=,𝐷01_𝑦1=𝐷^𝑦₂=𝐷^𝑒₁=𝐷^𝑒₂=01,𝐹^𝑦₁=𝐹^𝑦₂=𝐹^𝑒₁=𝐹^𝑒₂=[0].𝑑₁=0.5cos240𝜋𝑘+3𝜋4()()()𝑑+0.4sin240𝜋𝑘+0.3cos240𝜋𝑘0.5sin240𝜋𝑘volts,₂=0.5cos240𝜋𝑘+3𝜋40.5sin(240𝜋𝑘)volts. (6.1) The switching rule is defined as 𝑟=^{1if𝑒≤−6volts}_{2if𝑒>−6volts}. To construct a parameterized set of stabilizing controllers, the gains 𝐾_𝑟, 𝐿_𝑟, 𝑟∈{1,2}, designed based on (3.14) are given by: 𝐾₁=[−159.1746504.2805]×10⁻³,𝐾₂=[−126.6187407.4727]×10⁻³,𝐿₁=[873.4887564.5477]^𝑇×10⁻³,𝐿₂=[931.2618245.8963]^𝑇×10⁻³. (6.2) The set of controllers to be designed is obtained by considering 𝑄_𝑟 parameters of the form 𝑄_𝑟=((0.46𝑧+0.45)/(𝑧²−1.4𝑧+0.9))((𝜃¹_𝑟/𝑧)+(𝜃²_𝑟/𝑧²)+(𝜃³_𝑟/𝑧³)+(𝜃⁴_𝑟/𝑧⁴))×10³. Since 𝛿<0, condition (4.1) needs to consider only the system Σ₂ and yields: 𝐴_𝜃2=0.86198.825012.00448.676611.97398.1385−0.1084−8.2966×10³,𝐵_𝜃2=305.1971−261.1491×10⁻³. (6.3) First, without considering the 𝐻₂ performance constraint in the regulator synthesis procedure, the regulator is designed based only on the regulation conditions (4.4), (5.2), and (5.3). We have 𝐶_𝑄1=0−1.04480.4263−0.09050.0975×10⁻³,𝐶_𝑄2=0−0.5360−0.51211.3472−1.3250×10⁻³. (6.4) The experimental results corresponding to the above designed controller are shown in Figures 4, 5, 6, and 7, which include the two cases of with and without the unknown random disturbance 𝑑_𝑤. It can be seen that the resulting system with the designed controller can exactly reject the known disturbance in the presence of switching (contacting), however it cannot attenuate the disturbance 𝑑_𝑤 properly. By considering the 𝐻₂ constraint in the regulator synthesis procedure, we obtain 𝐶_𝑄1=00.1376−0.24520.16650.0208×10⁻³,𝐶_𝑄2=00.1129−0.23340.2105−0.1002×10⁻³,𝛾²=0.118. (6.5) The experimental results are shown in Figures 8 and 9, which indicate that the disturbance 𝑑_𝑤 has been attenuated effectively and the tip of the suspension beam tracks the contact surface while maintaining the desired separation between the tip and the contact surface.

Figure 3 

Images of the experimental setup.

(a)

(b)

Figure 4 

Experimental results of the performance variable and the positions of the tip and of the contact surface for the case of 𝑑𝑤=0 and obtained using the controller designed without accounting for the 𝐻2 performance constraint.

(a)

(b)

Figure 5 

Experimental results showing the random disturbance 𝑑𝑤 and the total current in the electromagnetic actuator coil obtained using the controller designed without accounting for the 𝐻2 performance constraint.

(a)

(b)

Figure 6 

Experimental results of the performance variable and the positions of the tip and of the contact surface in the presence of 𝑑𝑤 and obtained using the controller designed without accounting for the 𝐻2 performance constraint.

(a)

(b)

Figure 7 

Experimental results showing the random disturbance 𝑑𝑤 and the total current in the electromagnetic actuator coil obtained using the controller designed without accounting for the 𝐻2 performance constraint.

(a)

(b)

Figure 8 

Experimental results of the performance variable and the positions of the tip and of the contact surface in the presence of 𝑑𝑤 and obtained using the controller designed by accounting for the 𝐻2 performance constraint.

(a)

(b)

Figure 9 

Experimental results showing the random disturbance 𝑑𝑤 and the total current in the electromagnetic actuator coil obtained using the controller designed by accounting for the 𝐻2 performance constraint.

7. Conclusion

The problem of regulation in discrete-time bimodal switched systems against partially known deterministic exogenous inputs and unknown random inputs is treated. A regulator design approach based on the parameterization of a set of stabilizing controllers for the switched system is presented. First, regulation conditions for the switched system are derived based on the Youla parameterization of a set of stabilizing controllers, where a set of 𝑄 parameters is constructed to achieve regulation against the deterministic exogenous inputs and also against the unknown random exogenous input. Then, the regulator synthesis algorithm is developed based on solving a set of properly formulated linear matrix inequalities. Finally, a switched bimodal mechanical system experimental setup involving a flexible beam subject to contact vibrations is used to successfully demonstrate the performance of the proposed regulator.

Acknowledgments

The authors would like to thank the associate editor and the anonymous reviewers for their comments. This work was supported by the National Natural Science Foundation of China (51075254), the Shanghai Pujiang Program (11PJ1404000), the Innovation Program of Shanghai Municipal Education Commission (11YZ16), and the Natural Sciences and Engineering Research Council of Canada.