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An observer-based control scheme using negative-imaginary theory
Automatica 81 (2017) 196–202
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Automatica journal homepage: www.elsevier.com/locate/automatica
Brief paper
An observer-based control scheme using negative-imaginary theory✩ Parijat Bhowmick 1 , Sourav Patra Indian Institute of Technology Kharagpur, Kharagpur-721302, India
article
info
Article history: Received 16 May 2016 Received in revised form 12 November 2016 Accepted 27 February 2017
Keywords: Strongly strict negative-imaginary systems Positive feedback Observer Linear matrix inequality Robustness
abstract This paper presents a full-order state observer-based scheme that transforms a causal, LTI, minimally realized system into a strongly strict negative-imaginary system by defining an auxiliary output based on the observed states. The auxiliary output is used for closed-loop control with positive feedback invoking the internal stability condition of negative-imaginary theory. A set of LMI conditions is derived that determines the value of a design parameter µ required for the proposed scheme to transform a given system with a minimal state-space realization into a strongly strict negative-imaginary system. The proposed scheme is applicable for both stable/unstable and square/non-square systems. In this paper, the framework is further explored for robustness analysis of uncertain systems and numerical examples are given to elucidate effectiveness of the proposed results. © 2017 Elsevier Ltd. All rights reserved.
1. Introduction The negative-imaginary systems theory has rapidly attracted the interest of the control community due to its simple internal stability condition for interconnected systems that depends only on the DC loop gain, and its wide applicability in different areas of control systems engineering. For example, such systems arise while considering the transfer functions from input voltage to output voltage in active electrical filters (Patra & Lanzon, 2011), from input voltage to shaft rotational velocity in DC servo motor (Song, Lanzon, Patra, & Petersen, 2012a), from pump input voltage to water level in a linearized coupled tank system. Other applications include nano-positioning systems, flexible spacecrafts, robotic manipulator arms (Mabrok, Kallapur, Petersen, & Lanzon, 2014), etc. The notion of negative-imaginary (NI) and strictly negative-imaginary (SNI) systems has been introduced in Lanzon and Petersen (2008) for robust control of highly resonant flexible structures with colocated position sensors and force actuators (Bhikkaji, Moheimani, & Petersen, 2012; Mabrok et al., 2014). An NI system is a Lyapunov-stable system with equal number of inputs and outputs, having a real-rational, proper transfer function matrix R(s) that satisfies the frequency domain condition j[R(jω) −
✩ The material in this paper was not presented at any conference. This paper was recommended for publication in revised form by Associate Editor James Lam under the direction of Editor Ian R. Petersen. E-mail addresses: [email protected] (P. Bhowmick), [email protected] (S. Patra). 1 Fax: +91 3222 282262.
http://dx.doi.org/10.1016/j.automatica.2017.03.024 0005-1098/© 2017 Elsevier Ltd. All rights reserved.
R⋆ (jω)] ≥ 0 ∀ω ∈ (0, ∞). In a SISO setting, a real-rational, proper transfer function is said to have the NI (respectively, SNI) properties if its Nyquist plot lies below (respectively, strictly below) the real axis in the open positive frequency interval (Lanzon & Petersen, 2008; Petersen & Lanzon, 2010). So far in the NI literature, the state-space characterizations of LTI NI and SNI systems have been studied in Lanzon and Petersen (2008), Petersen and Lanzon (2010) and Xiong, Petersen, and Lanzon (2010); NI systems with single and double poles at the origin in Ferrante, Lanzon, and Ntogramatzidis (2016), Ferrante and Ntogramatzidis (2013) and Mabrok et al. (2014); strongly strict negative-imaginary (SSNI) systems in Ferrante et al. (2016) and Lanzon, Song, Patra, and Petersen (2011); lossless NI system properties in Xiong, Petersen, and Lanzon (2012); controller synthesis, robustness and performance analysis in Bhikkaji et al. (2012), Engelken, Patra, Lanzon, and Petersen (2010), Mabrok, Kallapur, Petersen, and Lanzon (2015) and Song, Lanzon, Patra, and Petersen (2010); Song et al. (2012a); Song, Lanzon, Patra, and Petersen (2012b). In recent times, the NI theory has been extended for symmetric and non-rational transfer functions (Ferrante et al., 2016; Ferrante & Ntogramatzidis, 2013) and also, the cooperative control strategy is being applied to control multiple NI agents connected through a communication network (Wang, Lanzon, & Petersen, 2015). The NI stability result for interconnected systems has drawn profound interest from control theoretic perspective since the stability criteria depends only on the loop gains at zero and infinite frequencies. However, the theory remains inapplicable if the interconnected systems are non-NI which occurs quite often in control theoretic applications. Therefore, if a scheme can be developed which transforms any LTI non-NI system into NI
P. Bhowmick, S. Patra / Automatica 81 (2017) 196–202
Fig. 1. An observer-based control scheme for stable LTI plants with positive feedback using NI theory.
class then the existing NI theory can be applied to synthesize a stabilizing controller depending on the loop gains at zero and infinite frequencies. So far in the NI literature, to the best of the authors’ knowledge, no such procedure is reported by which a nonNI system can be transformed into an NI system. This motivates us to propose a scheme by which an LTI system can be transformed into a negative-imaginary system facilitating the applicability of the NI theory for analysis and controller synthesis, especially widening the applicability of this theory in the presence of unstable and/or non-square plants. In this paper, we introduce a scheme, as shown in Fig. 1, which transforms a causal, LTI, minimally realized system into a strongly strict negative-imaginary (SSNI) system (Lanzon et al., 2011), a subset of the strictly negative-imaginary (SNI) class (Xiong et al., 2010), by defining an auxiliary output based on the observed states. The proposed observer-based SSNI scheme can also be used to stabilize a system G(s) in closed-loop when the auxiliary output z is positively fed back (see Fig. 1) along with satisfying a particular DC ¯ (s). This scheme loop gain condition on the transformed system G does not require the original system to be stable or square. In this scheme, a full-order state observer is placed in cascade with the plant and a new output z is defined that depends only on the estimated states xˆ . For a particular structure of the transformation matrix H, the combined system from input u to the defined output z := H xˆ exhibits SSNI properties. A set of LMI-based conditions, involving a design parameter µ > 0, is formulated that yields a feasible finite positive µ value required to design an observerbased control scheme for a given plant. In this work, we have exploited the fact that the SSNI system properties can be defined for uncontrollable systems while maintaining the observability and the full normal rank conditions (Patra & Lanzon, 2011). So far, various observer-based control schemes have been introduced in the literature. But none of these techniques is dedicated for the NI framework. To mention a few: An observerbased state feedback control scheme is designed in Arcak and Kokotovic (2001) to ensure global asymptotic stability of linear systems with slope-restricted nonlinearities in feedback; Lyapunov stability theory and LMI-based approach have been adopted in Lien (2004) to design observer-based control for a class of linear systems with state perturbations where the controller and observer gains can be obtained from the LMI formulations; A state observer-based robust trajectory-tracking control scheme is proposed in Oya, Su, and Kobayashi (2004) for electrically driven robot manipulators where the observer is used to estimate the joint angular velocities; An observer-based robust control technique is designed in Li and Yang (2012) that also facilitates the acquisition of information used for fault detection in feedback control systems; Further in Collado, Lozano, and Johansson (2007), Johansson and Robertsson (2002), the authors have proposed an observerbased transformation scheme in strictly positive-real framework. In this paper, the proposed observer-based transformation and the associated feedback control scheme broaden the scope of applicability of the NI theory for robust control analysis and synthesis. The remainder of the paper proceeds as follows: in Section 2, useful notations are given. Section 3 illustrates a few necessary
197
definitions, lemmas and theorems which streamline the main results of this paper. Section 4 deals with the observer-based scheme that transforms a causal, LTI system into an SSNI system. Section 5 formulates a set of LMI-based conditions by which the proposed observer-based feedback control scheme guarantees internal stability of LTI systems with positive feedback. In Section 6, robust stability analysis of LTI uncertain systems using the proposed observer-based control strategy is explained through an example. Section 7 concludes the paper. 2. Notations The notations and acronyms are standard throughout. The fields of real and complex numbers are denoted by R and C, respectively. The sets of real and complex matrices of dimension (n × n) are denoted by Rn×n and Cn×n , respectively. The maximum and the minimum eigenvalues of a matrix A ∈ Cn×n that has only real eigenvalues are indicated by λmax (A) and λmin (A), respectively. The determinant and the rank of a matrix A are denoted by det[A] and rank[A], respectively. The real and imaginary parts are represented by ℜ(·) and ℑ(·), respectively. AT , A⋆ and A¯ denote the transpose, the complex-conjugate transpose and the complex-
⋆
T
conjugate of a matrix A. The shorthands for A−1 and A−1 are represented by A−⋆ and A−T , respectively. The terminology that R(s) is a ‘proper’ transfer function matrix includes the cases R(∞) ≡ 0, det[R(∞)] = 0 and det[R(∞)] ̸= 0. The set of all proper, real-rational transfer function matrices is denoted by R n×n whereas the set of all proper, real-rational and asymptotically stable transfer function matrices is denoted by RH n∞×n , both of dimensions (n × n). For a transfer function matrix R(s), R⋆ (jω) = RT (−jω) and R∼ (s) = RT (¯s) where s¯ denotes the complexconjugate of s. The minimal state-space realization of a system R(s) A B is represented by R(s) = . C D min 3. Preliminaries In this section, some preliminary results are given which provide a background to establish the main results of this paper. We now describe the NI, SNI and SSNI systems through the following definitions and the lemma. For complete details of the NI theory and its applications literatures (Bhikkaji et al., 2012; Bhowmick & Patra, 2016, 2017; Dey, Patra, & Sen, 2016; Lanzon & Petersen, 2008; Lanzon et al., 2011; Mabrok et al., 2014, 2015; Petersen & Lanzon, 2010; Xiong et al., 2010) may be referred. Definition 1 (NI System (Mabrok et al., 2014, 2015)). A square, causal, real-rational, proper transfer function matrix R(s) is negative-imaginary (NI) if the following conditions are satisfied: (1) R(s) has no poles in ℜ[s] > 0; (2) j[R(jω) − R⋆ (jω)] ≥ 0 ∀ω ∈ (0, ∞) except the values of ω where jω is a pole of R(s); (3) If jω0 , ω0 ∈ (0, ∞), is a pole of R(s), it is at most a simple pole and the residue matrix K0 := lims→jω0 (s − jω0 )jR(s) is positive semidefinite Hermitian; (4) If s = 0 is a pole of R(s) then, lims→0 sk R(s) = 0 for all k ≥ 3 and lims→0 s2 R(s) is positive semidefinite Hermitian. According to Definition 1, the NI class includes systems having complex pole-pair on the jω axis or pole(s) at the origin, however, in this paper, we have considered only asymptotically stable NI systems. Definition 2 (SNI System (Petersen & Lanzon, 2010; Xiong et al., 2010)). A square, causal, real-rational, proper transfer function matrix R(s) is strictly negative-imaginary (SNI) if the following conditions are satisfied:
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P. Bhowmick, S. Patra / Automatica 81 (2017) 196–202
Fig. 3. An observer-based scheme to transform a stable LTI system into an SSNI system.
Fig. 2. Interconnection of NI systems with positive feedback.
(1) R(∞) = RT (∞); (2) R(s) has no poles in ℜ[s] ≥ 0; (3) j[R(jω) − R⋆ (jω)] > 0 ∀ω ∈ (0, ∞).
where x ∈ Rn , u ∈ Rm , y ∈ Rp and m ≤ n. We further consider the full-order Luenberger observer (Ogata, 2010) dynamics as:
The frequency-domain condition j[R(jω) − R⋆ (jω)] ≥ 0 (> 0, respectively) ∀ω ∈ (0, ∞) given in Definitions 1 and 2 implies that, for SISO case, the polar plots of NI (SNI) transfer functions lie below (strictly below) the real axis of the Nyquist plane. The following lemma is presented here to characterize strongly strict negative-imaginary (SSNI) systems, a subset of the SNI class (Xiong et al., 2010). The SSNI Lemma will be invoked later to prove the main results of this paper.
A B Lemma 1 (SSNI Lemma 1 (Lanzon et al., 2011)). Let C D be a state-space realization of a causal, real-rational, proper transfer function matrix R(s) ∈ R m×m . Assume, R(s) + R∼ (s) has full normal rank m and the pair (A, C ) is observable. Then, A is Hurwitz and R(s) is SNI with ⋆
lim jω R(jω) − R (jω) > 0 and
ω→∞
1
lim j
ω
ω→0
R(jω) − R⋆ (jω) > 0,
B = −AYC T .
(2)
(3)
4. An observer-based scheme to transform an LTI system into NI class In this section, we present an observer-based scheme that transforms a causal, LTI, minimally realized system into an SSNI system by defining an auxiliary output that depends on the observed states (see Fig. 3). The scheme is applicable for both stable/unstable and square/non-square systems. Let a causal, LTI, minimally realized system be given as: G:
x˙ = Ax + Bu, y = Cx + Du,
x (0) = x0 ;
where L ∈ Rp×n is the observer-gain matrix to be designed such that the observer dynamics must be stable and faster than the system dynamics. Fig. 3 shows the proposed observer-based scheme, the main contribution of this paper, where an auxiliary output z is defined based on the observed states xˆ given as z := H xˆ where H ∈ Rm×n is a transformation matrix to be designed. Introducing the state estimation error variable x˜ := xˆ − x, we ¯ (s) from input u to the auxiliary represent the combined system G output z (see Fig. 3) in a compact state-space form:
x˙ A 0 x B = + u, x˙˜ 0 0 (A − LT C ) x˜ ¯: G z = H H x . ˜ Denoting AL := A − LT C , A¯ :=
Theorem 1 (Lanzon & Petersen, 2008). Given a stable NI transfer function matrix R1 (s) and an SNI transfer function matrix R2 (s) that satisfy R1 (∞)R2 (∞) = 0 and R2 (∞) ≥ 0. Then, the positive feedback interconnection of R1 (s) and R2 (s), in Fig. 2, is internally stable if and only if
(5)
(6)
x
It is worth noting that the minimal state-space realization is not a mandatory criteria to define SSNI systems but the full normal rank condition must be satisfied. We now present a necessary and sufficient condition for internal stability of an NI interconnection with positive feedback as shown in Fig. 2. The following theorem will be invoked in Section 5 to establish internal stability of the proposed closed-loop control scheme.
λmax [R1 (0)R2 (0)] < 1.
xˆ (0) = 0;
(1)
if and only if, D = DT and there exists a real matrix Y = Y T > 0 such that AY + YAT < 0 and
T ˙ˆ ˆ ˆ Σo : x = Ax + Bu + L (y − y), yˆ = C xˆ + Du,
(4)
H
A 0
0 AL
, B¯ :=
B 0
and C¯ :=
H , the transfer function mapping from u to z becomes
¯ (s) := C¯ (sI − A¯ )−1 B¯ = H (sI − A)−1 B. It is worth noting that G B¯ A¯ is not minimal since the the state-space realization C¯ 0 modes associated with the block (A − LT C ) are not controllable. We first consider the observer-based scheme for stable plants. The ¯ (s) satisfies following lemma shows that the transformed system G the SSNI properties for a particular choice of H matrix.
A B Lemma 2. Let G(s) = ∈ RH p∞×m be a causal LTI plant C D min with rank[B] = m and m ≤ n where n is the number of states. Assume an observer-gain matrix L ∈ Rp×n is given such that AL := A − LT C is T Hurwitz and two symmetric positive definite matrices Y11 = Y11 >0 T and Y22 = Y22 > 0 are respectively the solutions of the Lyapunov equations AY11 + Y11 AT = −Q1 AL Y22 +
Y22 ATL
and
(7)
= −Q2
for given Q1 = Q1T > 0 and Q2 = Q2T > 0. Assume µ is a positive scalar that satisfies the following set of LMIs:
µY11 − Y22 > 0, µQ1 (AY22 + Y22 ATL )T
(8a)
(AY22 +
Y22 ATL
Q2
)
> 0,
(8b)
and z := H xˆ where H = BT A−T (Y22 − µ Y11 )−1 . Suppose further ¯ B¯ ¯ ( s) = A that the combined system G from input u to the C¯ 0 auxiliary output z, shown in Fig. 3, is completely observable. Then, ¯ (s) is SSNI. G
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Proof. Since, rank[B] = m, m ≤ n, where n is the number of states and A is Hurwitz, using the standard matrix rank inequality (Bernstein, 2009) we get, rank[A−1 B] = m. Further, we have rank[(Y22 − µ Y11 )−1 ] = n since (µ Y11 − Y22 ) > 0 from (8a) and rank[(sI − A)−1 ] = n since A is Hurwitz. Combining these ¯ (s) := H (sI − A)−1 B = (A−1 B)T (Y22 − results we can conclude G µ Y11 )−1 (sI − A)−1 B has full normal rank m. Since, A and A − LT C are both Hurwitz for a given L, therefore, for any Q1 = Q1T > 0 and Q2 = Q2T > 0 there always exist positive definite solutions T T Y11 = Y11 > 0 and Y22 = Y22 > 0 of the Lyapunov equations (7). We now show, with these assumptions, that there always exists a positive scalar µ that satisfies (8a)–(8b). From (8a), we have µ Y11 − Y22 > 0 ⇔ µ Y11 > Y22 ⇔ µλi [Y11 ] > λi [Y22 ] ∀i and from (8b), µ Q1 − (AY22 + Y22 ATL )Q2−1 (AY22 + Y22 ATL )T > 0 ⇔ µλi [Q1 ] > λi [(AY22 + Y22 ATL )Q2−1 (AY22 + Y22 ATL )T ] ∀i. Since Y11 > 0, Y22 > 0, Q1 > 0, Q2 > 0, (AY22 + Y22 ATL )Q2−1 (AY22 + Y22 ATL )T ≥ 0 and µ is associated with the lefthand terms (i.e., Y11 and Q1 ) in both the inequalities, there always exists a µ > 0 for a given system such that the eigenvalues of the left-hand terms are greater than the eigenvalues of the right-hand terms in the above inequalities. Therefore, we have a µ > 0 that satisfies (8a)–(8b). Following ¯ ¯ the assumption that (A, C ) is completely observable and defining µY
−Y22
Y := −Y11 22
Y22
> 0, we compute
¯ + Y A¯ T AY =
A 0
0 AL
µY11 −Y22
−Y22 Y22
µ AY11 + Y11 AT = T − AY22 + Y22 AL T µQ1 T =− AY22 + Y22 AL T
µY11 −Y22 A + −Y22 Y22 0 T − AY22 + Y22 AL AL Y22 + Y22 AL T AY22 + Y22 AL T <0
0 AL
being SSNI.
1 0 1 0 −4.25 −6 −2.75 0.5 that satisfies the assumptions −1 0 4 −1 0.5 0 0 0 rank[B] = m = 1 < n = 3. We choosethe observer poles at (−3, −6, −9) arbitrarily and obtain L = 24.0, −36.0, 8.0 by using Ackermann’s formula (Ogata, 2010). Further we take Q1 = 2 I3 , Q2 = 0.5 I equations (7) 3 and by solving the Lyapunov
Q2
6.1333 −1.1583 −1.1583 0.6854 −7.1333 0.6583 0.0095 −0.0095 > 0. For 0.1644
we get, Y11 =
0.0236
0.0170 0.0095
0.0170 0.0849 −0.0095
−7.1333 0.6583 10.7667
> 0 and Y22 =
these L, Q1 , Q2 , Y11 and Y22 ,
the set of LMIs (8a)–(8b) yields µ = 0.8723 using the CVX toolbox ¯ ¯ (Grant & Boyd, 2014). It is verified that the pair (A, C ) is observable, µY
−Y22
Y := −Y11 22
Y22
¯ + Y A¯ T < 0 and B¯ = −AY ¯ C¯ T . > 0, AY 0.35(s2 +1.444s+1.373) s3 +6s2 +11s+6
¯ (s) = H (sI − A)−1 B = Therefore, G
0
H = BT A−T (Y22 − µY11 )−1 = 0.4947,
[since H = B A (Y22 − µY11 ) ] B =− = −B¯ . −1
0.7022,
is SSNI with 0.2535 . The
¯ (s) is shown in Fig. 4. polar plot of G
0
Thus, it is proved that the state-space realization
0.35(s2 +1.444s+1.373) s3 +6s2 +11s+6
has to be searched for, which makes the pair (A¯ , C¯ ) completely observable. To this end, one may think of the other relaxed assumption on minimality of the state-space realization of an SSNI system (Lanzon et al., 2011, Theorem 3.4) which requires there ¯ (s). But it would be exist no observable uncontrollable modes of G more restrictive to satisfy the latter condition, as in A¯ all the modes associated with A − LT C are uncontrollable.
¯ C¯ T is simplified to Again, AY A 0 µY11 −Y22 H T 0 AL −Y22 Y22 HT A (µY11 − Y22 ) H T = AL (−Y22 + Y22 ) H T A (Y22 − µY11 ) (Y22 − µY11 )−1 A−1 B =− −T
¯ (s ) = Fig. 4. Polar plot of the transformed system G
Example 1. Let us consider a stable, non-NI transfer function G(s) = s3 +6s21+11s+6 with a minimal state-space realization
T
[using(8b)].
T
199
A¯ C¯
B¯ 0
Example 2. We consider a causal, stable, non-square, non-NI plant
of
¯ (s), given in (6), satisfies AY ¯ + Y A¯ T < 0 and the combined system G ¯B = −AY ¯ C¯ T where Y = Y T > 0. Therefore, G¯ (s) is SSNI according to Lemma 1. This completes the proof. Remark 1. In Lemma 2, it is established that for any given stable plant with a minimal state-space realization, there always exists a positive scalar µ that satisfies the set of LMIs (8a)–(8b). However, A¯ B¯ ¯ ( s) = the transformed system G may not always C¯ 0 be completely observable for all given plants, especially in case ¯ (s) to be of non-square plants. This violates the assumption for G SSNI. In that case, an appropriate observer-gain matrix L ∈ Rp×n
(s G(s) =
−13 17 −7 0.75 −2
2
s−1 + 8s + 20) with a minimal state-space realization s−1 (s + 3)
−5
0 1 5 0 −1 −3 −3 2 , rank[B] = m = 1 < n = 3. −0.25 0 0 −2 −2 1 We arbitrarily choose the observer poles at (−12, −12, −9) and obtain L =
9.5172 1.7931
−23.4483 −7.6207
13.9310 1.3276
. Further choosing Q1 =
Q2 = 2 I3 , we get Y11 > 0 and Y22 > 0 by solving (7) and based on these matrices, the set of LMIs (8a)–(8b) yields µ = 1.3357 using the CVX toolbox (Grant & Boyd, 2014). Now, following the ¯ ( s) = same steps of Example 1 we find the transformed system G
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P. Bhowmick, S. Patra / Automatica 81 (2017) 196–202
exactly that of Lemma 2 considering A¯ =
remaining the same. Hence, the proof is done.
AK 0
−BK AL
and B¯ , C¯
5. An observer-based closed-loop control scheme using negative-imaginary framework
¯ (s) = Fig. 5. Polar plot of the transformed transfer function G
3.4517(s2 +6.934s+17.3) (s+3)(s2 +8s+20)
being SSNI.
In the previous section, we have introduced a state observerbased technique that transforms a causal, LTI, minimally realized system (stable/unstable and square/non-square system) into an SSNI system by defining an auxiliary output that depends on the observed states. In this section, we derive a set of LMI-based conditions that yields a solution for the design parameter µ ¯ > 0 ¯ (s) from u to z (shown in for which the transformed system G Fig. 1) becomes SSNI and simultaneously satisfies the condition λmax [G¯ (0)] < 1 which, in turn, ensures internal stability of the proposed closed-loop control scheme via Theorem 1.
A B Theorem 2. Let G(s) = ∈ RH p∞×m be a causal, LTI C D min plant with rank[B] = m and m ≤ n where n is the number of states. Given an observer-gain matrix L ∈ Rp×n such that AL := A − LT C is T T Hurwitz, assume Y11 = Y11 > 0 and Y22 = Y22 > 0 are respectively the solutions of the Lyapunov equations AY11 + Y11 AT = −Q1 AL Y22 + Fig. 6. An observer-based scheme to transform an LTI, open-loop unstable system into an SSNI system.
H (sI − A)−1 B =
3.4517(s2 +6.934s+17.3) (s+3)(s2 +8s+20)
which is SSNI via Lemma 1.
¯ (s). Fig. 5 shows the polar plot of G
Note that Lemma 2 is given for stable systems. We now extend this result for unstable systems, presented below in Lemma 3, where a similar transformation scheme will be pursued preceded by an observer-based state feedback that first stabilizes the given unstable plant. The proposed scheme is shown in Fig. 6.
A B ∈ R p×m be a causal, LTI, unstable Lemma 3. Let G(s) = C D min plant with rank[B] = m and m ≤ n where n is the number of states. Assume an observer-gain matrix L ∈ Rp×n and a state feedback gain matrix K ∈ Rm×n are given such that AL := A − LT C and AK := A − BK T T are Hurwitz and Y11 = Y11 > 0 and Y22 = Y22 > 0 are respectively the solutions of the Lyapunov equations AK Y11 + Y11 ATK = −Q1 AL Y22 +
Y22 ATL
and
(9)
= −Q2
µ Y11 − Y22 > 0, µQ1 − BKY22 − Y22 K T BT T T L CY22 − Q2
(10a)
LT CY22 − Q2 > 0,
(10b)
Q2
T and z := H xˆ where H = BT A− (Y22 − µ Y11 )−1 . Suppose further K ¯ B¯ ¯ ( s) = A from input v to the that the combined system G C¯ 0 auxiliary output z, as shown in Fig. 6, is completely observable. Then, ¯ (s) is SSNI. G
Proof. In this lemma, K is a predetermined state-feedback gain matrix such that AK := A − BK is Hurwitz. Rest of the proof follows
and
(11)
= −Q2
for given Q1 = Q1T > 0 and Q2 = Q2T > 0. If there exists a scalar µ ¯ > 0 such that the following set of LMIs is satisfied:
µ ¯ Y11 − Y22 > 0, µ ¯ Q1 T AY22 + Y22 AL T T I
A−1 B
(12a)
AY22 + Y22 AL
T
Q2
> 0,
B A−T > 0, ¯ Y11 − Y22 ) (µ
(12b)
¯ (s) = and the combined system G
(12c)
A¯
B¯
from u to z := H xˆ = 0 C¯ BT A−T (Y22 − µ ¯ Y11 )−1 xˆ , as shown in Fig. 1, is completely observable, then the closed-loop control system is internally stable. Proof. We assume that there exists a µ ¯ > 0 such that the set of LMIs (12a)–(12c) holds true. The first two inequalities, together with the assumption that (A¯ , C¯ ) is completely observable, enforces ¯ (s) := H (sI − A)−1 B to be an SSNI system following Lemma 2. G While the third inequality (12c) is equivalent to the DC-gain ¯ (s), as shown below: condition on G
for given Q1 = Q1T > 0 and Q2 = Q2T > 0. Assume, µ > 0 is a scalar that satisfies the following set of LMIs:
Y22 ATL
I
A−1 B
BT A−T >0 ¯ Y11 − Y22 ) (µ
⇔ I − BT A−T (µ ¯ Y11 − Y22 )−1 A−1 B > 0 ⇔ I − H (µ ¯ Y11 − Y22 ) H T > 0 [since H = BT A−T (Y22 − µ ¯ Y11 )−1 ] T µ ¯ Y11 −Y22 H ⇔ H H
P. Bhowmick, S. Patra / Automatica 81 (2017) 196–202
201
Fig. 7. An observer-based control scheme for unstable LTI plants with positive feedback using NI theory.
¯ (0)] < 1, the closed-loop system with unity positive and λmax [G feedback is internally stable according to Theorem 1. This completes the proof.
Fig. 8. An observer-based control scheme for LTI uncertain systems using NI framework.
Remark 2. Using Lemma 3 and Theorem 2, we can develop an observer-based closed-loop control scheme for open-loop unstable systems as shown in Fig. 7. This idea is very similar to Lemma 3. Here, the scheme is discussed through an example given below (see Example 3). It is worth noting that, in the proposed closedloop scheme, in order to achieve some performance and robustness objectives, one can also introduce a compensator in the loop that must satisfy either NI or SNI properties along with the DC loop gain condition for internal stability. Example 3. Let us consider a (2 × 1) unstable system G(s) =
min
0
1 0 0 0 0 0 0 0 1 0 0 0 1 0 0 0 0 0 0 0 1 0 4. We choose the closed-loop system poles and observer poles arbitrarily at (−3, −4, −5, −6) and (−6, −8, −10, −12), respectively and obtain K = [−157.9496, −35.4312, −36 .6972, −34.8624]
0 −1 0 with rank[B] = m = 1 and n = 0.5
20.6 0 −0.4905
18.5560 1.9594
and L =
103.6738 17.6676
1.8833 17.4440
16.3878 72.9264
. Further choosing
Q1 = 0.5 I4 , Q2 = 0.003 I4 and by solving the Lyapunov equations T T (9) we get, Y11 = Y11 > 0 and Y22 = Y22 > 0. Based on these predetermined matrices, feasibility of the following set of LMIs is tested for µ ¯ >0
µ ¯ Y11 − Y22 > 0, T µ ¯ Q1 − BKY22 L CY − Q 22 2 − Y22 K T BT > 0, T T L CY22 − Q2 Q2 T −T I B AK > 0. 1 A− ¯ Y11 − Y22 ) (µ K B
Fig. 9. Simplified structure to obtain nominal closed-loop transfer function M (s).
of a square, LTI, uncertain system G(s) that can be modeled as a nominal plant Ga (s) with an input multiplicative uncertainty ∆(s) having stable negative-imaginary characteristic. The scheme ¯ a (s) := H (sI − A)−1 B is designed to is shown in Fig. 8 where G ¯ be an SSNI system with Ga (0) < I following Theorem 2. Again, ¯ a (s)]−1 G¯ a (s) (see Fig. 9) is SNI via (Petersen & M (s) = [I − G ¯ a (s) is SSNI. Now, M (s) is SNI, Lanzon, 2010, Theorem 2) since G ∆(s) is NI and M (∞) = [I − G¯ a (∞)]−1 G¯ a (∞) = 0, therefore, it is straightforward to conclude via Theorem 1 that the proposed observer-based control scheme guarantees robust stability of the closed-loop uncertain system for any stable NI uncertainty ∆(s) satisfying the condition λmax [M (0)∆(0)] < 1. This scheme is also applicable for unstable plants for which an additional observerbased state feedback is required to first stabilize the plant. To show the usefulness of the proposed scheme, we consider the example of DC servo motor taken from Gu, Petkov, and Konstantinov (2013) where it is modeled as G(s) = s(1+α sTk)(1+sT ) with 0 < α ≪ 1 being unknown and T ∈ (0, ∞), k ∈ (0, ∞) being the known parameters. G(s) can be decomposed as Ga (s)[1 + ∆(s)] where Ga (s) = s(1+k sT ) is treated as the nominal plant model and
∆(s) :=
−α sT 1+α sT
=
−s s+ α1T
is the uncertain part. Thus, it fits into the
Fig. 7) is internally stable using Theorem 1.
structure shown in Fig. 8. It is evident that ∆(s) is an SNI transfer function with zero DC-gain for any T ∈ (0, ∞) and 0 < α ≪ 1. Now, the observer-based transformation transforms Ga (s) into an ¯ a (s) with G¯ a (0) < I and eventually, M (s) becomes SSNI system G SNI with M (∞) = 0. Therefore, robust stability of the closedloop system can be guaranteed due to Theorem 1 for any T ∈ (0, ∞) and 0 < α ≪ 1 since the interconnected system satisfies λmax [M (0)∆(0)] = 0 < 1 as ∆(0) = 0. It can be noted that in the present scheme, input u to the nominal system Ga (s) is required to be accessible for measurement so that it can be fed to the observer block. In case of DC servo motor, the signal u to the block Ga (s) = s(1+k sT ) is the armature current ia which can be measured physically.
6. Robustness analysis: an example
7. Conclusions
In this section, it is shown how the proposed observer-based feedback control scheme can be used to analyze robust stability
This paper introduces an observer-based control scheme that widens the applicability of negative-imaginary theory for control
It gives µ ¯ = 1.1962 using the CVX toolbox (Grant & Boyd, 2014). It isverified that the pair (A¯ , C¯ ) is observable, Y = µ ¯ Y11 −Y22
A¯ =
−Y22
¯ + Y A¯ T < 0 and B¯ = −AY ¯ C¯ T where > 0, AY −BK ¯ B T ,B = and C¯ := H H with H := BT A− K (Y22 −
Y22
AK 0
AL
0
µ ¯ Y11 )−1 = [−0.4072, −0.1360, −0.1643, −0.1424]. Therefore, ¯G(s) = H (sI − AK )−1 B = H (sI − A + BK )−1 B = 0.064772 (s+1.534)(s2 +3.484s+16.23) ¯ (0) = 0.0045 < 1. is SSNI with G (s+6)(s+5)(s+4)(s+3) ¯ (s) (see Hence, the unity positive feedback interconnection of G
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analysis and synthesis, especially in the presence of unstable and non-square plants. The LMI-based formulation facilitates easy application of this scheme. Robustness analysis of LTI uncertain systems with stable NI uncertainties can further be explored using the proposed technique. The scheme can also be extended for absolute stability analysis of the Lur’e system with positive feedback. Achieving closed-loop performance using the present framework is also a topic for future research. References Arcak, M., & Kokotovic, P. (2001). Observer-based control of systems with slope-restricted nonlinearities. IEEE Transactions on Automatic Control, 46(7), 1146–1150. Bernstein, D. S. (2009). Matrix mathematics: theory, facts and formulas (2nd ed.). Princeton University Press. Bhikkaji, B., Moheimani, S. O. R., & Petersen, I. R. (2012). A negative-imaginary approach to modeling and control of a colocated structure. IEEE/ASME Transactions on Mechatronics, 17(4), 717–727. Bhowmick, P., & Patra, S. (2016). On input–output negative-imaginary systems and an output strict negative-imaginary lemma. In Proceedings of 2nd Indian control conference (ICC 2016), Hyderabad, India, pp. 176–181. http://dx.doi.org/10.1109/INDIANCC.2016.7441125. Bhowmick, P., & Patra, S. (2017). On LTI output strictly negative-imaginary systems. Systems & Control Letters, 100, 32–42. Collado, J., Lozano, R., & Johansson, R. (2007). Using an observer to transform linear systems into strictly positive-real systems. IEEE Transactions on Automatic Control, 52(6), 1082–1088. Dey, A., Patra, S., & Sen, S. (2016). Absolute stability analysis for negative-imaginary systems. Automatica, 67, 107–113. Engelken, S., Patra, S., Lanzon, A., & Petersen, I. R. (2010). Stability analysis of negative-imaginary systems with real parametric uncertainty - the SISO case. IET Control Theory & Applications, 4(11), 2631–2638. Ferrante, A., Lanzon, A., & Ntogramatzidis, L. (2016). Foundations of not necessarily rational negative-imaginary systems theory: Relations between classes of negative-imaginary and positive-real systems. IEEE Transactions on Automatic Control, 61(10), 3052–3057. Ferrante, A., & Ntogramatzidis, L. (2013). Some new results in the theory of negative-imaginary systems with symmetric transfer matrix function. Automatica, 49(7), 2138–2144. Grant, M., & Boyd, S. (2014). CVX: Matlab software for disciplined convex programming, version 2.1. http://cvxr.com/cvx, March. Gu, D.-W., Petkov, P. Hr., & Konstantinov, M. M. (2013). Robust control design with matlab (1st ed.). London: Springer-Verlag. Johansson, R., & Robertsson, A. (2002). Observer-based strict positive real (SPR) feedback control system design. Automatica, 38, 1557–1564. Lanzon, A., & Petersen, I. R. (2008). Stability robustness of a feedback interconnection of systems with negative-imaginary frequency response. IEEE Transactions on Automatic Control, 53(4), 1042–1046. Lanzon, A., Song, Z., Patra, S., & Petersen, I. R. (2011). A strongly strict negative-imaginary lemma for non-minimal linear systems. Communications in Information and Systems, 11(2), 139–152. Li, X.-J., & Yang, G.-H. (2012). Dynamic observer-based robust control and fault detection for linear systems. IET Control Theory & Applications, 6(17), 2657–2666. Lien, C. (2004). Robust observer-based control of systems with state perturbations via LMI approach. IEEE Transactions on Automatic Control, 49(8), 1365–1370. Mabrok, M. A., Kallapur, A. G., Petersen, I. R., & Lanzon, A. (2014). Generalizing negative-imaginary systems theory to include free body dynamics: control of highly resonant structures with free body motion. IEEE Transactions on Automatic Control, 59(10), 2692–2707.
Mabrok, M. A., Kallapur, A. G., Petersen, I. R., & Lanzon, A. (2015). A generalized negative-imaginary lemma and Riccati-based static state-feedback negativeimaginary synthesis. Systems & Control Letters, 77, 63–68. Ogata, K. (2010). Modern control engineering (5th ed.). New Jersey: Prentice Hall. Oya, M., Su, C., & Kobayashi, T. (2004). State observer-based robust control scheme for electrically driven robot manipulators. IEEE Transactions on Robotics, 20(4), 796–804. Patra, S., & Lanzon, A. (2011). Stability analysis of interconnected systems with ‘mixed’ negative-imaginary and small-gain properties. IEEE Transactions on Automatic Control, 56(6), 1395–1400. Petersen, I. R., & Lanzon, A. (2010). Feedback control of negative-imaginary systems. IEEE Control Systems Magazine, 30(5), 54–72. Song, Z., Lanzon, A., Patra, S., & Petersen, I. R. (2010). Towards controller synthesis for systems with negative-imaginary frequency response. IEEE Transactions on Automatic Control, 55(6), 1506–1511. Song, Z., Lanzon, A., Patra, S., & Petersen, I. R. (2012a). Robust performance analysis for uncertain negative-imaginary systems. International Journal of Robust and Nonlinear Control, 22(3), 262–281. Song, Z., Lanzon, A., Patra, S., & Petersen, I. R. (2012b). A negative-imaginary lemma without minimality assumptions and robust state-feedback synthesis for uncertain negative-imaginary systems. Systems & Control Letters, 61(12), 1269–1276. Wang, J., Lanzon, A., & Petersen, I. R. (2015). Robust output feedback consensus for networked negative-imaginary systems. IEEE Transactions on Automatic Control, 60(9), 2547–2552. Xiong, J., Petersen, I. R., & Lanzon, A. (2010). A negative-imaginary lemma and the stability of interconnections of linear negative-imaginary systems. IEEE Transactions on Automatic Control, 55(10), 2342–2347. Xiong, J., Petersen, I. R., & Lanzon, A. (2012). On lossless negative-imaginary systems. Automatica, 48(6), 1213–1217.
Parijat Bhowmick received his B.Tech. degree in Electrical Engineering from West Bengal University of Technology, Kolkata in 2008 and received his M.E. degree in Control Systems Engineering from Jadavpur University, Kolkata in 2012. Currently, he is pursuing his Ph.D. degree in Control Systems specialization in the Department of Electrical Engineering, Indian Institute of Technology Kharagpur, West Bengal, India. His research interests include negative-imaginary systems theory, passivitybased control and decentralized integral control.
Sourav Patra received the Ph.D. degree in Electrical Engineering from the Indian Institute of Technology Kharagpur, India in 2009. He is currently appointed as an Assistant Professor at the Department of Electrical Engineering, Indian Institute of Technology Kharagpur, India, having previously worked as a Postdoctoral Research Associate at the Control Systems Centre, School of Electrical and Electronic Engineering, University of Manchester, UK. Prior to this position, he was appointed as a Reader in the Department of Avionics, Indian Institute of Space Science and Technology Trivandrum, India. His major research area is robust control, and other research interests include the actuator saturator control, time-delay systems, nonlinear systems and systems biology.
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Brief paper
An observer-based control scheme using negative-imaginary theory✩ Parijat Bhowmick 1 , Sourav Patra Indian Institute of Technology Kharagpur, Kharagpur-721302, India
article
info
Article history: Received 16 May 2016 Received in revised form 12 November 2016 Accepted 27 February 2017
Keywords: Strongly strict negative-imaginary systems Positive feedback Observer Linear matrix inequality Robustness
abstract This paper presents a full-order state observer-based scheme that transforms a causal, LTI, minimally realized system into a strongly strict negative-imaginary system by defining an auxiliary output based on the observed states. The auxiliary output is used for closed-loop control with positive feedback invoking the internal stability condition of negative-imaginary theory. A set of LMI conditions is derived that determines the value of a design parameter µ required for the proposed scheme to transform a given system with a minimal state-space realization into a strongly strict negative-imaginary system. The proposed scheme is applicable for both stable/unstable and square/non-square systems. In this paper, the framework is further explored for robustness analysis of uncertain systems and numerical examples are given to elucidate effectiveness of the proposed results. © 2017 Elsevier Ltd. All rights reserved.
1. Introduction The negative-imaginary systems theory has rapidly attracted the interest of the control community due to its simple internal stability condition for interconnected systems that depends only on the DC loop gain, and its wide applicability in different areas of control systems engineering. For example, such systems arise while considering the transfer functions from input voltage to output voltage in active electrical filters (Patra & Lanzon, 2011), from input voltage to shaft rotational velocity in DC servo motor (Song, Lanzon, Patra, & Petersen, 2012a), from pump input voltage to water level in a linearized coupled tank system. Other applications include nano-positioning systems, flexible spacecrafts, robotic manipulator arms (Mabrok, Kallapur, Petersen, & Lanzon, 2014), etc. The notion of negative-imaginary (NI) and strictly negative-imaginary (SNI) systems has been introduced in Lanzon and Petersen (2008) for robust control of highly resonant flexible structures with colocated position sensors and force actuators (Bhikkaji, Moheimani, & Petersen, 2012; Mabrok et al., 2014). An NI system is a Lyapunov-stable system with equal number of inputs and outputs, having a real-rational, proper transfer function matrix R(s) that satisfies the frequency domain condition j[R(jω) −
✩ The material in this paper was not presented at any conference. This paper was recommended for publication in revised form by Associate Editor James Lam under the direction of Editor Ian R. Petersen. E-mail addresses: [email protected] (P. Bhowmick), [email protected] (S. Patra). 1 Fax: +91 3222 282262.
http://dx.doi.org/10.1016/j.automatica.2017.03.024 0005-1098/© 2017 Elsevier Ltd. All rights reserved.
R⋆ (jω)] ≥ 0 ∀ω ∈ (0, ∞). In a SISO setting, a real-rational, proper transfer function is said to have the NI (respectively, SNI) properties if its Nyquist plot lies below (respectively, strictly below) the real axis in the open positive frequency interval (Lanzon & Petersen, 2008; Petersen & Lanzon, 2010). So far in the NI literature, the state-space characterizations of LTI NI and SNI systems have been studied in Lanzon and Petersen (2008), Petersen and Lanzon (2010) and Xiong, Petersen, and Lanzon (2010); NI systems with single and double poles at the origin in Ferrante, Lanzon, and Ntogramatzidis (2016), Ferrante and Ntogramatzidis (2013) and Mabrok et al. (2014); strongly strict negative-imaginary (SSNI) systems in Ferrante et al. (2016) and Lanzon, Song, Patra, and Petersen (2011); lossless NI system properties in Xiong, Petersen, and Lanzon (2012); controller synthesis, robustness and performance analysis in Bhikkaji et al. (2012), Engelken, Patra, Lanzon, and Petersen (2010), Mabrok, Kallapur, Petersen, and Lanzon (2015) and Song, Lanzon, Patra, and Petersen (2010); Song et al. (2012a); Song, Lanzon, Patra, and Petersen (2012b). In recent times, the NI theory has been extended for symmetric and non-rational transfer functions (Ferrante et al., 2016; Ferrante & Ntogramatzidis, 2013) and also, the cooperative control strategy is being applied to control multiple NI agents connected through a communication network (Wang, Lanzon, & Petersen, 2015). The NI stability result for interconnected systems has drawn profound interest from control theoretic perspective since the stability criteria depends only on the loop gains at zero and infinite frequencies. However, the theory remains inapplicable if the interconnected systems are non-NI which occurs quite often in control theoretic applications. Therefore, if a scheme can be developed which transforms any LTI non-NI system into NI
P. Bhowmick, S. Patra / Automatica 81 (2017) 196–202
Fig. 1. An observer-based control scheme for stable LTI plants with positive feedback using NI theory.
class then the existing NI theory can be applied to synthesize a stabilizing controller depending on the loop gains at zero and infinite frequencies. So far in the NI literature, to the best of the authors’ knowledge, no such procedure is reported by which a nonNI system can be transformed into an NI system. This motivates us to propose a scheme by which an LTI system can be transformed into a negative-imaginary system facilitating the applicability of the NI theory for analysis and controller synthesis, especially widening the applicability of this theory in the presence of unstable and/or non-square plants. In this paper, we introduce a scheme, as shown in Fig. 1, which transforms a causal, LTI, minimally realized system into a strongly strict negative-imaginary (SSNI) system (Lanzon et al., 2011), a subset of the strictly negative-imaginary (SNI) class (Xiong et al., 2010), by defining an auxiliary output based on the observed states. The proposed observer-based SSNI scheme can also be used to stabilize a system G(s) in closed-loop when the auxiliary output z is positively fed back (see Fig. 1) along with satisfying a particular DC ¯ (s). This scheme loop gain condition on the transformed system G does not require the original system to be stable or square. In this scheme, a full-order state observer is placed in cascade with the plant and a new output z is defined that depends only on the estimated states xˆ . For a particular structure of the transformation matrix H, the combined system from input u to the defined output z := H xˆ exhibits SSNI properties. A set of LMI-based conditions, involving a design parameter µ > 0, is formulated that yields a feasible finite positive µ value required to design an observerbased control scheme for a given plant. In this work, we have exploited the fact that the SSNI system properties can be defined for uncontrollable systems while maintaining the observability and the full normal rank conditions (Patra & Lanzon, 2011). So far, various observer-based control schemes have been introduced in the literature. But none of these techniques is dedicated for the NI framework. To mention a few: An observerbased state feedback control scheme is designed in Arcak and Kokotovic (2001) to ensure global asymptotic stability of linear systems with slope-restricted nonlinearities in feedback; Lyapunov stability theory and LMI-based approach have been adopted in Lien (2004) to design observer-based control for a class of linear systems with state perturbations where the controller and observer gains can be obtained from the LMI formulations; A state observer-based robust trajectory-tracking control scheme is proposed in Oya, Su, and Kobayashi (2004) for electrically driven robot manipulators where the observer is used to estimate the joint angular velocities; An observer-based robust control technique is designed in Li and Yang (2012) that also facilitates the acquisition of information used for fault detection in feedback control systems; Further in Collado, Lozano, and Johansson (2007), Johansson and Robertsson (2002), the authors have proposed an observerbased transformation scheme in strictly positive-real framework. In this paper, the proposed observer-based transformation and the associated feedback control scheme broaden the scope of applicability of the NI theory for robust control analysis and synthesis. The remainder of the paper proceeds as follows: in Section 2, useful notations are given. Section 3 illustrates a few necessary
197
definitions, lemmas and theorems which streamline the main results of this paper. Section 4 deals with the observer-based scheme that transforms a causal, LTI system into an SSNI system. Section 5 formulates a set of LMI-based conditions by which the proposed observer-based feedback control scheme guarantees internal stability of LTI systems with positive feedback. In Section 6, robust stability analysis of LTI uncertain systems using the proposed observer-based control strategy is explained through an example. Section 7 concludes the paper. 2. Notations The notations and acronyms are standard throughout. The fields of real and complex numbers are denoted by R and C, respectively. The sets of real and complex matrices of dimension (n × n) are denoted by Rn×n and Cn×n , respectively. The maximum and the minimum eigenvalues of a matrix A ∈ Cn×n that has only real eigenvalues are indicated by λmax (A) and λmin (A), respectively. The determinant and the rank of a matrix A are denoted by det[A] and rank[A], respectively. The real and imaginary parts are represented by ℜ(·) and ℑ(·), respectively. AT , A⋆ and A¯ denote the transpose, the complex-conjugate transpose and the complex-
⋆
T
conjugate of a matrix A. The shorthands for A−1 and A−1 are represented by A−⋆ and A−T , respectively. The terminology that R(s) is a ‘proper’ transfer function matrix includes the cases R(∞) ≡ 0, det[R(∞)] = 0 and det[R(∞)] ̸= 0. The set of all proper, real-rational transfer function matrices is denoted by R n×n whereas the set of all proper, real-rational and asymptotically stable transfer function matrices is denoted by RH n∞×n , both of dimensions (n × n). For a transfer function matrix R(s), R⋆ (jω) = RT (−jω) and R∼ (s) = RT (¯s) where s¯ denotes the complexconjugate of s. The minimal state-space realization of a system R(s) A B is represented by R(s) = . C D min 3. Preliminaries In this section, some preliminary results are given which provide a background to establish the main results of this paper. We now describe the NI, SNI and SSNI systems through the following definitions and the lemma. For complete details of the NI theory and its applications literatures (Bhikkaji et al., 2012; Bhowmick & Patra, 2016, 2017; Dey, Patra, & Sen, 2016; Lanzon & Petersen, 2008; Lanzon et al., 2011; Mabrok et al., 2014, 2015; Petersen & Lanzon, 2010; Xiong et al., 2010) may be referred. Definition 1 (NI System (Mabrok et al., 2014, 2015)). A square, causal, real-rational, proper transfer function matrix R(s) is negative-imaginary (NI) if the following conditions are satisfied: (1) R(s) has no poles in ℜ[s] > 0; (2) j[R(jω) − R⋆ (jω)] ≥ 0 ∀ω ∈ (0, ∞) except the values of ω where jω is a pole of R(s); (3) If jω0 , ω0 ∈ (0, ∞), is a pole of R(s), it is at most a simple pole and the residue matrix K0 := lims→jω0 (s − jω0 )jR(s) is positive semidefinite Hermitian; (4) If s = 0 is a pole of R(s) then, lims→0 sk R(s) = 0 for all k ≥ 3 and lims→0 s2 R(s) is positive semidefinite Hermitian. According to Definition 1, the NI class includes systems having complex pole-pair on the jω axis or pole(s) at the origin, however, in this paper, we have considered only asymptotically stable NI systems. Definition 2 (SNI System (Petersen & Lanzon, 2010; Xiong et al., 2010)). A square, causal, real-rational, proper transfer function matrix R(s) is strictly negative-imaginary (SNI) if the following conditions are satisfied:
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Fig. 3. An observer-based scheme to transform a stable LTI system into an SSNI system.
Fig. 2. Interconnection of NI systems with positive feedback.
(1) R(∞) = RT (∞); (2) R(s) has no poles in ℜ[s] ≥ 0; (3) j[R(jω) − R⋆ (jω)] > 0 ∀ω ∈ (0, ∞).
where x ∈ Rn , u ∈ Rm , y ∈ Rp and m ≤ n. We further consider the full-order Luenberger observer (Ogata, 2010) dynamics as:
The frequency-domain condition j[R(jω) − R⋆ (jω)] ≥ 0 (> 0, respectively) ∀ω ∈ (0, ∞) given in Definitions 1 and 2 implies that, for SISO case, the polar plots of NI (SNI) transfer functions lie below (strictly below) the real axis of the Nyquist plane. The following lemma is presented here to characterize strongly strict negative-imaginary (SSNI) systems, a subset of the SNI class (Xiong et al., 2010). The SSNI Lemma will be invoked later to prove the main results of this paper.
A B Lemma 1 (SSNI Lemma 1 (Lanzon et al., 2011)). Let C D be a state-space realization of a causal, real-rational, proper transfer function matrix R(s) ∈ R m×m . Assume, R(s) + R∼ (s) has full normal rank m and the pair (A, C ) is observable. Then, A is Hurwitz and R(s) is SNI with ⋆
lim jω R(jω) − R (jω) > 0 and
ω→∞
1
lim j
ω
ω→0
R(jω) − R⋆ (jω) > 0,
B = −AYC T .
(2)
(3)
4. An observer-based scheme to transform an LTI system into NI class In this section, we present an observer-based scheme that transforms a causal, LTI, minimally realized system into an SSNI system by defining an auxiliary output that depends on the observed states (see Fig. 3). The scheme is applicable for both stable/unstable and square/non-square systems. Let a causal, LTI, minimally realized system be given as: G:
x˙ = Ax + Bu, y = Cx + Du,
x (0) = x0 ;
where L ∈ Rp×n is the observer-gain matrix to be designed such that the observer dynamics must be stable and faster than the system dynamics. Fig. 3 shows the proposed observer-based scheme, the main contribution of this paper, where an auxiliary output z is defined based on the observed states xˆ given as z := H xˆ where H ∈ Rm×n is a transformation matrix to be designed. Introducing the state estimation error variable x˜ := xˆ − x, we ¯ (s) from input u to the auxiliary represent the combined system G output z (see Fig. 3) in a compact state-space form:
x˙ A 0 x B = + u, x˙˜ 0 0 (A − LT C ) x˜ ¯: G z = H H x . ˜ Denoting AL := A − LT C , A¯ :=
Theorem 1 (Lanzon & Petersen, 2008). Given a stable NI transfer function matrix R1 (s) and an SNI transfer function matrix R2 (s) that satisfy R1 (∞)R2 (∞) = 0 and R2 (∞) ≥ 0. Then, the positive feedback interconnection of R1 (s) and R2 (s), in Fig. 2, is internally stable if and only if
(5)
(6)
x
It is worth noting that the minimal state-space realization is not a mandatory criteria to define SSNI systems but the full normal rank condition must be satisfied. We now present a necessary and sufficient condition for internal stability of an NI interconnection with positive feedback as shown in Fig. 2. The following theorem will be invoked in Section 5 to establish internal stability of the proposed closed-loop control scheme.
λmax [R1 (0)R2 (0)] < 1.
xˆ (0) = 0;
(1)
if and only if, D = DT and there exists a real matrix Y = Y T > 0 such that AY + YAT < 0 and
T ˙ˆ ˆ ˆ Σo : x = Ax + Bu + L (y − y), yˆ = C xˆ + Du,
(4)
H
A 0
0 AL
, B¯ :=
B 0
and C¯ :=
H , the transfer function mapping from u to z becomes
¯ (s) := C¯ (sI − A¯ )−1 B¯ = H (sI − A)−1 B. It is worth noting that G B¯ A¯ is not minimal since the the state-space realization C¯ 0 modes associated with the block (A − LT C ) are not controllable. We first consider the observer-based scheme for stable plants. The ¯ (s) satisfies following lemma shows that the transformed system G the SSNI properties for a particular choice of H matrix.
A B Lemma 2. Let G(s) = ∈ RH p∞×m be a causal LTI plant C D min with rank[B] = m and m ≤ n where n is the number of states. Assume an observer-gain matrix L ∈ Rp×n is given such that AL := A − LT C is T Hurwitz and two symmetric positive definite matrices Y11 = Y11 >0 T and Y22 = Y22 > 0 are respectively the solutions of the Lyapunov equations AY11 + Y11 AT = −Q1 AL Y22 +
Y22 ATL
and
(7)
= −Q2
for given Q1 = Q1T > 0 and Q2 = Q2T > 0. Assume µ is a positive scalar that satisfies the following set of LMIs:
µY11 − Y22 > 0, µQ1 (AY22 + Y22 ATL )T
(8a)
(AY22 +
Y22 ATL
Q2
)
> 0,
(8b)
and z := H xˆ where H = BT A−T (Y22 − µ Y11 )−1 . Suppose further ¯ B¯ ¯ ( s) = A that the combined system G from input u to the C¯ 0 auxiliary output z, shown in Fig. 3, is completely observable. Then, ¯ (s) is SSNI. G
P. Bhowmick, S. Patra / Automatica 81 (2017) 196–202
Proof. Since, rank[B] = m, m ≤ n, where n is the number of states and A is Hurwitz, using the standard matrix rank inequality (Bernstein, 2009) we get, rank[A−1 B] = m. Further, we have rank[(Y22 − µ Y11 )−1 ] = n since (µ Y11 − Y22 ) > 0 from (8a) and rank[(sI − A)−1 ] = n since A is Hurwitz. Combining these ¯ (s) := H (sI − A)−1 B = (A−1 B)T (Y22 − results we can conclude G µ Y11 )−1 (sI − A)−1 B has full normal rank m. Since, A and A − LT C are both Hurwitz for a given L, therefore, for any Q1 = Q1T > 0 and Q2 = Q2T > 0 there always exist positive definite solutions T T Y11 = Y11 > 0 and Y22 = Y22 > 0 of the Lyapunov equations (7). We now show, with these assumptions, that there always exists a positive scalar µ that satisfies (8a)–(8b). From (8a), we have µ Y11 − Y22 > 0 ⇔ µ Y11 > Y22 ⇔ µλi [Y11 ] > λi [Y22 ] ∀i and from (8b), µ Q1 − (AY22 + Y22 ATL )Q2−1 (AY22 + Y22 ATL )T > 0 ⇔ µλi [Q1 ] > λi [(AY22 + Y22 ATL )Q2−1 (AY22 + Y22 ATL )T ] ∀i. Since Y11 > 0, Y22 > 0, Q1 > 0, Q2 > 0, (AY22 + Y22 ATL )Q2−1 (AY22 + Y22 ATL )T ≥ 0 and µ is associated with the lefthand terms (i.e., Y11 and Q1 ) in both the inequalities, there always exists a µ > 0 for a given system such that the eigenvalues of the left-hand terms are greater than the eigenvalues of the right-hand terms in the above inequalities. Therefore, we have a µ > 0 that satisfies (8a)–(8b). Following ¯ ¯ the assumption that (A, C ) is completely observable and defining µY
−Y22
Y := −Y11 22
Y22
> 0, we compute
¯ + Y A¯ T AY =
A 0
0 AL
µY11 −Y22
−Y22 Y22
µ AY11 + Y11 AT = T − AY22 + Y22 AL T µQ1 T =− AY22 + Y22 AL T
µY11 −Y22 A + −Y22 Y22 0 T − AY22 + Y22 AL AL Y22 + Y22 AL T AY22 + Y22 AL T <0
0 AL
being SSNI.
1 0 1 0 −4.25 −6 −2.75 0.5 that satisfies the assumptions −1 0 4 −1 0.5 0 0 0 rank[B] = m = 1 < n = 3. We choosethe observer poles at (−3, −6, −9) arbitrarily and obtain L = 24.0, −36.0, 8.0 by using Ackermann’s formula (Ogata, 2010). Further we take Q1 = 2 I3 , Q2 = 0.5 I equations (7) 3 and by solving the Lyapunov
Q2
6.1333 −1.1583 −1.1583 0.6854 −7.1333 0.6583 0.0095 −0.0095 > 0. For 0.1644
we get, Y11 =
0.0236
0.0170 0.0095
0.0170 0.0849 −0.0095
−7.1333 0.6583 10.7667
> 0 and Y22 =
these L, Q1 , Q2 , Y11 and Y22 ,
the set of LMIs (8a)–(8b) yields µ = 0.8723 using the CVX toolbox ¯ ¯ (Grant & Boyd, 2014). It is verified that the pair (A, C ) is observable, µY
−Y22
Y := −Y11 22
Y22
¯ + Y A¯ T < 0 and B¯ = −AY ¯ C¯ T . > 0, AY 0.35(s2 +1.444s+1.373) s3 +6s2 +11s+6
¯ (s) = H (sI − A)−1 B = Therefore, G
0
H = BT A−T (Y22 − µY11 )−1 = 0.4947,
[since H = B A (Y22 − µY11 ) ] B =− = −B¯ . −1
0.7022,
is SSNI with 0.2535 . The
¯ (s) is shown in Fig. 4. polar plot of G
0
Thus, it is proved that the state-space realization
0.35(s2 +1.444s+1.373) s3 +6s2 +11s+6
has to be searched for, which makes the pair (A¯ , C¯ ) completely observable. To this end, one may think of the other relaxed assumption on minimality of the state-space realization of an SSNI system (Lanzon et al., 2011, Theorem 3.4) which requires there ¯ (s). But it would be exist no observable uncontrollable modes of G more restrictive to satisfy the latter condition, as in A¯ all the modes associated with A − LT C are uncontrollable.
¯ C¯ T is simplified to Again, AY A 0 µY11 −Y22 H T 0 AL −Y22 Y22 HT A (µY11 − Y22 ) H T = AL (−Y22 + Y22 ) H T A (Y22 − µY11 ) (Y22 − µY11 )−1 A−1 B =− −T
¯ (s ) = Fig. 4. Polar plot of the transformed system G
Example 1. Let us consider a stable, non-NI transfer function G(s) = s3 +6s21+11s+6 with a minimal state-space realization
T
[using(8b)].
T
199
A¯ C¯
B¯ 0
Example 2. We consider a causal, stable, non-square, non-NI plant
of
¯ (s), given in (6), satisfies AY ¯ + Y A¯ T < 0 and the combined system G ¯B = −AY ¯ C¯ T where Y = Y T > 0. Therefore, G¯ (s) is SSNI according to Lemma 1. This completes the proof. Remark 1. In Lemma 2, it is established that for any given stable plant with a minimal state-space realization, there always exists a positive scalar µ that satisfies the set of LMIs (8a)–(8b). However, A¯ B¯ ¯ ( s) = the transformed system G may not always C¯ 0 be completely observable for all given plants, especially in case ¯ (s) to be of non-square plants. This violates the assumption for G SSNI. In that case, an appropriate observer-gain matrix L ∈ Rp×n
(s G(s) =
−13 17 −7 0.75 −2
2
s−1 + 8s + 20) with a minimal state-space realization s−1 (s + 3)
−5
0 1 5 0 −1 −3 −3 2 , rank[B] = m = 1 < n = 3. −0.25 0 0 −2 −2 1 We arbitrarily choose the observer poles at (−12, −12, −9) and obtain L =
9.5172 1.7931
−23.4483 −7.6207
13.9310 1.3276
. Further choosing Q1 =
Q2 = 2 I3 , we get Y11 > 0 and Y22 > 0 by solving (7) and based on these matrices, the set of LMIs (8a)–(8b) yields µ = 1.3357 using the CVX toolbox (Grant & Boyd, 2014). Now, following the ¯ ( s) = same steps of Example 1 we find the transformed system G
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P. Bhowmick, S. Patra / Automatica 81 (2017) 196–202
exactly that of Lemma 2 considering A¯ =
remaining the same. Hence, the proof is done.
AK 0
−BK AL
and B¯ , C¯
5. An observer-based closed-loop control scheme using negative-imaginary framework
¯ (s) = Fig. 5. Polar plot of the transformed transfer function G
3.4517(s2 +6.934s+17.3) (s+3)(s2 +8s+20)
being SSNI.
In the previous section, we have introduced a state observerbased technique that transforms a causal, LTI, minimally realized system (stable/unstable and square/non-square system) into an SSNI system by defining an auxiliary output that depends on the observed states. In this section, we derive a set of LMI-based conditions that yields a solution for the design parameter µ ¯ > 0 ¯ (s) from u to z (shown in for which the transformed system G Fig. 1) becomes SSNI and simultaneously satisfies the condition λmax [G¯ (0)] < 1 which, in turn, ensures internal stability of the proposed closed-loop control scheme via Theorem 1.
A B Theorem 2. Let G(s) = ∈ RH p∞×m be a causal, LTI C D min plant with rank[B] = m and m ≤ n where n is the number of states. Given an observer-gain matrix L ∈ Rp×n such that AL := A − LT C is T T Hurwitz, assume Y11 = Y11 > 0 and Y22 = Y22 > 0 are respectively the solutions of the Lyapunov equations AY11 + Y11 AT = −Q1 AL Y22 + Fig. 6. An observer-based scheme to transform an LTI, open-loop unstable system into an SSNI system.
H (sI − A)−1 B =
3.4517(s2 +6.934s+17.3) (s+3)(s2 +8s+20)
which is SSNI via Lemma 1.
¯ (s). Fig. 5 shows the polar plot of G
Note that Lemma 2 is given for stable systems. We now extend this result for unstable systems, presented below in Lemma 3, where a similar transformation scheme will be pursued preceded by an observer-based state feedback that first stabilizes the given unstable plant. The proposed scheme is shown in Fig. 6.
A B ∈ R p×m be a causal, LTI, unstable Lemma 3. Let G(s) = C D min plant with rank[B] = m and m ≤ n where n is the number of states. Assume an observer-gain matrix L ∈ Rp×n and a state feedback gain matrix K ∈ Rm×n are given such that AL := A − LT C and AK := A − BK T T are Hurwitz and Y11 = Y11 > 0 and Y22 = Y22 > 0 are respectively the solutions of the Lyapunov equations AK Y11 + Y11 ATK = −Q1 AL Y22 +
Y22 ATL
and
(9)
= −Q2
µ Y11 − Y22 > 0, µQ1 − BKY22 − Y22 K T BT T T L CY22 − Q2
(10a)
LT CY22 − Q2 > 0,
(10b)
Q2
T and z := H xˆ where H = BT A− (Y22 − µ Y11 )−1 . Suppose further K ¯ B¯ ¯ ( s) = A from input v to the that the combined system G C¯ 0 auxiliary output z, as shown in Fig. 6, is completely observable. Then, ¯ (s) is SSNI. G
Proof. In this lemma, K is a predetermined state-feedback gain matrix such that AK := A − BK is Hurwitz. Rest of the proof follows
and
(11)
= −Q2
for given Q1 = Q1T > 0 and Q2 = Q2T > 0. If there exists a scalar µ ¯ > 0 such that the following set of LMIs is satisfied:
µ ¯ Y11 − Y22 > 0, µ ¯ Q1 T AY22 + Y22 AL T T I
A−1 B
(12a)
AY22 + Y22 AL
T
Q2
> 0,
B A−T > 0, ¯ Y11 − Y22 ) (µ
(12b)
¯ (s) = and the combined system G
(12c)
A¯
B¯
from u to z := H xˆ = 0 C¯ BT A−T (Y22 − µ ¯ Y11 )−1 xˆ , as shown in Fig. 1, is completely observable, then the closed-loop control system is internally stable. Proof. We assume that there exists a µ ¯ > 0 such that the set of LMIs (12a)–(12c) holds true. The first two inequalities, together with the assumption that (A¯ , C¯ ) is completely observable, enforces ¯ (s) := H (sI − A)−1 B to be an SSNI system following Lemma 2. G While the third inequality (12c) is equivalent to the DC-gain ¯ (s), as shown below: condition on G
for given Q1 = Q1T > 0 and Q2 = Q2T > 0. Assume, µ > 0 is a scalar that satisfies the following set of LMIs:
Y22 ATL
I
A−1 B
BT A−T >0 ¯ Y11 − Y22 ) (µ
⇔ I − BT A−T (µ ¯ Y11 − Y22 )−1 A−1 B > 0 ⇔ I − H (µ ¯ Y11 − Y22 ) H T > 0 [since H = BT A−T (Y22 − µ ¯ Y11 )−1 ] T µ ¯ Y11 −Y22 H ⇔ H H
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201
Fig. 7. An observer-based control scheme for unstable LTI plants with positive feedback using NI theory.
¯ (0)] < 1, the closed-loop system with unity positive and λmax [G feedback is internally stable according to Theorem 1. This completes the proof.
Fig. 8. An observer-based control scheme for LTI uncertain systems using NI framework.
Remark 2. Using Lemma 3 and Theorem 2, we can develop an observer-based closed-loop control scheme for open-loop unstable systems as shown in Fig. 7. This idea is very similar to Lemma 3. Here, the scheme is discussed through an example given below (see Example 3). It is worth noting that, in the proposed closedloop scheme, in order to achieve some performance and robustness objectives, one can also introduce a compensator in the loop that must satisfy either NI or SNI properties along with the DC loop gain condition for internal stability. Example 3. Let us consider a (2 × 1) unstable system G(s) =
min
0
1 0 0 0 0 0 0 0 1 0 0 0 1 0 0 0 0 0 0 0 1 0 4. We choose the closed-loop system poles and observer poles arbitrarily at (−3, −4, −5, −6) and (−6, −8, −10, −12), respectively and obtain K = [−157.9496, −35.4312, −36 .6972, −34.8624]
0 −1 0 with rank[B] = m = 1 and n = 0.5
20.6 0 −0.4905
18.5560 1.9594
and L =
103.6738 17.6676
1.8833 17.4440
16.3878 72.9264
. Further choosing
Q1 = 0.5 I4 , Q2 = 0.003 I4 and by solving the Lyapunov equations T T (9) we get, Y11 = Y11 > 0 and Y22 = Y22 > 0. Based on these predetermined matrices, feasibility of the following set of LMIs is tested for µ ¯ >0
µ ¯ Y11 − Y22 > 0, T µ ¯ Q1 − BKY22 L CY − Q 22 2 − Y22 K T BT > 0, T T L CY22 − Q2 Q2 T −T I B AK > 0. 1 A− ¯ Y11 − Y22 ) (µ K B
Fig. 9. Simplified structure to obtain nominal closed-loop transfer function M (s).
of a square, LTI, uncertain system G(s) that can be modeled as a nominal plant Ga (s) with an input multiplicative uncertainty ∆(s) having stable negative-imaginary characteristic. The scheme ¯ a (s) := H (sI − A)−1 B is designed to is shown in Fig. 8 where G ¯ be an SSNI system with Ga (0) < I following Theorem 2. Again, ¯ a (s)]−1 G¯ a (s) (see Fig. 9) is SNI via (Petersen & M (s) = [I − G ¯ a (s) is SSNI. Now, M (s) is SNI, Lanzon, 2010, Theorem 2) since G ∆(s) is NI and M (∞) = [I − G¯ a (∞)]−1 G¯ a (∞) = 0, therefore, it is straightforward to conclude via Theorem 1 that the proposed observer-based control scheme guarantees robust stability of the closed-loop uncertain system for any stable NI uncertainty ∆(s) satisfying the condition λmax [M (0)∆(0)] < 1. This scheme is also applicable for unstable plants for which an additional observerbased state feedback is required to first stabilize the plant. To show the usefulness of the proposed scheme, we consider the example of DC servo motor taken from Gu, Petkov, and Konstantinov (2013) where it is modeled as G(s) = s(1+α sTk)(1+sT ) with 0 < α ≪ 1 being unknown and T ∈ (0, ∞), k ∈ (0, ∞) being the known parameters. G(s) can be decomposed as Ga (s)[1 + ∆(s)] where Ga (s) = s(1+k sT ) is treated as the nominal plant model and
∆(s) :=
−α sT 1+α sT
=
−s s+ α1T
is the uncertain part. Thus, it fits into the
Fig. 7) is internally stable using Theorem 1.
structure shown in Fig. 8. It is evident that ∆(s) is an SNI transfer function with zero DC-gain for any T ∈ (0, ∞) and 0 < α ≪ 1. Now, the observer-based transformation transforms Ga (s) into an ¯ a (s) with G¯ a (0) < I and eventually, M (s) becomes SSNI system G SNI with M (∞) = 0. Therefore, robust stability of the closedloop system can be guaranteed due to Theorem 1 for any T ∈ (0, ∞) and 0 < α ≪ 1 since the interconnected system satisfies λmax [M (0)∆(0)] = 0 < 1 as ∆(0) = 0. It can be noted that in the present scheme, input u to the nominal system Ga (s) is required to be accessible for measurement so that it can be fed to the observer block. In case of DC servo motor, the signal u to the block Ga (s) = s(1+k sT ) is the armature current ia which can be measured physically.
6. Robustness analysis: an example
7. Conclusions
In this section, it is shown how the proposed observer-based feedback control scheme can be used to analyze robust stability
This paper introduces an observer-based control scheme that widens the applicability of negative-imaginary theory for control
It gives µ ¯ = 1.1962 using the CVX toolbox (Grant & Boyd, 2014). It isverified that the pair (A¯ , C¯ ) is observable, Y = µ ¯ Y11 −Y22
A¯ =
−Y22
¯ + Y A¯ T < 0 and B¯ = −AY ¯ C¯ T where > 0, AY −BK ¯ B T ,B = and C¯ := H H with H := BT A− K (Y22 −
Y22
AK 0
AL
0
µ ¯ Y11 )−1 = [−0.4072, −0.1360, −0.1643, −0.1424]. Therefore, ¯G(s) = H (sI − AK )−1 B = H (sI − A + BK )−1 B = 0.064772 (s+1.534)(s2 +3.484s+16.23) ¯ (0) = 0.0045 < 1. is SSNI with G (s+6)(s+5)(s+4)(s+3) ¯ (s) (see Hence, the unity positive feedback interconnection of G
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analysis and synthesis, especially in the presence of unstable and non-square plants. The LMI-based formulation facilitates easy application of this scheme. Robustness analysis of LTI uncertain systems with stable NI uncertainties can further be explored using the proposed technique. The scheme can also be extended for absolute stability analysis of the Lur’e system with positive feedback. Achieving closed-loop performance using the present framework is also a topic for future research. References Arcak, M., & Kokotovic, P. (2001). Observer-based control of systems with slope-restricted nonlinearities. IEEE Transactions on Automatic Control, 46(7), 1146–1150. Bernstein, D. S. (2009). Matrix mathematics: theory, facts and formulas (2nd ed.). Princeton University Press. Bhikkaji, B., Moheimani, S. O. R., & Petersen, I. R. (2012). A negative-imaginary approach to modeling and control of a colocated structure. IEEE/ASME Transactions on Mechatronics, 17(4), 717–727. Bhowmick, P., & Patra, S. (2016). On input–output negative-imaginary systems and an output strict negative-imaginary lemma. In Proceedings of 2nd Indian control conference (ICC 2016), Hyderabad, India, pp. 176–181. http://dx.doi.org/10.1109/INDIANCC.2016.7441125. Bhowmick, P., & Patra, S. (2017). On LTI output strictly negative-imaginary systems. Systems & Control Letters, 100, 32–42. Collado, J., Lozano, R., & Johansson, R. (2007). Using an observer to transform linear systems into strictly positive-real systems. IEEE Transactions on Automatic Control, 52(6), 1082–1088. Dey, A., Patra, S., & Sen, S. (2016). Absolute stability analysis for negative-imaginary systems. Automatica, 67, 107–113. Engelken, S., Patra, S., Lanzon, A., & Petersen, I. R. (2010). Stability analysis of negative-imaginary systems with real parametric uncertainty - the SISO case. IET Control Theory & Applications, 4(11), 2631–2638. Ferrante, A., Lanzon, A., & Ntogramatzidis, L. (2016). Foundations of not necessarily rational negative-imaginary systems theory: Relations between classes of negative-imaginary and positive-real systems. IEEE Transactions on Automatic Control, 61(10), 3052–3057. Ferrante, A., & Ntogramatzidis, L. (2013). Some new results in the theory of negative-imaginary systems with symmetric transfer matrix function. Automatica, 49(7), 2138–2144. Grant, M., & Boyd, S. (2014). CVX: Matlab software for disciplined convex programming, version 2.1. http://cvxr.com/cvx, March. Gu, D.-W., Petkov, P. Hr., & Konstantinov, M. M. (2013). Robust control design with matlab (1st ed.). London: Springer-Verlag. Johansson, R., & Robertsson, A. (2002). Observer-based strict positive real (SPR) feedback control system design. Automatica, 38, 1557–1564. Lanzon, A., & Petersen, I. R. (2008). Stability robustness of a feedback interconnection of systems with negative-imaginary frequency response. IEEE Transactions on Automatic Control, 53(4), 1042–1046. Lanzon, A., Song, Z., Patra, S., & Petersen, I. R. (2011). A strongly strict negative-imaginary lemma for non-minimal linear systems. Communications in Information and Systems, 11(2), 139–152. Li, X.-J., & Yang, G.-H. (2012). Dynamic observer-based robust control and fault detection for linear systems. IET Control Theory & Applications, 6(17), 2657–2666. Lien, C. (2004). Robust observer-based control of systems with state perturbations via LMI approach. IEEE Transactions on Automatic Control, 49(8), 1365–1370. Mabrok, M. A., Kallapur, A. G., Petersen, I. R., & Lanzon, A. (2014). Generalizing negative-imaginary systems theory to include free body dynamics: control of highly resonant structures with free body motion. IEEE Transactions on Automatic Control, 59(10), 2692–2707.
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Parijat Bhowmick received his B.Tech. degree in Electrical Engineering from West Bengal University of Technology, Kolkata in 2008 and received his M.E. degree in Control Systems Engineering from Jadavpur University, Kolkata in 2012. Currently, he is pursuing his Ph.D. degree in Control Systems specialization in the Department of Electrical Engineering, Indian Institute of Technology Kharagpur, West Bengal, India. His research interests include negative-imaginary systems theory, passivitybased control and decentralized integral control.
Sourav Patra received the Ph.D. degree in Electrical Engineering from the Indian Institute of Technology Kharagpur, India in 2009. He is currently appointed as an Assistant Professor at the Department of Electrical Engineering, Indian Institute of Technology Kharagpur, India, having previously worked as a Postdoctoral Research Associate at the Control Systems Centre, School of Electrical and Electronic Engineering, University of Manchester, UK. Prior to this position, he was appointed as a Reader in the Department of Avionics, Indian Institute of Space Science and Technology Trivandrum, India. His major research area is robust control, and other research interests include the actuator saturator control, time-delay systems, nonlinear systems and systems biology.