Generalizing Negative Imaginary Systems Theory to Include Free Body Dynamics:Control of Highly Resonant Structures with Free Body Motion

Generalizing Negative Imaginary Systems Theory to Include Free Body Dynamics: Control of Highly Resonant Structures with Free Body Motion

M. A. Mabrok, A. G. Kallapur, I. R. Petersen and A. Lanzon M. Mabrok, A. Kallapur and I. R. Petersen are with the School of Engineering and Information Technology, University of New South Wales at the Australian Defence Force Academy, Canberra ACT 2600, Australia, email:abdallamath@gmail.com, abhijit.kallapur@gmail.com, i.r.petersen@gmail.com.A. Lanzon is with the Control Systems Centre, School of Electrical and Electronic Engineering, University of Manchester, Manchester M13 9PL, United Kingdom, email:Alexander.Lanzon@manchester.ac.uk.This research was supported by the Australian Research Council and the EPSRC.
Abstract

Negative imaginary (NI) systems play an important role in the robust control of highly resonant flexible structures. In this paper, a generalized NI system framework is presented. A new NI system definition is given, which allows for flexible structure systems with colocated force actuators and position sensors, and with free body motion. This definition extends the existing definitions of NI systems. Also, necessary and sufficient conditions are provided for the stability of positive feedback control systems where the plant is NI according to the new definition and the controller is strictly negative imaginary. The stability conditions in this paper are given purely in terms of properties of the plant and controller transfer function matrices, although the proofs rely on state space techniques. Furthermore, the stability conditions given are independent of the plant and controller system order. As an application of these results, a case study involving the control of a flexible robotic arm with a piezo-electric actuator and sensor is presented.

Negative imaginary systems, flexible structures, free body motion.

I Introduction

Flexible structure dynamics arise in many areas such as flexible robot manipulators [1], ground and aerospace vehicles [2], atomic force microscopes (AFMs) [3, 4] and other nano-positioning systems [5, 6, 7, 8]. Flexible structures can be modeled as infinite dimensional distributed parameter systems [9]. However, finite dimensional models are often used for the purpose of designing controllers [10, 11, 9, 12]. In designing controllers for these flexible systems, it is important to consider the effect of highly resonant modes. Such resonant modes are known to adversely affect the stability and performance of flexible structure feedback control systems [12, 13, 14], and are often very sensitive to changes in environmental variables. For instance, a small change in the environment of the system such as changing temperature, can lead to significant changes in the resonant frequencies of such systems. These changes in resonant frequencies can lead to large changes in the gain and phase of the system frequency response at a given frequency, which may lead to instability or poor performance in the corresponding feedback system. In addition, highly resonant modes lead to vibrational effects which limit the ability of control systems to achieve desired levels of performance in many applications such as precision instrumentation, optical systems, precision machine tools, wafer steppers, telescopes, and atomic force microscopes [12]. These issues arising from the presence of highly resonant modes in flexible structures motivate the need for tools to guarantee robust stability and performance in flexible structure control systems.

One common solution to issues of robustness, stability, and performance in the control of highly resonant flexible structures is to use force actuators combined with colocated measurements of velocity, position, or acceleration [12, 13, 14]. Colocated control with velocity measurements, known as negative-velocity feedback, can be used to directly increase the effective damping in the system, thereby facilitating the design of controllers that can guarantee closed-loop stability in the presence of parameter variations and unmodeled plant dynamics [12]. Similarly, a class of colocated controllers with position measurements, known as positive-position feedback controllers, where velocity sensors are replaced with position sensors, can also be used to increase damping in flexible systems as discussed in [13, 15]. Also, positive-position feedback controllers are robust against uncertainties in resonant frequencies as well as unmodeled plant dynamics, in a similar way to negative-velocity feedback controllers [13, 16, 14].

The properties of negative-velocity feedback has been studied using passivity theory and the theory of positive real (PR) linear time invariant (LTI) systems; e.g., see [17, 18]. However, PR theory cannot be used directly when using position or acceleration measurements [14]. This drawback is important in applications to the field of nanotechnology, especially for nano-positioning systems, where position measurements are widely used; see e.g., [3, 19, 20, 5, 6, 21, 7, 22, 23, 8]. Similar issues also arise in application to the area of robotics where position measurements are also widely used.

Lanzon and Petersen introduced a notion of negative imaginary (NI) systems in [16, 14] for the robust control of flexible structures with force actuators combined with position or acceleration sensors. (SISO) case, NI systems are defined by considering the properties of the imaginary part of the system frequency response and requiring the condition for all . The NI property arises in many practical systems. For example, such systems arise when considering the transfer function from a force actuator to a corresponding colocated position sensor (for instance, a piezoelectric sensor) in a lightly damped structure [13, 14, 3, 24, 25]. Another area where the underlying system dynamics are NI, in the area of nano-positioning systems; see e.g., [3, 19, 20, 5, 6, 21, 7, 22, 23, 8]. Also, the positive-position feedback control scheme in [13, 26], can be considered using the NI framework. Furthermore, other control methodologies in the literature such as integral resonant control (IRC) [27] and resonant feedback control [28, 29], fit into the NI framework and their stability robustness properties can be explained by NI systems theory.

The stability robustness of interconnected NI systems has been studied in [16, 14]. In these papers, it is shown that a necessary and sufficient condition for the internal stability of a positive-feedback control system (see Fig. 1) consisting of an NI plant with transfer function matrix and a strictly negative imaginary (SNI) controller with transfer function matrix is given by the DC gain condition

 λmax(G(0)¯G(0))<1, (1)

where the notation denotes the maximum eigenvalue of a matrix with only real eigenvalues. This stability result has been used in a number of practical applications [25, 3, 4, 30, 31, 8]. For example in [25], this stability result is applied to the problem of decentralized control of large vehicle platoons. In [3, 4], the NI stability result is applied to nanopositioning in an atomic force microscope. A positive position feedback control scheme based on the NI stability result provided in [16, 14] is used to design a novel compensation method for a coupled fuselage-rotor mode of a rotary wing unmanned aerial vehicle in [30]. In [8], an IRC scheme based on the stability results provided in [16, 14] is used to design an active vibration control system for the mitigation of human induced vibrations in light-weight civil engineering structures, such as floors and footbridges via proof-mass actuators. An identification algorithm which enforces the NI constraint is proposed in [31] for estimating model parameters, following which an Integral resonant controller is designed for damping vibrations in flexible structures. In addition, it is shown in [32] that the class of linear systems having NI transfer function matrices is closely related to the class of linear Hamiltonian input-output systems. Also, an extension of the NI systems theory to infinite-dimensional systems is presented in [33].

The NI framework presented in [16, 14] considers systems with poles in the open left half of the complex plane. This theory has been extended in [34] to include NI systems with poles in the closed left half of the complex plane, except at the origin. Also, further extensions to NI systems theory include the study of NI controller synthesis [35, 36], connections between NI systems analysis and -analysis [37], and conditions for robust stability analysis of mixed NI and bounded-real classes of uncertainties [38]. Furthermore, the concept of lossless NI transfer functions is introduced in [39], an algebraic approach to the realization of a lossless NI behavior is presented in [40], and a spectral characterization of NI descriptor systems is discussed in [41]. The NI systems theory can be extended to nonlinear systems using the concept of counter-clockwise input-output dynamics as presented in [42, 43, 44]. In [44], a sufficient conditions under which a semilinear Duhem model is counter-clockwise is given, where the counter-clockwise input-output system is restricted to periodic input signals. positive feedback interconnection for SISO linear case is provided in [44].

Despite generalizations of the NI systems framework presented in [34], an important class of systems, that cannot be captured by the existing NI systems framework, corresponds to flexible systems with free body motion. These systems arise in areas such as rotating flexible spacecraft [45], rotary cranes [46], robotics and flexible link manipulators [27, 29, 47], and dual-stage hard disk drives [48, 49, 50, 51]. Flexible structures with free body motion lead to dynamical models including poles at the origin, which is not covered in earlier work on NI systems theory. In particular, the stability condition (1) is not well defined in the case of flexible structures with free body motion which results in poles at the origin, since in this case, the plant DC gain will be infinite. However, control systems involving flexible structures with free body motion arising in these important application areas still suffer from the stability and performance issues mentioned above. Thus we are motivated to extend the NI robust stability theory developed in [16, 14, 34] so that it can be applied to control systems involving highly resonant flexible structures with free body motion.

Fig. 2 shows a block diagram of a system which includes a flexible structure with free body motion that arises in a problem of disk-drive control; see [52]. Here, a voice coil motor (VCM) is used to actuate the arm of the reader head. The free body motion of the reader head leads to a transfer function from the input to the VCM output which has poles at the origin. Furthermore, the overall system satisfies the NI frequency response property and includes poles at the origin. However, the NI stability results presented in [16, 14, 34] do not allow for poles at the origin and cannot be applied to control systems such as this disk-drive control system.

In this paper, we present a new generalized definition of NI systems which allows for flexible structures with colocated force actuators and position sensors and with free body motion. This definition extends the previous definitions of NI systems presented in [16, 14, 34] to allow for up to two poles at the origin. We also derive new generalized stability conditions for positive-feedback control systems involving an NI plant and an SNI controller.

As in [16, 14, 34], the stability conditions presented in this paper, are given purely in terms of properties of the plant and controller transfer function matrices, although the proofs rely on state space techniques. Furthermore, the stability conditions given are independent of the plant and controller system order and can be stated without using the fact that the plant and the controller transfer function matrices are rational. However, the proofs given in this paper only apply to the rational case.

Preliminary conference versions of the stability results presented in this paper were presented in [53, 54]. However, in this paper, much more general versions of these stability results are presented in Theorems 1 - 4 and Corollaries 1, 2, which allow for the existence of free body motion in some but not all input-output channels. This is important since multivariable control systems involving flexible structures with free body motion usually include free body motion in some but not all input-output channels. Also, this paper includes a case study involving the control of a flexible robotic arm, which has not been considered in the previous conference versions of the paper.

This paper is further organized as follows: Section II recalls the existing definition for NI systems and outlines the notation that will be used in the rest of the paper. Section III introduces the new generalized definition for NI systems, which allows for systems with free body dynamics. Also in this section, we present the main stability results in Theorems 1 - 4 and Corollaries 1-2. Section IV presents a case study, which involves a flexible robotic arm, as an application of the NI theory presented in this paper. The paper is concluded with a summary and remarks on future work in Section V. All proofs of the presented theorems, lemmas and corollaries are given in the Appendix.

Ii Preliminaries and Notation

In this section, we recall the existing definitions of NI and SNI systems as given in [34] for systems with poles in the closed left half of the complex plane, except at the origin. We will use this existing definition of SNI (first introduced in [16]) systems but in the next section we will present our new definition of generalized NI systems. We also define notation used to describe positive feedback interconnections and internal stability, which will be used to present the main results in this paper.

Consider the following LTI system,

 ˙x(t)=Ax(t)+Bu(t), (2) y(t)=Cx(t)+Du(t), (3)

where and with the square transfer function matrix . The transfer function matrix is said to be strictly proper if . We will use the notation to denote the state space realization (2), (3).

The existing definition of NI systems states that a square transfer function matrix is NI if the following conditions are satisfied [34]:

1. has no pole at the origin and in .

2. The corresponding frequency response is such that

 j(G(jω)−G(jω)∗)≥0,

for all where is not a pole of .

3. If with is a pole of , it is at most a simple pole and the residue matrix is positive semidefinite Hermitian.

Definition 1

[34] A square transfer function matrix is SNI if the following conditions are satisfied:

1. has no pole in .

2. For all , .

Now, consider a positive feedback interconnection between an NI system with transfer function matrix and an SNI system with transfer function matrix as shown in Fig. 1. Also, suppose that the transfer function matrix has a minimal state space realization and has a minimal state space realization Furthermore, it is assumed that the matrix is nonsingular. Then the closed system has a system matrix given by

 ˘A=[A+B¯D(I−D¯D)−1CB¯C+B¯D(I−D¯D)−1D¯C¯B(I−D¯D)−1C¯A+¯B(I−D¯D)−1D¯C]. (4)

Moreover, the positive feedback interconnection between and as shown in Fig. 1 and denoted is said to be internally stable if the closed-loop system matrix in (4) is Hurwitz; e.g., see [55].

Iii Main results

The main contribution of this paper is a generalization of the framework for NI systems presented in [34]. We introduce a new definition of NI systems that will allow for systems with free body dynamics. This generalized definition will be used in a new set of stability conditions that will allow for NI systems with free body motion to be included into the framework of NI systems theory. Henceforth, when a system is said to be NI, we will mean NI as defined below, not NI as defined in earlier papers.

Definition 2

A square transfer function matrix is NI if the following conditions are satisfied:

1. has no pole in .

2. For all such that is not a pole of ,

3. If with is a pole of , then it is a simple pole and the residue matrix is Hermitian and positive semidefinite.

4. If is a pole of , then for all and is Hermitian and positive semidefinite.

Here, is the frequency response corresponding to the transfer function . Unlike the NI definition presented in [34], Definition 2 allows for poles at the origin. In this case, we cannot use the existing stability results presented in [14, 34, 35], because the stability condition in (1) is not defined. The inclusion of poles at the origin extends the NI systems theory to include flexible systems with free body dynamics. In order to derive a set of stability conditions that allow for NI systems with free body motion, we define the following constant matrices for a given NI transfer function matrix

 G2 =lims⟶0s2G(s), G1 =lims⟶0s(G(s)−G2s2), G0 =lims⟶0(G(s)−G2s2−G1s). (5)

These matrices are the first three coefficients in the Laurent series expansion of the transfer function . These matrices carry information about properties of the free body motion of the system under consideration and will be used in stability conditions for the positive feedback interconnection of NI and SNI systems. Note that the DC gain condition (1) cannot be defined for an NI system with transfer function matrix unless , which reduces to the case where the dynamical system has no free body motion. From Condition 4) in Definition 2, the matrix is required to be Hermitian and positive semidefinite. Hence, it follows (e.g., see [56]) that if , it can be decomposed in the form

 G2=JJT, (6)

where is a full column rank matrix.

We now present conditions for the stability of a positive feedback control system involving an NI plant with free body motion. These conditions are stated using the quantities defined in (5). First, we define the Hankel matrix as

 Γ=[G1G2G20]. (7)

Suppose that . Using the singular value decomposition (SVD), we can decompose the Hankel matrix as

 Γ =[H1H2][S000][VT1VT2] =H1SVT1=UVT1=[U1U2]VT1, (8)

where are unitary matrices, , , and the matrices and each have orthogonal columns. Furthermore, we can decompose the matrix using the SVD as

 UT1U2=^U^S^VT=^U[S1000][^V1T^V2T], (9)

where and are orthogonal matrices, and .

We now introduce some notation which will be used throughout the paper. Given matrices and such that , then the matrix valued function is defined by

 P(X,Y)≜X−XY(YTXY)−1YTX. (10)

Using this notation, we define the matrix

 Nf=P(¯G(0),F), (11)

where the matrix is given by

 F=U1^V2, (12)

and we will assume that .

We will use the following condition in the theorem which follows:

 FT¯G(0)F<0. (13)

Also, for the case in which is positive semidefinite, we will use the condition

 I−N12fG0N12f−N12fG1J(JTJ)−2JTGT1N12f>0. (14)

Moreover, for the case in which is negative semidefinite, we will use the condition

 det(I+~NfG0~Nf+~NfG1J(JTJ)−2JTGT1~Nf)≠0. (15)

Here, and matrices and are defined in (5), (6), (11), and (12) respectively. Also, denotes the square root of a positive semidefinite matrix.

The following theorem is our first main stability result for the case in which . That is, the system has double poles at the origin.

Theorem 1

Suppose that the square transfer function matrix is strictly proper and NI with , and the transfer function matrix is SNI. Also, suppose that the matrix is non-singular. If is positive semidefinite, then the closed-loop positive-feedback interconnection between and as shown in Fig. 1 is internally stable if and only if conditions (13) and (14) are satisfied. Furthermore, if is negative semidefinite, then the closed-loop positive-feedback interconnection between and is internally stable if and only if conditions (13) and (15) are satisfied.

The proof of this and subsequent theorems and corollaries are presented in Appendix B.

We now present a corollary to this theorem which considers the special case in which none of the free body modes of the plant have frictional force present; i.e., . In order to present this corollary, we define the matrix as follows:

 N2=P(¯G(0),J), (16)

where we assume that the matrix is non-singular.

We will use the following condition in the next corollary, which corresponds to condition (13) in Theorem 1:

 JT¯G(0)J<0. (17)

Also, for the case in which is positive semidefinite, we will use the following condition which corresponds to condition (14) in Theorem 1:

 I−N122G0N122>0. (18)

Moreover, for the case in which is negative semidefinite, we will use the following condition which corresponds to condition (15) in Theorem 1:

 det(I+~N2G0~N2)≠0, (19)

where .

Corollary 1

Suppose that the transfer function matrix is SNI and the strictly proper transfer function matrix is NI with and . Also, suppose that the matrix is non-singular. If is positive semidefinite, then the closed-loop positive-feedback interconnection between and is internally stable if and only if conditions (17) and (18) are satisfied. Furthermore, if is negative semidefinite, then the closed-loop positive-feedback interconnection between and is internally stable if and only if conditions (17) and (19) are satisfied.

The following theorem imposes some extra conditions on the matrix which enables us to relax the sign definiteness condition on the matrix . This then leads to a simplified stability condition.

Theorem 2

Suppose that the transfer function matrix is SNI and the strictly proper transfer function matrix is NI with and . Also, suppose that , where denotes the null space of a matrix. Then the closed-loop positive-feedback interconnection between and is internally stable if and only if condition (17) is satisfied.

In Theorem 3, Theorem 4 and Corollary 2, we consider cases which correspond to free body motion with frictional force present. As in Theorem 1, these cases allow for fact that the free body motion may not be present in all input-output channels.

In order to present Theorem 3 and Theorem 4, suppose that and . This corresponds to the case when the system has a single pole at the origin. Then we consider the following SVD decomposition of the matrix defined in (5):

 G1=[~F1~F2][S2000][VT1VT2]=F1VT1, (20)

where , and the matrices and each have orthogonal columns. Also, we define the matrix as follows:

 N1=P(¯G(0),F1), (21)

where the matrix is assumed to be non-singular.

We will use the following condition in Theorem 3 and Corollary 2 which corresponds to condition (13) in Theorem 1:

 FT1¯G(0)F1<0. (22)

For the case in which is positive semidefinite, we also will use the following condition which corresponds to condition (14) in Theorem 1:

 I−N121G0N121>0. (23)

Moreover, for the case in which is negative semidefinite, we will use the following condition which corresponds to condition (15) in Theorem 1:

 det(I+~N1G0~N1)≠0, (24)

where .

Theorem 3

Suppose that the transfer function matrix is SNI and the strictly proper transfer function matrix is NI with and . Also, suppose that the matrix non-singular. If is positive semidefinite, then the closed-loop positive-feedback interconnection between and is internally stable if and only if conditions (22) and (23) are satisfied. Furthermore, if is negative semidefinite, then the closed-loop positive-feedback interconnection between and is internally stable if and only if conditions (22) and (24) are satisfied.

The following theorem imposes some extra conditions on the matrix which enables us to relax the sign definiteness condition on the matrix . This then leads to a simplified stability condition.

Theorem 4

Suppose that the transfer function matrix is SNI and the strictly proper transfer function matrix is NI with and . Also, suppose that . Then the closed-loop positive-feedback interconnection between and is internally stable if and only if condition (22) is satisfied.

The following corollary presents an important special case of Theorem 2 and 4.

Corollary 2

Suppose that the transfer function matrix is SNI and the strictly proper transfer function matrix is NI with either and invertible or and . Then, the closed-loop positive-feedback interconnection between and is internally stable if and only if

Remark 1

The case where and corresponds to the existing stability results presented in [14, 34, 35]. In this case, the stability condition reduces to This condition can be obtained from (23) using the fact in this case. Also, we require the assumption . Hence,

 I−N121G0N121>0, ⇔ N−11−G0>0, ⇔ λmax(¯G(0)G0)<1.

Note that using a similar argument to the proof of Theorem 3, we can obtain a similar result under the assumption that

Iv Case Study: Control of Flexible robotic arm

In this section, we present an application of the stability results presented in this paper to the control of a flexible robotic arm system. The robotic arm is pinned to a motor at one end. For the purposes of modeling the flexible robotic arm, we use an equivalent slewing beam model as depicted in Fig. 3; see [57].

The motor allows the robotic arm to traverse in the vertical plane. Two piezoelectric patches are attached to the arm on either side. Here, one piezoelectric patch acts as an actuator while the other is a sensor. The robotic arm system has two inputs and two outputs: the inputs are the voltage applied to the piezoelectric actuator and the torque applied by the motor, whereas the outputs are the voltage produced by the piezoelectric sensor and the motor hub angle . The fact that this system involves colocated “force” actuators and “position” sensors indicates that the system will be NI; e.g., see [14].

Iv-a Mathematical model for the robotic arm

The beam in Fig. 3 is modeled using the Bernoulli-Euler equations of motion for a beam with actuating and sensing piezoelectric elements as in [57]:

 ∂2∂x2[EI∂2∂y2y(x,t)−CaVa(x,t)]+ρA∂2∂t2y(x,t)=0. (25)

Here, is Young’s modulus and is the second moment of inertia of the beam, is the density of the beam, and is the area of the composite beam. If the thickness of the piezoelectric films are comparable to the thickness of the beam, then the products and would be different in the laminated and non-laminated areas of the beam. However, since piezoelectric films used in practical applications are often thin compared to the thickness of the beam, these differences will be neglected. Assuming that the products and are uniform over the length of the beam simplifies the modeling procedure.

Now we consider various boundary conditions in modeling the beam. These are given as

 y(0,t)=0, (26) EI∂2∂x2y(0,t)−Ih∂3∂t2∂xy(0,t)+τ(t)=0, (27) EI∂2∂x2y(L,t)+It∂3∂t2∂xy(L,t)=0, (28) EI∂3∂y3y(L,t)−Mt∂3∂t2∂xy(L,t)=0. (29)

Here, and are the mass and inertia of the tip, which will be neglected in this paper.Also, (26) represents the inability of the motor joint to undergo transverse motion. As in [57], the time domain beam equation (25) with boundary conditions (26)-(29) can be transformed into an equivalent Laplace domain representation as

 Y′′′′(x,s)−β4Y(x,s)=CaV′′a(x,s)EI (30)

with boundary conditions

 Y(0,s)=0, (31) EIY′′(0,s)−Ihs2Y′(0,s)+τ(s)=0, (32) EIY′′(L,s)+Its2Y′(L,s)=0, (33) EIY′′′(L,s)−Mts2Y(L,s)=0, (34)

where the primes indicate spatial derivatives and

 β4(s)=−ρAs2EI. (35)

Note that (30) is the Laplace domain equivalent of the Bernoulli-Euler beam equation with as a forcing input. Together, (30)-(34) represent a set of linear ordinary differential equations with mixed boundary conditions: two at and two at . A state space representation for the system can be formed from equations (30)-(34) as in [57]:

 ⎡⎢ ⎢ ⎢ ⎢⎣Y′(x,s)Y′′(x,s)Y′′′(x,s)Y′′′′(x,s)⎤⎥ ⎥ ⎥ ⎥⎦= ⎡⎢ ⎢ ⎢ ⎢⎣010000100001β4000⎤⎥ ⎥ ⎥ ⎥⎦⎡⎢ ⎢ ⎢ ⎢⎣Y(x,s)Y′(x,s)Y′′(x,s)Y′′′(x,s)⎤⎥ ⎥ ⎥ ⎥⎦ +⎡⎢ ⎢ ⎢⎣0001⎤⎥ ⎥ ⎥⎦CaVa(s)EI2∑i=1δ(x−xi)(−1)i+1 (36)

where represents the Dirac delta function. The equation (IV-A) can be written in the general form

 Z′(x,s)=¯AZ(x,s)+¯BU(x,s), (37)

the solution to which is given by

 Z(x,s) =e¯AxZ(0,s)+[¯Ae¯A(x−x1)¯B−¯Ae¯A(x−x2)¯B]CaVa(s)EI. (38)

Once the boundary conditions and are known, (IV-A) will depend upon three conditions for , namely, . For further details see [57].

Iv-B Infinite Dimensional Transfer function Model

Here, we present the input-output relationship between the two inputs and , and the corresponding collocated outputs and in the form of the transfer function matrix,

 [θ(s)Vs(s)]=G(s)[τ(s)Va(s)], (39)

where and each of the elements of this transfer function matrix is an infinite dimensional transfer function defined in terms of transcendental functions of . Indeed, each of the four transfer functions in (39) can be written as a ratio of numerator and denominator functions computed as

 Gτ,θ(s)=Nτ,θ(s)D(s)=Y′(0,s)T∣∣Va(s)=0, (40) GVa,θ(s)=NVa,θ(s)D(s)=Y′(0,s)Va∣∣τ(s)=0, (41) Gτ,Vs(s)=Nτ,Vs(s)D(s)=Cs(Y′(x2,s)−Y′(x1,s))τ(s)∣∣Va(s)=0, (42) GVa,Vs(s)=NVa,Vs(s)D(s)=Cs(Y′(x2,s)−Y′(x1,s))Va(s)∣∣T(s)=0. (43)

Here,

 D(s)= 4βEI(ρA(cos(βl)sinh(βl)−cosh(βl)sin(βl))) −4β4EIIh(1+cos(βl))cosh(βl)), (44)

where is the hub inertia. Also, the functions , are given by very complicated expressions which can be found in equations (26)-(28) in [57].

We now compute the transfer functions in (40)-(43) for the case where the piezoelectric actuators and sensors span the entire length of the beam. This corresponds to the substitutions: and . The resulting transfer functions have been verified in [58] for an experimented robotic arm system.

Despite the fact that we have not defined the NI property for infinite dimensional transfer functions, we will provide some calculations which indicate that the infinite dimensional transfer function matrix defined in (39)-(43) satisfies the NI conditions given in Definition 2. Since the infinite dimensional transfer function matrix is actually a transcendental function of , , then Condition 2) in Definition 2 is equivalent to the condition

 j(~G(β(jω))−~G(β(jω))∗)≥0 (45)

for all where is given by (35). Indeed, it is straightforward to verify from the formulas for the transfer function matrix (39)-(43) that for all . Also, the function given in (IV-B) has an infinite numbers of roots. However, we can check Condition 3) in Definition 2 for a finite number of these roots on the imaginary axis. To do so, we have calculated the first eleven -axis roots of numerically. At each of these roots , the corresponding residue matrix is calculated using L’Hopital’s rule as follows:

 K(jω0) =lims⟶jω0(s−jω0)jG(s) =lims⟶jω0(s−jω0)j⎡⎢ ⎢⎣Nτ,θ(s)D(s)NVa,θ(s)D(s)Nτ,Vs(s)D(s)NVa,Vs(s)D(s)⎤⎥ ⎥⎦ =j⎡⎢ ⎢⎣Nτ,θ(jω0)D′(jω0)NVa,θ(jω0)D′(jω0)Nτ,Vs(jω0)D′(jω0)NVa,Vs(jω0)D′(jω0)⎤⎥ ⎥⎦, (46)

where denotes the first derivative of with respect to .

In this case study, the parameter values for the robotic arm are taken from [58]. These parameter values are shown in the Table I.

Table II shows the calculated roots of and the minimum eigenvalue of the corresponding residue matrix given in (IV-B). Also, the matrix is found to be which is positive semidefinite.

These results show that the infinite dimensional transfer function matrix satisfies the conditions of Definition 2, at least for the first ten resonant modes.

Also in the Fig. 4, we plot the log of the minimum eigenvalue of the matrix as a function of frequency , where is a positive constant. This plot also indicates that the residue matrix defined in (IV-B) will be positive semidefinite at the system poles within the frequency range of interest.

Iv-C Approximate Finite-dimensional Transfer Function Matrix

The transfer functions in (40)-(43) are irrational functions of . We now approximate these transfer functions by rational functions in in order to design a suitable controller for the robotic arm system and to simulate its performance. Various methods such as the Maclaurin series expansion presented in [59], the Rayleigh-Ritz method [60], and the assumed modes method [60] are available in literature for the finite dimensional approximation of such an infinite dimensional model. Here, we adopt a partial fraction approach to obtain a finite dimensional approximation of . This method is similar to the assumed modes technique described in [60]. The finite dimensional model can be written as

 Gf(s) =[Gfτ,θ(s)GfVa,θ(s)Gfτ,Vs(s)GfVa,Vs(s)] =⎡⎢ ⎢⎣Nτ,θ(s)D(s)NVa,θ(s)D(s)Nτ,Vs(s)D(s)NVa,Vs(s)D(s)⎤⎥ ⎥⎦ =n∑i=01k⎡⎢ ⎢⎣ais2+p2ibis2+p2icis2+p2idis2+p2i⎤⎥ ⎥⎦. (47)

Here in the infinite dimensional model is approximated by

 Df(s)=kn∏i=0(s2+p2j), (48)

where, are the first n -axis roots of . Also, the coefficient matrices are computed using a partial fraction expansion method. That is,

 Ci=1k∏nj=0,j≠i(−p2i+p2j)[Nτ,θ(jpi)NVa,θ(jpi)Nτ,Vs(jpi)NVa,Vs(jpi)]. (49)

The constant is chosen so that

 D(jω0)=kn∏i=0(−ω20+p2i), (50)

where is such that is not a root of . We consider the first resonant mode; i.e., for the controller design. The corresponding coefficient matrices were computed and were found to be and . Also, the poles were computed to be .

The finite dimensional model in (IV-C) is NI, since for all , where is not a pole for . This follows because in this example, is real and symmetric for all such that is not a pole of . Also, the coefficient matrices are positive semidefinite which implies that Condition 3) in Definition 2 is satisfied. Moreover, which implies that Condition 4) in Definition 2 is satisfied.

Iv-D Controller design

According to Theorem 1 if a plant is NI, any SNI controller which satisfies the conditions of Theorem 1 will stabilize the system. The fact that the robotic arm plant involves colocated “force” actuators and “position” sensors indicates that this plant should be NI. In particular, the finite dimensional approximation to the robotic arm model derived in Subsection IV-C was shown to be NI. We will now use a finite dimensional model of the form (IV-C) to design a controller for the system. First, we compute the matrices , and in (5), for the finite dimension approximate system where n=1 in (IV-C) to obtain

 G2=[0.14000]⩾0;G1=[0000]; G0=[0.412530830.00003190.00003190.15672805]. (51)

This implies that we can use Corollary 1 to guarantee the stability of the positive feedback interconnection between the plant and an SNI controller.

In this case study, an integral resonant controller (IRC) is chosen to stabilize the system; e.g., see [14]. An IRC is a first order controller which takes the form

 ¯G(s)=(sI+ΓΦ)−1Γ−Δ. (52)

This controller is SNI if and is a symmetric matrix [14]. Now, we chose the controller matrices and such that the conditions of Corollary 1 are satisfied. We choose the controller matrices as follows:

 Γ=[35151520];Φ=[0.7450.5210.5211.021];Δ=[4.2900002.22]. (53)

This leads to a controller DC gain matrix of . To check the stability conditions in Corollary 1, we first compute the matrix in (6) using in (IV-D). This yields Also, the matrix in (16) is calculated as , which is negative semidefinite. Then we conclude where Also, Thus, the conditions of Corollary 1 are satisfied.

To verify the performance of the closed loop system, we simulate the response of this system corresponding to a step change in the reference position of the robotic arm; see Fig. 5. This step response is shown in Fig. 6. Also, the corresponding response of the piezo sensor output is shown in Fig. 7. Here, the step responses were calculated using finite dimensional plant models defined in (IV-C) for different numbers of modes, n=2,3…7.

To this end, we have used the proposed controller which is designed for the finite dimensional model with n=1 when applied to the plant with finite dimensional model where n=2,3…7 in order to check the performance and robustness of the proposed controller. In fact, the performance of the closed loop system is found to improve by increasing the number of modes; see Fig 6 and Fig 7.

Note that the controller parameters in (53) were chosen by process of trial and error to obtain good closed loop performance the case of the nominal plant model, . An alternative approach, which would be useful in the case of a more complicated SNI controller structure, would be to use an optimization procedure to obtain the controller parameters; e.g., see [61].

V Conclusion

In this paper, new stability results for the positive-feedback interconnection of negative imaginary systems have been derived. A new NI definition is presented, which allows for systems having free body dynamics to be considered as NI systems. This work can be used in controller design to allow for a broader class of NI systems than considered in previous work. The application of the main results in this paper has been illustrated via a case study involving the control of a flexible robotic arm.

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