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Robust output regulation for state feedback descriptor systems with nonovershooting behavior
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Robust output regulation for state feedback descriptor systems with nonovershooting behavior Praveen S Babu, Nithin Xavier, Bijnan Bandyopadhyay∗ Systems and Control Engineering, Indian Institute of Technology Bombay, Mumbai 400 076, India
a r t i c l e
i n f o
Article history: Received 3 November 2018 Revised 4 June 2019 Accepted 30 August 2019 Available online xxx Recommended by Eduardo Costa Keywords: Descriptor systems Output regulation Robust control Tracking
a b s t r a c t In this paper, the design of robust non-overshooting (NO) controllers for output regulation is presented for a continuous linear time-invariant (LTI), multi-input multi-output (MIMO) state feedback based descriptor system. In this method of controller design for descriptor system, the integral sliding mode (ISM) technique along with generalised Moore’s eigen structure assignment method and output regulation theory for descriptor system is invoked to ensure robust output regulation with NO behavior. The efficiency of the proposed controller is also demonstrated with the help of a numerical example and simulation results presented at the end. © 2019 European Control Association. Published by Elsevier Ltd. All rights reserved.
1. Introduction The systems in which the relationship among the state variables is governed by both differential and algebraic equations are often called as descriptor systems. Hence, for such a system, if it is LTI, the state space representation called as generalised state space representation is given by E x˙ (t ) = Ax(t ) + Bu(t ), y(t ) = Cx(t ) [6,7], where E is a singular matrix. If E is non-singular and an identity matrix, then we can call such systems as the explicit systems, and these are the systems with which we are quite familiar. As in the case of explicit systems, intensive research has been taking place in the area of continuous LTI descriptor systems, also related to observer development (see for example [11,12] etc.), controller development, etc. As the descriptor system can contain both differential and algebraic equations, compared to the explicit systems, care should be taken while designing the controller to quench the impulse behavior [6,7], they should be regular to guarantee existence and uniqueness of solution [6,7] and they should also satisfy the required admissibility conditions related to control in addition to providing usual requirements like stability, performance guaranteeing, etc. An important aspect of research related to the controller design for the descriptor system is with respect to the development of step tracking controllers that can provide good transient behavior during tracking. Designing a tracking controller which gives
∗
Corresponding author. E-mail addresses: [email protected], [email protected] (B. Bandyopadhyay).
smaller rise time along with zero overshoot is a difficult task to achieve at the same time. In [14] one can see the design of such a type of controller for MIMO descriptor systems called NO controller which was constructed using generalised Moore’s eigen structure assignment method. This NO controller will give good transient behavior, especially with zero overshoot while tracking with arbitrary small rise time. To improve the robustness of this NO controllers [14] an ISM based robust control strategy was developed in [2] for the descriptor system. For the explicit system, a robust NO controller design was proposed in [27]. A common short coming with all these control methods is that they all ensured zero overshoot only for step reference tracking. The classical output regulation [8,13,18,21] theory aims to provide internal stability of the closed loop system, disturbance rejection and asymptotic tracking for a class of target inputs and disturbances. Hence, output regulation problem aims to provide asymptotic tracking of a more general class of time-varying references like sine, cosine etc. generated by an exo-system even in presence of disturbance along with guaranteeing internal stability of the closed loop system. For MIMO state feedback based explicit systems, a controller for output regulation with NO behavior was proposed in [23] based on output regulation theory for explicit systems. But such a method that provides output regulation with NO behavior for the MIMO descriptor system is not addressed in the literature. Such a type of controller is required in many fields like robotics, aerospace etc. where the system may be modeled as a descriptor system [9,19] and the requirements will be to track a given time varying trajectory as fast as possible with no
https://doi.org/10.1016/j.ejcon.2019.08.008 0947-3580/© 2019 European Control Association. Published by Elsevier Ltd. All rights reserved.
Please cite this article as: P.S. Babu, N. Xavier and B. Bandyopadhyay, Robust output regulation for state feedback descriptor systems with nonovershooting behavior, European Journal of Control, https://doi.org/10.1016/j.ejcon.2019.08.008
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overshoot. A characteristic feature and shortcoming associated with the state feedback controller in [23] was that we need to assume that all states of the exo-system (a mathematical model for exogenous signal generator for reference and disturbance, which will be discussed in the sequel) are measurable and are available to frame the control law. Also, in such a method, one can say the designer is aware of the exact frequencies of the disturbance to be rejected. Hence, if disturbance’s whose frequencies are unknown is present in the system, then the performance of such a control system may deteriorate. More over this controller was designed for the explicit systems. If matched disturbance are to be rejected and even if the frequencies of the disturbance to be rejected is unknown, then sliding mode controller [4,25] is a good choice. In [3,24] we can see the sliding mode control being employed for output regulation in case of usual explicit systems. Also, a robust controller was proposed in [1] based on the output regulation theory for a single input single output (SISO) descriptor system [18] using sliding mode. A common shortcoming with all these control methods is that they did not ensure an NO behavior. Hence in this paper, as a first step we try to extend the results in [23] applicable for an explicit system with full state information feedback, to a descriptor system when there is no disturbance. This means that initially we show, how the state feedback controller can be designed to ensure output regulation with NO behavior for a descriptor system when there is no disturbance and this will be done using generalised Moore’s eigen structure assignment method and output regulation theory for descriptor systems. The shortcoming with this controller will be that it cannot assure an output regulation with NO behavior when a disturbance whose information is not available for feedback from the exo-system is present in the system. Hence as a next step, to retain the output regulation with NO behavior and to reject the matched disturbance whose information is not available for feedback from the exo-system, we develop a controller using ISM technique. Thus, the proposed controller developed for a MIMO state feedback descriptor system, will provide a good transient behavior with no overshoot by rejecting the matched disturbance while tracking a time varying reference generated by an exo-system, under reasonable assumptions. Also, the control law designed here will be continuous in order to satisfy the admissibility requirements in control for descriptor systems. We call this as an integral sliding mode based NO controller for robust output regulation (ISMRO). The organization of rest of the paper is as follows: Problem formulation and preliminaries are given in Section 2. Section 3 discusses in detail ISMRO development for the descriptor systems. Numerical examples along with simulation results are presented in Section 4. In Section 5 the paper is concluded.
Here, A11 , A12 , A21 , A22 , B1 , B2 , C1 and C2 are constant matrices of appropriate dimensions and xd and xa represents the differential and algebraic state variables, respectively. Assumption 2.1. 1. The system (1) (a) has rank(E ) = q < n. (b) is regular, i.e. there exists a λ ∈ C such that det (λE − A ) = 0 [7]. ¯ + , for finite λ (c) is stabilisable, i.e. rank [λE − A B] = n, ∀λ ⊂ C [7]. (d) is impulse controllable, i.e. rank
E
0 A E
0 B
= rank(E ) + n
[7]. (e) has B and C matrices with full column and row rank respectively. 2. The disturbance, d (t ) ∈ Rm in (1) is bounded, continuous with d (0 ) = 0, where d (t ) := [d1 (t ), . . . , dm (t )]T such that d˙i (t ) exist and d˙i (t ) ≤ Dmax for i ∈ {1, . . . , m}, where Dmax is a known quantity. Let the autonomous exo-system which generates the time varying reference, r(t) to be tracked be represented [1] as follows:
e :
w˙ (t ) = Sw(t ) r (t ) = yexo (t ) = L1 w(t )
(3)
where w(t ) represents the states of the exo-system and yexo (t) represents the output of the exosytem which itself is taken as r(t). Here, S, L1 are constant matrices of suitable dimensions. The entries in S, L1 and w(0 ) are so chosen to produce the required references. Without loss of generality, the eigen values of S are assumed to have non-negative real parts [18]. Let the system (1) together with the exo-system (3) be represented as follows:
⎧ ⎪ ⎨E x˙ (t ) = Ax(t ) + Bu(t ) + Bd (t ) w˙ (t ) = Sw(t ) e 1 : ⎪e(t ) := y(t ) − r (t ) = Cx(t ) − L1 w(t ) ⎩ = Cx(t ) + Q1 w(t )
(4)
where, Q1 := −L1 and e(t) denotes the error which gives the difference between y(t), and r(t) generated as an output from (3). Let, (E x0 , w0 ) represents an initial condition of (4) where, Ex0 := Ex(0) and w0 := w(0 ) represents an initial condition of (1) and (3), respectively. Definition 2.1. [14] The descriptor system (1) with d (t ) = 0 is said to provide a NO response for a constant reference, r ∈ R p from an initial condition Ex0 , if the error, e(t ) := y(t )−r tends to zero as t goes to infinity without sign change in any of its error components.
(1)
Definition 2.2. The descriptor system (1) with d (t ) = 0 together with an exosystem (3) which generates a time varying reference, r (t ) ∈ R p being represented by (4) with d (t ) = 0, is said to provide an output regulation with NO from an initial condition (E x0 , w0 ), if the error, e(t ) := y(t )−r (t ) tends to zero as t goes to infinity without sign change in any of its error components.
where x ∈ Rn , u ∈ Rm are state and control input, respectively. In (1), y ∈ R p represents the controlled output vector and d ∈ Rm represents the matched disturbance vector. E, A ∈ Rn×n , B ∈ Rn×m , C ∈ R p×n are constant matrices. We assume that system (1) is already in a form with E, A, B, C as in (2)
Definition 2.3. The descriptor system (1) together with an exosystem (3) which generates a time varying reference, r (t ) ∈ R p being represented by (4), is said to provide a robust output regulation with NO from an initial condition (E x0 , w0 ), if the error, e(t ) := y(t )−r (t ) tends to zero as t goes to infinity without sign change in any of its error components.
2. Problem formulation and preliminaries Consider a continuous MIMO, LTI descriptor system given by
E x˙ (t ) = Ax(t ) + Bu(t ) + Bd (t ) y(t ) = Cx(t )
1 :
E=
Iq 0
B=
0 A11 ,A = 0n−q A21
B1 , C = C1 B2
C2 , x =
A12 A22
xd xa
(2)
Problem Statement: Given the descriptor system (1) which satisfies Assumption 2.1 and an exo-system (3) which generates the time-varying references, r(t), which together being represented as in (4), the objective here is to design a robust state feedback controller, u(t) which is continuous using ISM technique, in order to
Please cite this article as: P.S. Babu, N. Xavier and B. Bandyopadhyay, Robust output regulation for state feedback descriptor systems with nonovershooting behavior, European Journal of Control, https://doi.org/10.1016/j.ejcon.2019.08.008
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3
ensure a robust output regulation with NO by rejecting the disturbance, d(t) from an intial condition (E x0 , w0 ).
E S = A + B
2.1. Preliminaries
2.2. Base system
In this subsection, we give a brief description about the admissibility requirements in control for descriptor systems, some preliminaries related to the output regulation theory for descriptor systems and also give details about a system called as the base system.
This subsection discusses the construction of a system called as the base system, b similar to [23]. The design of such a base system, b is needed here because in the due course it will be shown that the system (4) with d (t ) = 0, will provide an output regulation with zero overshoot, if the base system gives an NO behavior. The base system, b can be obtained from (4) by removing the exo-system which generate the reference and by making disturbance, d (t ) = 0. The state, control and error vector of the base system can be denoted with x˜(t ), u˜ (t ) and e˜(t ), respectively. Hence, the base system can be represented as follows:
2.1.1. Control admissibility requirement If system (1) is impulse free and regular, then there will exist two non-singular matrices N, M [7] such that for the state equations in (1), doing the transformation, x = Ng, where g = [gT1 gT2 ]T and then its pre-multiplication by M will result in the following form [10]:
g˙ 1 (t ) = A1 g1 (t ) + B11 u(t ) + B11 d (t ) W g˙ 2 (t ) = g2 (t ) + B21 u(t ) + B21 d (t )
(5)
where W here will be a zero matrix. In such a case, for solution of g2 to be continuous, it is required that the control and disturbance should be at least continuous [4]. By Assumption 2.1–2, the disturbance, d(t) is already continuous. Therefore, now the control should also be designed in such a way that it satisfies this admissibility requirement. 2.1.2. Output regulation theory for descriptor systems This subsection gives a brief description about the output regulation problem, relevant definitions and conditions under which output regulation problem can be solved for a state feedback descriptor system. To present these details and for further analysis, we need the following system representation which is obtained by applying a control u = F x + Gw to (4).
E x˙ (t ) = (A + BF )x(t ) + BGw(t ) + Bd (t ) cl1 : w˙ (t ) = Sw(t ) e(t ) = Cx(t ) + Q1 w(t )
(6)
For system of the form (4) and (6) when d (t ) = 0, following [18], the below given definitions can be stated: Definition 2.4. Internal stability: The system (6) is said to be internally stable, when w(t ) = 0 and d (t ) = 0 if (6) is asymptotically stable, i.e. roots of det (λE − (A + BF ))⊂ C− . For system (4), with d (t ) = 0, the problem of output regulation via full state information feedback, u = F x+Gw aims at finding gain matrices F and G such that for the resulting system (6) with d (t ) = 0, the roots of det (λE − (A + BF ))⊂ C− (for internal stability) and Lim e(t ) = 0 for any initial condition [18].
t→∞
Definition 2.5. Solvability: For system (4) with d (t ) = 0, the problem of output regulation via full state information feedback, u = F x + Gw is said to be solvable if there exist gain matrices F and G such that for the resulting system (6) with d (t ) = 0, the roots of det (λE − (A + BF ))⊂ C− (for internal stability) and Lim e(t ) = 0 for t→∞
any initial condition. Following [18], one can say that for the system (4) with d (t ) = 0, the problem of output regulation via full state information feedback, u = F x + Gw is solvable if and only if there exist two matrices and such that it satisfies the following equations (regulator equations) given by (7), for which G is to be chosen as G = − F .
0 = C + Q1
˙ b : E x˜(t ) = Ax˜(t ) + Bu˜ (t ) e˜(t ) = C x˜(t )
(7)
(8)
Let E x˜0 := E x˜(0 ) represents an initial condition of (8). Let (8), satisfies Assumption 2.1. If we apply u˜ = F x˜(t ), then, it becomes
E x˜˙ (t ) = (A + BF )x˜(t ), e˜(t ) = C x˜(t )
(9)
As per Assumption 2.1-1(c), (8) is stabilizable which in turn implies the existence of an F so that (E, A + BF ) is asymptotically stable. Therefore, as t → ∞, x˜(t ) → 0 and hence e˜(t ) → 0. 2.2.1. Non-overshooting tracking controller for the base system As per Assumption 2.1-1(c), the base system (8) is stabilisable and more over as per Assumption 2.1-1(d), the system is impulse controllable as well. Hence, because of these two assumptions, there will exist a gain feedback matrix, F such that the on applying, u˜ = F x˜(t ) to (8), the resulting closed loop system, (9) satisfies the following two properties: (a) roots of det (λE − (A + BF ))⊂ C− – which will guarantee asymptotic stability of (9) and (b) deg(det (λE − (A + BF ))) = rank(E ) – which will guarantee impulse free behavior in the responses of (9)[7,14]. Selecting a gain feedback matrix, F which satisfies only these two properties are not enough to satisfy the current design requirements. This is because, more importantly here while applying u˜ = F x˜(t ) to (8) e˜(t ) → 0 as t → ∞ with NO behavior from an initial condition E x˜0 . Therefore, to get these desired characteristics, the design of F has to be carried out in a particular way. In [14], we can see the details for constructing such an F which make use of Moore’s [20] eigen value and vector assignment method, generalised for descriptor systems so as to get a NO behavior. Remark 2.1. For system (1) we assumed that the E, A, B, C matrices have a form as in (2). Then, one can use the methods available in [14] to construct an F that can assure NO behavior for (8) from an initial condition E x˜0 without any additional co-ordinate transformation. If (1) is not in a form with matrices as in (2) then one can always find two non-singular matrices M1 and N1 [7,14] such that doing the transformation, x = N1 [xTd xTa ]T and pre-multiplication of the state equation by M1 will result in a system with matrices having a similar form as in (2) and can continue with the design procedure. 3. ISM based robust output regulation strategy with NO The main results of the paper consisting of the design of ISMRO that can solve our problem statement, by providing robust output regulation with NO by rejecting the matched disturbance entering the system, is discussed in this section. Here, in our design compared to the output regulation strategy in [23] for full state feedback, no exo-system component will generate the disturbance and hence we do not have any disturbance
Please cite this article as: P.S. Babu, N. Xavier and B. Bandyopadhyay, Robust output regulation for state feedback descriptor systems with nonovershooting behavior, European Journal of Control, https://doi.org/10.1016/j.ejcon.2019.08.008
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related information available for measurement and feedback from the exo-system. This is very clear from the exo-system dynamics given by (3) in which the exo-system is generating only the reference signal, r(t). Hence, with the so chosen exo-system, if we select u = F x(t ) + Gw(t ) as the control law to solve our problem statement, then closed loop system may provide an exact tracking without overshoot only in the absence of matched disturbance. It may fail to reject the disturbance once the disturbance is present. Hence, to reject the matched disturbance in such a case we utilise the sliding mode control (SMC) part of the integral sliding mode controller. The combined system and exo-system representation is given by (4) for which we need to design the controller. Motivated from [23], for designing this controller and to state and prove Lemma 3.1, first we design a feedback matrix, F using Moore’s algorithm, generalised for descriptor system described in [14]. Then, if such an F exists, apply u˜ = F x˜(t ) to (8). If it results in assymptotic convergence of e˜(t ) to zero with NO from E x˜0 , then use this F to frame an NO control law, denoted by uNOISM = F x(t ) + Gw(t ),where G = − F for the system (4), for which we need to attain the robust output regulation without any overshoot. The following lemma help us to arrive at a condition on d(t) under which the system (4) will achieve output regulation without any overshoot. Lemma 3.1. Consider a continuous MIMO LTI descriptor system (1) which satisfies Assumption 2.1. Let the system (1) along with the exo-system generator (3) can together be represented as in (4) with an initial condition (Ex0 , w0 ). Let the eigen values of S matrix of exo-system generator (3) has non-negative real parts and there is and which satisfies (7). Also let d (t ) = 0 and there is an F such that, on applying u˜ = F x˜(t ) to (8), with the initial condition E x˜0 = E (x0 − w0 ), will result in asymptotic convergence of e˜(t ) to zero with an NO behavior. Then, for this F and G = − F , the control law uNOISM = F x(t ) + Gw(t ) applied to (4) will provide output regulation with an NO behavior from (Ex0 , w0 ). Proof. On applying u˜ = F x˜(t ) to (8) with an initial condition E x˜0 = E (x0 − w0 ) will result in (9). By assumption, there exist such an F, so that Limt→∞ e˜(t ) = 0 without sign change in any of its error components. Now using this F, applying u = uNOISM = F x(t ) + Gw(t ) to (4) will result in (6). Now, from (6), if σ (t ) = x(t ) − w(t ) and E σ (0 ) = E (x(0 ) − w(0 )) = E x˜0 , using G = − F we can obtain
E (x˙ (t ) − w˙ (t ) ) = (A + BF )x(t ) + (BG )w(t ) + Bd (t ) − E Sw(t ) E σ˙ (t ) = (A + BF )x(t ) + (B( − F ))w(t ) + Bd (t ) − E Sw(t ) = (A + BF )x(t ) + (B − E S )w(t ) + Bd (t ) − BF w(t ) (10) Using the fact that d (t ) = 0 as assumed and from (7), the expression for (10) becomes
E σ˙ (t ) = (A + BF )x(t ) − Aw(t ) − BF w(t ) (11)
As for the error, e(t) we have from (6)
e(t ) = Cx(t ) + Q1 w(t ) = Cx(t ) + Q1 w(t ) + C w(t ) − C w(t )
(12)
Now, rearranging the terms in (12) and from (7) we can obtain
e(t ) = C (x(t ) − w(t )) + (Q1 + C )w(t ) = C (x(t ) − w(t ))
e(t ) = C σ (t )
(14)
On comparing (14) with (9) we can see that both the systems are identical. Hence, Limt→∞ e(t ) = 0 happens without any sign change in any of its error components from (Ex0 , w0 ), because for (9) as per the assumption we already know the fact that Limt→∞ e˜(t ) = 0 happens without sign change in any of its error components. From Lemma 3.1, we were able to see that the output regulation with NO will happen for (4) by applying uNOISM alone, only when d (t ) = 0. Hence, through Lemma 3.1, using generalised Moore’s eigen structure assignment method and output regulation theory for descriptor systems, we extended the results in [23] applicable for an explicit system with full state information feedback, to a descriptor system, when d (t ) = 0. This means that, we had shown when there is no disturbance, how the state feedback controller can be designed to ensure output regulation with NO behavior for a descriptor system. On the other hand, if d(t), whose information is not available for feedback from the exo-system is present, selecting the control law as u = uNOISM cannot reject d(t) and hence they cannot ensure robust output regulation with NO behavior. Hence, here we are employing an ISM control which will handle the disturbance rejection aspect also. Like in any other ISM design [4,17,26], the main idea here is to design an integral sliding surface variable and a controller based on this sliding surface variable. 3.1. Integral sliding surface variable For the given MIMO descriptor system (1) with u ∈ Rm , the integral sliding surface variable, s(t ) := [s1 (t ), . . . , sm (t )]T ∈ Rm is designed as follows:
s(t ) = B+ (E x(t )−E x(0 ) −
0
t
(Ax(τ ) + BuNOISM (τ ))dτ )
(13)
As σ (t ) = x(t ) − w(t ), (11) and (13) can together be written as
(15)
where E, A ∈ Rn×n , B ∈ Rn×m are constant matrices as in (1). The matrix B+ ∈ Rm×n represents the pseudo-inverse of B. As per Assumption 2.1-1(e), B has full column rank and hence B+ exist. In (15), the term uNOISM is computed as uNOISM = F x + Gw, where the gain matrix F and G are obtained using Lemma 3.1. An integral sliding surface can be considered as a set of state variables, x(t) for which s(t) becomes zero. Here, the integral sliding surface variable, s(t) in (15) is designed in such a way that, it will become zero from initial time, t = 0 as in [2]. Hence, the state trajectories will start from the integral sliding surface. 3.2. Controller design For (1), the governing law for the control is chosen to be
uact = u = uNOISM + uSMC
= (A + BF )x(t ) − (A + BF )w(t ) = (A + BF )(x(t ) − w(t ))
E σ˙ (t ) = (A + BF )σ (t )
(16)
Here, uNOISM = F x(t ) + Gw(t ) will handle the asymptotic tracking along with NO characteristics of the system (1) whereas uSMC that represents the sliding mode control law, is to be designed such that it should reject the matched disturbance. As per the discussion in Section 2.1, the control needs to be at least continuous to guarantee continuity in the solutions for a regular impulse free descriptor system. Hence, we are selecting, as an uSMC , super twisting sliding mode control [5,16,22], uSTC which is continuous and which needs only the information on sliding surface variable, s(t) for its implementation as follows:
uSMC = uST C = −κ1 |s(t )|1/2 sgn(s(t )) + ω (t )
(17)
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where,
⎡
κ1 |s(t )| sgn(s(t )) := ⎣
k11 |s1 (t )|
1/2
1/2
k1m |sm (t )| and
⎡
.. .
sgn(s1 (t ))
1/2
⎤ ⎦
(18)
sgn(sm (t ))
⎤
−k21 sgn(s1 (t )) .. ⎦ ω˙ (t ) = −κ2 sgn(s(t )) := ⎣ . −k2m sgn(sm (t ))
Theorem 3.1. Consider a continuous MIMO LTI descriptor system (1) which satisfies Assumption 2.1. Let the system (1) along with the exo-system generator (3) can together be represented as in (4) with an initial condition (Ex0 , w0 ). Also let the eigen values of S matrix of exo-system generator (3) has non-negative real parts and there is and which satisfies (7). Assume that there is an F such that, on applying u˜ = F x˜(t ) to (8), with the initial condition E x˜0 = E (x0 − w0 ), will result in asymptotic convergence of e˜(t ) to zero with an NO behavior. Then, for this F and G = − F , the control law u = F x(t ) + Gw(t ) + uSMC = uNOISM + uSMC applied to (4) will provide a robust output regulation with NO from (Ex0 , w0 ) by rejecting the disturbance if sliding surface variable, s (t) is chosen as in (15) and uSMC is chosen as (17) with k1i = k1 >1.8 k2 + Dmax and k2i = k2 >Dmax for i ∈ {1, . . . , m} and ω (0 ) = 0. Proof. Let the sliding surface variable, s(t) be chosen as in (15) and uSMC as in (17) with k1i = k1 >1.8 k2 + Dmax and k2i = k2 >Dmax for i ∈ {1, . . . , m} and ω (0 ) = 0. By differentiating (15) we get
s˙ (t ) = B+ (Ax + Bu + Bd (t ) − Ax − BuNOISM ) s˙ (t ) = d (t ) + uSMC = d (t ) + −κ1 |s(t )|1/2 sgn(s(t )) + ω (t ) (t)]T ,
(20)
then (20) be-
Consider system (1) as a dc motor actuated pendulum [10], with E, A, B matrices as follows, which is obtained after substituting the values of parameters given in [10]
⎡
0 ⎢−47.5521 A=⎣ 0 0
1.0 0 0 0 −0.2424 0 0.0760
1 ⎢0 E=⎣ 0 0
0 0⎥ 0⎦ 0
⎡
0 1 0 0
B= 0
0
Remembering the fact that, κ1 := diag(k11 , . . . , k1m ), κ2 := diag(k21 , . . . , k2m ), d˙ (t ) := [d˙1 (t ), . . . , d˙m (t )]T and d˙i (t ) ≤ Dmax As per [15], if we choose, k1i = k1 > 1.8 k2 + Dmax and k2i = k2 >Dmax for i ∈ {1, . . . , m}, each pair of (si (t ), s˙ i (t )) for i ∈ {1, . . . , m} converges to zero in finite time. As, all (si (t ), s˙ i (t )) pair converges in finite time to zero, we can say in finite time (s(t ), s˙ (t )) converges to zero and sliding mode happens. Hence, during sliding mode, from (20), we can write, uSMC = −d (t ). Thus, the uSMC will be equal to the negative of the disturbance and hence uSMC will completely reject the disturbance from system (1). As ω(0) and d(0) are assumed to be zero, using (15) and (20), we can say that both s(t ), s˙ (t ) will be zero from the very beginning. Once the sliding mode happens, the disturbance rejection will take place and the resulting system will be similar to (1) without disturbance and sliding mode control. Similarly, in (6) there will not be any disturbance term. Hence, we have a (6) with no disturbance and hence, by Lemma 3.1, Limt→∞ e(t ) = 0 happens without any sign change in any of its error components from (Ex0 , w0 ). Hence, the control, u will provide a robust NO output regulation. Remark 3.1. If S is selected as a zero matrix in (3) with w(t ) ∈ R p and L1 = I and with the elements of w(0 ) ∈ R p being chosen as the
T
⎤
0 10.8095 0 0.10 0 0
1 ,
0
C= 1
0
0 0 ⎥ , 10.8696⎦ 1.0 0 0 0
0
0
(22)
Let the angular position (q), angular velocity(q˙ ), armature current (Ia ) and inductor voltage(VL ) be the state variables x1 , x2 , x3 and x4 respectively. Here, we are only interested in the tracking of angular position (q) and √ let the required reference profile for tracking √ be, r (t ) = −0.8 ∗ ((8 2/π 2 )sin(π t ) + (8 2/(9π 2 ))sin(3π t )) from an initial condition Ex0 = [−5 0.2 0.1 0]T . Let the matched disturbance be d (t ) = [0.5sin(π t )]T . The given system with matrices ¯ + , for as in (22) is stabilisable as rank([λE − A B] ) = 4 = n, ∀λ ⊂ C finite λ and regular as det (λE − A ) = 0 and impulse controllable as rank
E
A
0 E
0 B
= 7 = rank(E ) + n[7].
For the exo-system √(3), in order to √ generate the reference, r (t ) = −0.8 ∗ ((8 2/π 2 )sin(π t ) + (8 2/(9π 2 ))sin(3π t )), choose L1 = [−1 0 − 1 0], S and w(0 ) in (3) as follows:
S=
S1 0
0 w1 ( 0 ) , w (0 ) = S2 w2 ( 0 )
(21)
⎤
0 0 1 0
where,
s˙ (t ) = −κ1 |s(t )|1/2 sgn(s(t )) + z(t ) z˙ (t ) = −κ2 sgn(s(t )) + d˙ (t )
required magnitude of the reference step signals to be tracked and if there is a and that satisfies (7), then, this ISMRO scheme may be used for step tracking as well. In such a case the ISMRO scheme implemented here may be utilized as an alternate option for the controller developed in [2], where in which controller was designed to track a step reference signal. 4. Numerical example and simulation results
(19)
where, |s(t )|1/2 sgn(s(t )) := [|s1 (t )|1/2 sgn(s1 (t )), . . . , |sm (t )|1/2 sgn(sm (t ))]T where each si ∈R and i∈{1, . . . , m}, ω (t ) := [ω1 (t ), . . . , ωm (t )]T and κ1 := diag(k11 , . . . , k1m ), κ2 := diag(k21 , . . . , k2m ) with k1i , k2i for i ∈ {1, . . . , m} are the controller parameters to be designed.
Let, ω (t ) + d (t ) = z(t ), where z(t) := [z1 (t) z2 comes
5
√ 1 , w1 (0 ) = [0 (8 2/π 2 )0.8π ]T 0
0 S1 = − (π 2 )
0 S2 = −9π 2
(23)
√ 1 , w2 (0 ) = [0 (8 2/(9π 2 ))0.8 ∗ 3π ]T 0
Now, solving (7) for and we can get
⎡
−1.0 0 0 0 ⎢ 0 =⎣ −3.4861 0.0204
= 0.3282
0 −1.0 0 0 0 −0.0224 −0.3207
0.3990
−1.0 0 0 0 0 3.8183 0.1832
−0.5651
(24)
⎤
0 −1.0 0 0 0⎥ , −0.0224 ⎦ 0.3513
−0.2730
(25)
Now, the next step is to design F. Before designing an ISMRO, first of all we need to ensure that, with this F, when u = F x˜ is applied to the base system, (8) with E, A, B, C matrices as given by (22) will provide a NO response for e˜, from an initial condition, E x˜0 = E (x0 − w0 ). To design F, first we need to find the number of stable invariant zeros present in the base system, which will be same as the number of invariant zeros present in the actual system as E, A, B, C matrices for both are the same. From (22), the number of invariant zeros can be found to be zero. i.e. we say we have (q − 3 ∗ p) invariant zeros present in the system (remember, q = rank(E ) = 3 and no. of outputs, p = 1) because q − 3 p = 3 − 3 ∗ 1 = 0. Hence, for the design we can use the methodology
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Fig. 1. (a) Plot of e˜ response of base system (b) Tracking responses with ISMRO (c) Plot of sliding surface variable for ISMRO (d) Plot of tracking error under various conditions.
given in [2,14] for a q − l p case, where l = 3 here. As the system has no invariant zeros, we can arbitrarily place 3 poles which are distinct, real and stable and let this be at λ1 = −10, λ2 = −8 and λ3 = −6. Now, using Moore’s algorithm for generalized system [14], F can be obtained as
F = −5.5306
1.0696
2.0852
0
(26)
The design of a state feedback controller with this F will give an impulse free response as it satisfies deg(det (λE − (A + BF ))) = 3 = rank(E ) [7]. Fig. 1(a) shows that we will get an NO response for e˜, when we apply u = F x˜ to b with F as in (26). ISMRO Design. For the ISMRO design we need to design both uSMC and uNOISM . Here, we have a matched disturbance, d (t ) = [0.5sin(π t )]T . Now, for uSMC in (16), choose k11 = 10, k21 = 2 and ω (0 ) = 0. For designing uNOISM = F x(t ) + Gw(t ), F can be chosen as in (26). Now, with this F and , as in (25), by solving G = − F gives G as follows:
G = 2.0669
1.5153
−14.0577
0.8433
(27)
Fig. 1(b) shows the responses of y(t ) = q(t ) tracking the given reference profile using ISMRO design after rejecting the matched disturbance without any overshoot, which can be confirmed from the error response, e(t ) = y(t ) − r (t ) plot. Thus robust output regulation with NO is achieved for the given system (22). Fig. 1(c) gives the response of sliding surface variable and it assures that the sliding mode is happening from the very beginning itself. Fig. 1(d) shows the response of : (1) tracking error for the closed loop system with ISMRO in presence of disturbance, d(t) where the control is given by (16) (2) tracking error for the closed loop system with NO controller acting alone where the control is given by u = uNOISM = F x(t ) + Gw(t ) but with no d(t) in the system (3)tracking error for the closed loop system with u = uNOISM = F x(t ) + Gw(t ) alone, but with d(t) in the system (22). Hence, it is very evident from the plot that tracking performance of the system
(22) with the proposed ISMRO in presence of d(t) and that of NO controller without d(t) are one and the same. But in the absence of sliding mode controller, when the system (22) is controlled with NO controller alone in presence of d(t) the tracking response is becoming poor and hence robust output regulation with NO cannot be guaranteed. 5. Conclusion This paper presented the design of a robust output regulation control strategy with NO for the MIMO descriptor systems. The design strategy helped to track a time-varying reference generated by an exo-system by rejecting the matched disturbances even if their frequencies are unknown by employing an ISM based technique. Simulation results confirmed the effectiveness of the proposed method. As we have designed a sliding mode based control which is continuous, this will alleviate the harmful chattering effects of the actuator while implementation. Moreover in our simulation to obtain the feedback matrix F we have placed the stable and distinct poles arbitrarily on the real axis. By placing these stable and distinct poles farther from the imaginary axis we may be able to get a faster decay of error response to zero but the price we need to pay may be higher control effort. The control scheme discussed here is based on state feedback based technique. Hence, as a scope for future work one can try designing a robust NO controller using an output feedback technique. One can also try extending the idea of implementing the robust NO output regulation scheme for discrete-time systems. References [1] P.S. Babu, B. Bandyopadhyay, M. Thomas, Robust composite non-linear feedback control for descriptor systems with general reference tracking, in: Proceedings of the 44th Annual Conference of the IEEE Industrial Electronics Society, IECON, 2018.
Please cite this article as: P.S. Babu, N. Xavier and B. Bandyopadhyay, Robust output regulation for state feedback descriptor systems with nonovershooting behavior, European Journal of Control, https://doi.org/10.1016/j.ejcon.2019.08.008
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P.S. Babu, N. Xavier and B. Bandyopadhyay / European Journal of Control xxx (xxxx) xxx [2] P.S. Babu, N. Xavier, B. Bandyopadhyay, Robust non-overshooting tracking controller for descriptor systems, Eur. J. Control 43 (2018) 57–63. [3] C. Bonivento, L. Marconi, Z. Roberto, On the output regulation of nonlinear systems by sliding mode, IFAC Proc. Vol. 32 (2) (1999) 2719–2724. [4] F. Castaños, D. Hernández, L. Fridman, Integral sliding-mode control for linear time-invariant implicit systems, Automatica 50 (2014) 971–975. [5] A. Chalanga, S. Kamal, B. Bandyopadhyay, A new algorithm for continuous sliding mode control with implementation to industrial emulator setup, IEEE/ASME Trans. Mech. 20 (2015) 2194–2204. [6] L. Dai, Singular Control Systems, Springer Verlag, Berlin, 1989. [7] G.R. Duan, Analysis and Design of Descriptor Linear Systems, Springer, NewYork, 2010. [8] B.A. Francis, W.M. Wonham, The internal model principle for linear multivariable regulators, Appl. Math. Optim. 2 (2) (1975) 170–194. [9] H. Halalchi, E. Laroche, G.I. Bara, LPV modeling and control of a 2-dof robotic manipulator based on descriptor representation, in: Pan-American Congress of Applied Mechanics, Foz do Iguaçu, Brazil, 2010. [10] D. Hernández, L. Fridman, M. Golkani, Continuous sliding-mode control for singular systems, Int. J. Robust Nonlinear Control 28 (10) (2018) 3454–3474. [11] M. Hou, P. Muller, Observer design for descriptor systems, IEEE Trans. Autom. Control 44 (1) (1999) 164–169. [12] M. Hou, P. Muller, Design of a class of Luenberger observers for descriptor systems, IEEE Trans. Autom. Control 40 (1) (1995) 133–136. [13] J. Huang, Nonlinear Output Regulation: Theory and Applications, 8, SIAM, 2004. [14] C. Kazantzidou, R. Schmid, L. Ntogramatzidis, Nonovershooting state feedback and dynamic output feedback tracking controllers for descriptor systems, Int. J. Control 91 (8) (2018) 1785–1800. [15] P. Kumar, A.K. Behera, B. Bandyopadhyay, Robust finite-time tracking of stewart platform: a super-twisting like observer-based forward kinematics solution, IEEE Trans. Ind. Electron. 64 (5) (2017) 3776–3785. [16] A. Levant, Sliding order and sliding accuracy in sliding mode control, Int. J. Control 58 (1993) 1247–1263.
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[17] J. Li, Q. Zhang, X.-G. Yan, S.K. Spurgeon, Integral sliding mode control for Markovian jump T–S fuzzy descriptor systems based on the super-twisting algorithm, IET Control Theory Appl. 11 (8) (2017) 1134–1143. [18] W. Lin, L. Dai, Solutions to the output regulation problem of linear singular systems, Automatica 32 (12) (1996) 1713–1718. [19] I. Masubuchi, J. Kato, M. Saeki, A. Ohara, Gain-scheduled controller design based on descriptor representation of LPV systems: application to flight vehicle control, in: Proceedings of the 43rd IEEE Conference on Decision and Control, 1, 2004, pp. 815–820. [20] B.C. Moore, On the flexibility offered by state feedback in multivariable systems beyond closed loop eigenvalue assignment, IEEE Trans. Autom. Control 21 (5) (1976) 689–692. [21] A. Saberi, A.A. Stoorvogel, P. Sannuti, Control of Linear Systems with Regulation and Input Constraints, Springer Science & Business Media, 2012. [22] S. Sadala, B. Patre, A new continuous sliding mode control approach with actuator saturation for control of 2-dof helicopter system, ISA Trans. 74 (2018) 165–174. [23] R. Schmid, L. Ntogramatzidis, Nonovershooting and nonundershooting exact output regulation, Syst. Control Lett. 70 (2014) 30–37. [24] F. Shiravani, M. Shafiei, Robust output regulation via sliding mode control and disturbance observer: application in a forced van der pol chaotic oscillator, J. Dyn. Syst. Measur. Control 139 (9) (2017) 091015. [25] V.I. Utkin, Sliding mode control design principles and applications to electric drives, IEEE Trans. Ind. Electr. 40 (1) (1993) 23–36. [26] V. Utkin, J. Shi, Integral sliding mode in systems operating under uncertainty conditions, in: Proceedings of the 35th IEEE Conference on Decision and Control, 1996, pp. 4591–4596. [27] N. Xavier, B. Bandyopadhyay, R. Schmid, Robust non-overshooting tracking using continuous control for linear multivariable systems, IET Control Theory Appl. 12 (7) (2018) 1006–1011.
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Contents lists available at ScienceDirect
European Journal of Control journal homepage: www.elsevier.com/locate/ejcon
Robust output regulation for state feedback descriptor systems with nonovershooting behavior Praveen S Babu, Nithin Xavier, Bijnan Bandyopadhyay∗ Systems and Control Engineering, Indian Institute of Technology Bombay, Mumbai 400 076, India
a r t i c l e
i n f o
Article history: Received 3 November 2018 Revised 4 June 2019 Accepted 30 August 2019 Available online xxx Recommended by Eduardo Costa Keywords: Descriptor systems Output regulation Robust control Tracking
a b s t r a c t In this paper, the design of robust non-overshooting (NO) controllers for output regulation is presented for a continuous linear time-invariant (LTI), multi-input multi-output (MIMO) state feedback based descriptor system. In this method of controller design for descriptor system, the integral sliding mode (ISM) technique along with generalised Moore’s eigen structure assignment method and output regulation theory for descriptor system is invoked to ensure robust output regulation with NO behavior. The efficiency of the proposed controller is also demonstrated with the help of a numerical example and simulation results presented at the end. © 2019 European Control Association. Published by Elsevier Ltd. All rights reserved.
1. Introduction The systems in which the relationship among the state variables is governed by both differential and algebraic equations are often called as descriptor systems. Hence, for such a system, if it is LTI, the state space representation called as generalised state space representation is given by E x˙ (t ) = Ax(t ) + Bu(t ), y(t ) = Cx(t ) [6,7], where E is a singular matrix. If E is non-singular and an identity matrix, then we can call such systems as the explicit systems, and these are the systems with which we are quite familiar. As in the case of explicit systems, intensive research has been taking place in the area of continuous LTI descriptor systems, also related to observer development (see for example [11,12] etc.), controller development, etc. As the descriptor system can contain both differential and algebraic equations, compared to the explicit systems, care should be taken while designing the controller to quench the impulse behavior [6,7], they should be regular to guarantee existence and uniqueness of solution [6,7] and they should also satisfy the required admissibility conditions related to control in addition to providing usual requirements like stability, performance guaranteeing, etc. An important aspect of research related to the controller design for the descriptor system is with respect to the development of step tracking controllers that can provide good transient behavior during tracking. Designing a tracking controller which gives
∗
Corresponding author. E-mail addresses: [email protected], [email protected] (B. Bandyopadhyay).
smaller rise time along with zero overshoot is a difficult task to achieve at the same time. In [14] one can see the design of such a type of controller for MIMO descriptor systems called NO controller which was constructed using generalised Moore’s eigen structure assignment method. This NO controller will give good transient behavior, especially with zero overshoot while tracking with arbitrary small rise time. To improve the robustness of this NO controllers [14] an ISM based robust control strategy was developed in [2] for the descriptor system. For the explicit system, a robust NO controller design was proposed in [27]. A common short coming with all these control methods is that they all ensured zero overshoot only for step reference tracking. The classical output regulation [8,13,18,21] theory aims to provide internal stability of the closed loop system, disturbance rejection and asymptotic tracking for a class of target inputs and disturbances. Hence, output regulation problem aims to provide asymptotic tracking of a more general class of time-varying references like sine, cosine etc. generated by an exo-system even in presence of disturbance along with guaranteeing internal stability of the closed loop system. For MIMO state feedback based explicit systems, a controller for output regulation with NO behavior was proposed in [23] based on output regulation theory for explicit systems. But such a method that provides output regulation with NO behavior for the MIMO descriptor system is not addressed in the literature. Such a type of controller is required in many fields like robotics, aerospace etc. where the system may be modeled as a descriptor system [9,19] and the requirements will be to track a given time varying trajectory as fast as possible with no
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overshoot. A characteristic feature and shortcoming associated with the state feedback controller in [23] was that we need to assume that all states of the exo-system (a mathematical model for exogenous signal generator for reference and disturbance, which will be discussed in the sequel) are measurable and are available to frame the control law. Also, in such a method, one can say the designer is aware of the exact frequencies of the disturbance to be rejected. Hence, if disturbance’s whose frequencies are unknown is present in the system, then the performance of such a control system may deteriorate. More over this controller was designed for the explicit systems. If matched disturbance are to be rejected and even if the frequencies of the disturbance to be rejected is unknown, then sliding mode controller [4,25] is a good choice. In [3,24] we can see the sliding mode control being employed for output regulation in case of usual explicit systems. Also, a robust controller was proposed in [1] based on the output regulation theory for a single input single output (SISO) descriptor system [18] using sliding mode. A common shortcoming with all these control methods is that they did not ensure an NO behavior. Hence in this paper, as a first step we try to extend the results in [23] applicable for an explicit system with full state information feedback, to a descriptor system when there is no disturbance. This means that initially we show, how the state feedback controller can be designed to ensure output regulation with NO behavior for a descriptor system when there is no disturbance and this will be done using generalised Moore’s eigen structure assignment method and output regulation theory for descriptor systems. The shortcoming with this controller will be that it cannot assure an output regulation with NO behavior when a disturbance whose information is not available for feedback from the exo-system is present in the system. Hence as a next step, to retain the output regulation with NO behavior and to reject the matched disturbance whose information is not available for feedback from the exo-system, we develop a controller using ISM technique. Thus, the proposed controller developed for a MIMO state feedback descriptor system, will provide a good transient behavior with no overshoot by rejecting the matched disturbance while tracking a time varying reference generated by an exo-system, under reasonable assumptions. Also, the control law designed here will be continuous in order to satisfy the admissibility requirements in control for descriptor systems. We call this as an integral sliding mode based NO controller for robust output regulation (ISMRO). The organization of rest of the paper is as follows: Problem formulation and preliminaries are given in Section 2. Section 3 discusses in detail ISMRO development for the descriptor systems. Numerical examples along with simulation results are presented in Section 4. In Section 5 the paper is concluded.
Here, A11 , A12 , A21 , A22 , B1 , B2 , C1 and C2 are constant matrices of appropriate dimensions and xd and xa represents the differential and algebraic state variables, respectively. Assumption 2.1. 1. The system (1) (a) has rank(E ) = q < n. (b) is regular, i.e. there exists a λ ∈ C such that det (λE − A ) = 0 [7]. ¯ + , for finite λ (c) is stabilisable, i.e. rank [λE − A B] = n, ∀λ ⊂ C [7]. (d) is impulse controllable, i.e. rank
E
0 A E
0 B
= rank(E ) + n
[7]. (e) has B and C matrices with full column and row rank respectively. 2. The disturbance, d (t ) ∈ Rm in (1) is bounded, continuous with d (0 ) = 0, where d (t ) := [d1 (t ), . . . , dm (t )]T such that d˙i (t ) exist and d˙i (t ) ≤ Dmax for i ∈ {1, . . . , m}, where Dmax is a known quantity. Let the autonomous exo-system which generates the time varying reference, r(t) to be tracked be represented [1] as follows:
e :
w˙ (t ) = Sw(t ) r (t ) = yexo (t ) = L1 w(t )
(3)
where w(t ) represents the states of the exo-system and yexo (t) represents the output of the exosytem which itself is taken as r(t). Here, S, L1 are constant matrices of suitable dimensions. The entries in S, L1 and w(0 ) are so chosen to produce the required references. Without loss of generality, the eigen values of S are assumed to have non-negative real parts [18]. Let the system (1) together with the exo-system (3) be represented as follows:
⎧ ⎪ ⎨E x˙ (t ) = Ax(t ) + Bu(t ) + Bd (t ) w˙ (t ) = Sw(t ) e 1 : ⎪e(t ) := y(t ) − r (t ) = Cx(t ) − L1 w(t ) ⎩ = Cx(t ) + Q1 w(t )
(4)
where, Q1 := −L1 and e(t) denotes the error which gives the difference between y(t), and r(t) generated as an output from (3). Let, (E x0 , w0 ) represents an initial condition of (4) where, Ex0 := Ex(0) and w0 := w(0 ) represents an initial condition of (1) and (3), respectively. Definition 2.1. [14] The descriptor system (1) with d (t ) = 0 is said to provide a NO response for a constant reference, r ∈ R p from an initial condition Ex0 , if the error, e(t ) := y(t )−r tends to zero as t goes to infinity without sign change in any of its error components.
(1)
Definition 2.2. The descriptor system (1) with d (t ) = 0 together with an exosystem (3) which generates a time varying reference, r (t ) ∈ R p being represented by (4) with d (t ) = 0, is said to provide an output regulation with NO from an initial condition (E x0 , w0 ), if the error, e(t ) := y(t )−r (t ) tends to zero as t goes to infinity without sign change in any of its error components.
where x ∈ Rn , u ∈ Rm are state and control input, respectively. In (1), y ∈ R p represents the controlled output vector and d ∈ Rm represents the matched disturbance vector. E, A ∈ Rn×n , B ∈ Rn×m , C ∈ R p×n are constant matrices. We assume that system (1) is already in a form with E, A, B, C as in (2)
Definition 2.3. The descriptor system (1) together with an exosystem (3) which generates a time varying reference, r (t ) ∈ R p being represented by (4), is said to provide a robust output regulation with NO from an initial condition (E x0 , w0 ), if the error, e(t ) := y(t )−r (t ) tends to zero as t goes to infinity without sign change in any of its error components.
2. Problem formulation and preliminaries Consider a continuous MIMO, LTI descriptor system given by
E x˙ (t ) = Ax(t ) + Bu(t ) + Bd (t ) y(t ) = Cx(t )
1 :
E=
Iq 0
B=
0 A11 ,A = 0n−q A21
B1 , C = C1 B2
C2 , x =
A12 A22
xd xa
(2)
Problem Statement: Given the descriptor system (1) which satisfies Assumption 2.1 and an exo-system (3) which generates the time-varying references, r(t), which together being represented as in (4), the objective here is to design a robust state feedback controller, u(t) which is continuous using ISM technique, in order to
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ensure a robust output regulation with NO by rejecting the disturbance, d(t) from an intial condition (E x0 , w0 ).
E S = A + B
2.1. Preliminaries
2.2. Base system
In this subsection, we give a brief description about the admissibility requirements in control for descriptor systems, some preliminaries related to the output regulation theory for descriptor systems and also give details about a system called as the base system.
This subsection discusses the construction of a system called as the base system, b similar to [23]. The design of such a base system, b is needed here because in the due course it will be shown that the system (4) with d (t ) = 0, will provide an output regulation with zero overshoot, if the base system gives an NO behavior. The base system, b can be obtained from (4) by removing the exo-system which generate the reference and by making disturbance, d (t ) = 0. The state, control and error vector of the base system can be denoted with x˜(t ), u˜ (t ) and e˜(t ), respectively. Hence, the base system can be represented as follows:
2.1.1. Control admissibility requirement If system (1) is impulse free and regular, then there will exist two non-singular matrices N, M [7] such that for the state equations in (1), doing the transformation, x = Ng, where g = [gT1 gT2 ]T and then its pre-multiplication by M will result in the following form [10]:
g˙ 1 (t ) = A1 g1 (t ) + B11 u(t ) + B11 d (t ) W g˙ 2 (t ) = g2 (t ) + B21 u(t ) + B21 d (t )
(5)
where W here will be a zero matrix. In such a case, for solution of g2 to be continuous, it is required that the control and disturbance should be at least continuous [4]. By Assumption 2.1–2, the disturbance, d(t) is already continuous. Therefore, now the control should also be designed in such a way that it satisfies this admissibility requirement. 2.1.2. Output regulation theory for descriptor systems This subsection gives a brief description about the output regulation problem, relevant definitions and conditions under which output regulation problem can be solved for a state feedback descriptor system. To present these details and for further analysis, we need the following system representation which is obtained by applying a control u = F x + Gw to (4).
E x˙ (t ) = (A + BF )x(t ) + BGw(t ) + Bd (t ) cl1 : w˙ (t ) = Sw(t ) e(t ) = Cx(t ) + Q1 w(t )
(6)
For system of the form (4) and (6) when d (t ) = 0, following [18], the below given definitions can be stated: Definition 2.4. Internal stability: The system (6) is said to be internally stable, when w(t ) = 0 and d (t ) = 0 if (6) is asymptotically stable, i.e. roots of det (λE − (A + BF ))⊂ C− . For system (4), with d (t ) = 0, the problem of output regulation via full state information feedback, u = F x+Gw aims at finding gain matrices F and G such that for the resulting system (6) with d (t ) = 0, the roots of det (λE − (A + BF ))⊂ C− (for internal stability) and Lim e(t ) = 0 for any initial condition [18].
t→∞
Definition 2.5. Solvability: For system (4) with d (t ) = 0, the problem of output regulation via full state information feedback, u = F x + Gw is said to be solvable if there exist gain matrices F and G such that for the resulting system (6) with d (t ) = 0, the roots of det (λE − (A + BF ))⊂ C− (for internal stability) and Lim e(t ) = 0 for t→∞
any initial condition. Following [18], one can say that for the system (4) with d (t ) = 0, the problem of output regulation via full state information feedback, u = F x + Gw is solvable if and only if there exist two matrices and such that it satisfies the following equations (regulator equations) given by (7), for which G is to be chosen as G = − F .
0 = C + Q1
˙ b : E x˜(t ) = Ax˜(t ) + Bu˜ (t ) e˜(t ) = C x˜(t )
(7)
(8)
Let E x˜0 := E x˜(0 ) represents an initial condition of (8). Let (8), satisfies Assumption 2.1. If we apply u˜ = F x˜(t ), then, it becomes
E x˜˙ (t ) = (A + BF )x˜(t ), e˜(t ) = C x˜(t )
(9)
As per Assumption 2.1-1(c), (8) is stabilizable which in turn implies the existence of an F so that (E, A + BF ) is asymptotically stable. Therefore, as t → ∞, x˜(t ) → 0 and hence e˜(t ) → 0. 2.2.1. Non-overshooting tracking controller for the base system As per Assumption 2.1-1(c), the base system (8) is stabilisable and more over as per Assumption 2.1-1(d), the system is impulse controllable as well. Hence, because of these two assumptions, there will exist a gain feedback matrix, F such that the on applying, u˜ = F x˜(t ) to (8), the resulting closed loop system, (9) satisfies the following two properties: (a) roots of det (λE − (A + BF ))⊂ C− – which will guarantee asymptotic stability of (9) and (b) deg(det (λE − (A + BF ))) = rank(E ) – which will guarantee impulse free behavior in the responses of (9)[7,14]. Selecting a gain feedback matrix, F which satisfies only these two properties are not enough to satisfy the current design requirements. This is because, more importantly here while applying u˜ = F x˜(t ) to (8) e˜(t ) → 0 as t → ∞ with NO behavior from an initial condition E x˜0 . Therefore, to get these desired characteristics, the design of F has to be carried out in a particular way. In [14], we can see the details for constructing such an F which make use of Moore’s [20] eigen value and vector assignment method, generalised for descriptor systems so as to get a NO behavior. Remark 2.1. For system (1) we assumed that the E, A, B, C matrices have a form as in (2). Then, one can use the methods available in [14] to construct an F that can assure NO behavior for (8) from an initial condition E x˜0 without any additional co-ordinate transformation. If (1) is not in a form with matrices as in (2) then one can always find two non-singular matrices M1 and N1 [7,14] such that doing the transformation, x = N1 [xTd xTa ]T and pre-multiplication of the state equation by M1 will result in a system with matrices having a similar form as in (2) and can continue with the design procedure. 3. ISM based robust output regulation strategy with NO The main results of the paper consisting of the design of ISMRO that can solve our problem statement, by providing robust output regulation with NO by rejecting the matched disturbance entering the system, is discussed in this section. Here, in our design compared to the output regulation strategy in [23] for full state feedback, no exo-system component will generate the disturbance and hence we do not have any disturbance
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related information available for measurement and feedback from the exo-system. This is very clear from the exo-system dynamics given by (3) in which the exo-system is generating only the reference signal, r(t). Hence, with the so chosen exo-system, if we select u = F x(t ) + Gw(t ) as the control law to solve our problem statement, then closed loop system may provide an exact tracking without overshoot only in the absence of matched disturbance. It may fail to reject the disturbance once the disturbance is present. Hence, to reject the matched disturbance in such a case we utilise the sliding mode control (SMC) part of the integral sliding mode controller. The combined system and exo-system representation is given by (4) for which we need to design the controller. Motivated from [23], for designing this controller and to state and prove Lemma 3.1, first we design a feedback matrix, F using Moore’s algorithm, generalised for descriptor system described in [14]. Then, if such an F exists, apply u˜ = F x˜(t ) to (8). If it results in assymptotic convergence of e˜(t ) to zero with NO from E x˜0 , then use this F to frame an NO control law, denoted by uNOISM = F x(t ) + Gw(t ),where G = − F for the system (4), for which we need to attain the robust output regulation without any overshoot. The following lemma help us to arrive at a condition on d(t) under which the system (4) will achieve output regulation without any overshoot. Lemma 3.1. Consider a continuous MIMO LTI descriptor system (1) which satisfies Assumption 2.1. Let the system (1) along with the exo-system generator (3) can together be represented as in (4) with an initial condition (Ex0 , w0 ). Let the eigen values of S matrix of exo-system generator (3) has non-negative real parts and there is and which satisfies (7). Also let d (t ) = 0 and there is an F such that, on applying u˜ = F x˜(t ) to (8), with the initial condition E x˜0 = E (x0 − w0 ), will result in asymptotic convergence of e˜(t ) to zero with an NO behavior. Then, for this F and G = − F , the control law uNOISM = F x(t ) + Gw(t ) applied to (4) will provide output regulation with an NO behavior from (Ex0 , w0 ). Proof. On applying u˜ = F x˜(t ) to (8) with an initial condition E x˜0 = E (x0 − w0 ) will result in (9). By assumption, there exist such an F, so that Limt→∞ e˜(t ) = 0 without sign change in any of its error components. Now using this F, applying u = uNOISM = F x(t ) + Gw(t ) to (4) will result in (6). Now, from (6), if σ (t ) = x(t ) − w(t ) and E σ (0 ) = E (x(0 ) − w(0 )) = E x˜0 , using G = − F we can obtain
E (x˙ (t ) − w˙ (t ) ) = (A + BF )x(t ) + (BG )w(t ) + Bd (t ) − E Sw(t ) E σ˙ (t ) = (A + BF )x(t ) + (B( − F ))w(t ) + Bd (t ) − E Sw(t ) = (A + BF )x(t ) + (B − E S )w(t ) + Bd (t ) − BF w(t ) (10) Using the fact that d (t ) = 0 as assumed and from (7), the expression for (10) becomes
E σ˙ (t ) = (A + BF )x(t ) − Aw(t ) − BF w(t ) (11)
As for the error, e(t) we have from (6)
e(t ) = Cx(t ) + Q1 w(t ) = Cx(t ) + Q1 w(t ) + C w(t ) − C w(t )
(12)
Now, rearranging the terms in (12) and from (7) we can obtain
e(t ) = C (x(t ) − w(t )) + (Q1 + C )w(t ) = C (x(t ) − w(t ))
e(t ) = C σ (t )
(14)
On comparing (14) with (9) we can see that both the systems are identical. Hence, Limt→∞ e(t ) = 0 happens without any sign change in any of its error components from (Ex0 , w0 ), because for (9) as per the assumption we already know the fact that Limt→∞ e˜(t ) = 0 happens without sign change in any of its error components. From Lemma 3.1, we were able to see that the output regulation with NO will happen for (4) by applying uNOISM alone, only when d (t ) = 0. Hence, through Lemma 3.1, using generalised Moore’s eigen structure assignment method and output regulation theory for descriptor systems, we extended the results in [23] applicable for an explicit system with full state information feedback, to a descriptor system, when d (t ) = 0. This means that, we had shown when there is no disturbance, how the state feedback controller can be designed to ensure output regulation with NO behavior for a descriptor system. On the other hand, if d(t), whose information is not available for feedback from the exo-system is present, selecting the control law as u = uNOISM cannot reject d(t) and hence they cannot ensure robust output regulation with NO behavior. Hence, here we are employing an ISM control which will handle the disturbance rejection aspect also. Like in any other ISM design [4,17,26], the main idea here is to design an integral sliding surface variable and a controller based on this sliding surface variable. 3.1. Integral sliding surface variable For the given MIMO descriptor system (1) with u ∈ Rm , the integral sliding surface variable, s(t ) := [s1 (t ), . . . , sm (t )]T ∈ Rm is designed as follows:
s(t ) = B+ (E x(t )−E x(0 ) −
0
t
(Ax(τ ) + BuNOISM (τ ))dτ )
(13)
As σ (t ) = x(t ) − w(t ), (11) and (13) can together be written as
(15)
where E, A ∈ Rn×n , B ∈ Rn×m are constant matrices as in (1). The matrix B+ ∈ Rm×n represents the pseudo-inverse of B. As per Assumption 2.1-1(e), B has full column rank and hence B+ exist. In (15), the term uNOISM is computed as uNOISM = F x + Gw, where the gain matrix F and G are obtained using Lemma 3.1. An integral sliding surface can be considered as a set of state variables, x(t) for which s(t) becomes zero. Here, the integral sliding surface variable, s(t) in (15) is designed in such a way that, it will become zero from initial time, t = 0 as in [2]. Hence, the state trajectories will start from the integral sliding surface. 3.2. Controller design For (1), the governing law for the control is chosen to be
uact = u = uNOISM + uSMC
= (A + BF )x(t ) − (A + BF )w(t ) = (A + BF )(x(t ) − w(t ))
E σ˙ (t ) = (A + BF )σ (t )
(16)
Here, uNOISM = F x(t ) + Gw(t ) will handle the asymptotic tracking along with NO characteristics of the system (1) whereas uSMC that represents the sliding mode control law, is to be designed such that it should reject the matched disturbance. As per the discussion in Section 2.1, the control needs to be at least continuous to guarantee continuity in the solutions for a regular impulse free descriptor system. Hence, we are selecting, as an uSMC , super twisting sliding mode control [5,16,22], uSTC which is continuous and which needs only the information on sliding surface variable, s(t) for its implementation as follows:
uSMC = uST C = −κ1 |s(t )|1/2 sgn(s(t )) + ω (t )
(17)
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where,
⎡
κ1 |s(t )| sgn(s(t )) := ⎣
k11 |s1 (t )|
1/2
1/2
k1m |sm (t )| and
⎡
.. .
sgn(s1 (t ))
1/2
⎤ ⎦
(18)
sgn(sm (t ))
⎤
−k21 sgn(s1 (t )) .. ⎦ ω˙ (t ) = −κ2 sgn(s(t )) := ⎣ . −k2m sgn(sm (t ))
Theorem 3.1. Consider a continuous MIMO LTI descriptor system (1) which satisfies Assumption 2.1. Let the system (1) along with the exo-system generator (3) can together be represented as in (4) with an initial condition (Ex0 , w0 ). Also let the eigen values of S matrix of exo-system generator (3) has non-negative real parts and there is and which satisfies (7). Assume that there is an F such that, on applying u˜ = F x˜(t ) to (8), with the initial condition E x˜0 = E (x0 − w0 ), will result in asymptotic convergence of e˜(t ) to zero with an NO behavior. Then, for this F and G = − F , the control law u = F x(t ) + Gw(t ) + uSMC = uNOISM + uSMC applied to (4) will provide a robust output regulation with NO from (Ex0 , w0 ) by rejecting the disturbance if sliding surface variable, s (t) is chosen as in (15) and uSMC is chosen as (17) with k1i = k1 >1.8 k2 + Dmax and k2i = k2 >Dmax for i ∈ {1, . . . , m} and ω (0 ) = 0. Proof. Let the sliding surface variable, s(t) be chosen as in (15) and uSMC as in (17) with k1i = k1 >1.8 k2 + Dmax and k2i = k2 >Dmax for i ∈ {1, . . . , m} and ω (0 ) = 0. By differentiating (15) we get
s˙ (t ) = B+ (Ax + Bu + Bd (t ) − Ax − BuNOISM ) s˙ (t ) = d (t ) + uSMC = d (t ) + −κ1 |s(t )|1/2 sgn(s(t )) + ω (t ) (t)]T ,
(20)
then (20) be-
Consider system (1) as a dc motor actuated pendulum [10], with E, A, B matrices as follows, which is obtained after substituting the values of parameters given in [10]
⎡
0 ⎢−47.5521 A=⎣ 0 0
1.0 0 0 0 −0.2424 0 0.0760
1 ⎢0 E=⎣ 0 0
0 0⎥ 0⎦ 0
⎡
0 1 0 0
B= 0
0
Remembering the fact that, κ1 := diag(k11 , . . . , k1m ), κ2 := diag(k21 , . . . , k2m ), d˙ (t ) := [d˙1 (t ), . . . , d˙m (t )]T and d˙i (t ) ≤ Dmax As per [15], if we choose, k1i = k1 > 1.8 k2 + Dmax and k2i = k2 >Dmax for i ∈ {1, . . . , m}, each pair of (si (t ), s˙ i (t )) for i ∈ {1, . . . , m} converges to zero in finite time. As, all (si (t ), s˙ i (t )) pair converges in finite time to zero, we can say in finite time (s(t ), s˙ (t )) converges to zero and sliding mode happens. Hence, during sliding mode, from (20), we can write, uSMC = −d (t ). Thus, the uSMC will be equal to the negative of the disturbance and hence uSMC will completely reject the disturbance from system (1). As ω(0) and d(0) are assumed to be zero, using (15) and (20), we can say that both s(t ), s˙ (t ) will be zero from the very beginning. Once the sliding mode happens, the disturbance rejection will take place and the resulting system will be similar to (1) without disturbance and sliding mode control. Similarly, in (6) there will not be any disturbance term. Hence, we have a (6) with no disturbance and hence, by Lemma 3.1, Limt→∞ e(t ) = 0 happens without any sign change in any of its error components from (Ex0 , w0 ). Hence, the control, u will provide a robust NO output regulation. Remark 3.1. If S is selected as a zero matrix in (3) with w(t ) ∈ R p and L1 = I and with the elements of w(0 ) ∈ R p being chosen as the
T
⎤
0 10.8095 0 0.10 0 0
1 ,
0
C= 1
0
0 0 ⎥ , 10.8696⎦ 1.0 0 0 0
0
0
(22)
Let the angular position (q), angular velocity(q˙ ), armature current (Ia ) and inductor voltage(VL ) be the state variables x1 , x2 , x3 and x4 respectively. Here, we are only interested in the tracking of angular position (q) and √ let the required reference profile for tracking √ be, r (t ) = −0.8 ∗ ((8 2/π 2 )sin(π t ) + (8 2/(9π 2 ))sin(3π t )) from an initial condition Ex0 = [−5 0.2 0.1 0]T . Let the matched disturbance be d (t ) = [0.5sin(π t )]T . The given system with matrices ¯ + , for as in (22) is stabilisable as rank([λE − A B] ) = 4 = n, ∀λ ⊂ C finite λ and regular as det (λE − A ) = 0 and impulse controllable as rank
E
A
0 E
0 B
= 7 = rank(E ) + n[7].
For the exo-system √(3), in order to √ generate the reference, r (t ) = −0.8 ∗ ((8 2/π 2 )sin(π t ) + (8 2/(9π 2 ))sin(3π t )), choose L1 = [−1 0 − 1 0], S and w(0 ) in (3) as follows:
S=
S1 0
0 w1 ( 0 ) , w (0 ) = S2 w2 ( 0 )
(21)
⎤
0 0 1 0
where,
s˙ (t ) = −κ1 |s(t )|1/2 sgn(s(t )) + z(t ) z˙ (t ) = −κ2 sgn(s(t )) + d˙ (t )
required magnitude of the reference step signals to be tracked and if there is a and that satisfies (7), then, this ISMRO scheme may be used for step tracking as well. In such a case the ISMRO scheme implemented here may be utilized as an alternate option for the controller developed in [2], where in which controller was designed to track a step reference signal. 4. Numerical example and simulation results
(19)
where, |s(t )|1/2 sgn(s(t )) := [|s1 (t )|1/2 sgn(s1 (t )), . . . , |sm (t )|1/2 sgn(sm (t ))]T where each si ∈R and i∈{1, . . . , m}, ω (t ) := [ω1 (t ), . . . , ωm (t )]T and κ1 := diag(k11 , . . . , k1m ), κ2 := diag(k21 , . . . , k2m ) with k1i , k2i for i ∈ {1, . . . , m} are the controller parameters to be designed.
Let, ω (t ) + d (t ) = z(t ), where z(t) := [z1 (t) z2 comes
5
√ 1 , w1 (0 ) = [0 (8 2/π 2 )0.8π ]T 0
0 S1 = − (π 2 )
0 S2 = −9π 2
(23)
√ 1 , w2 (0 ) = [0 (8 2/(9π 2 ))0.8 ∗ 3π ]T 0
Now, solving (7) for and we can get
⎡
−1.0 0 0 0 ⎢ 0 =⎣ −3.4861 0.0204
= 0.3282
0 −1.0 0 0 0 −0.0224 −0.3207
0.3990
−1.0 0 0 0 0 3.8183 0.1832
−0.5651
(24)
⎤
0 −1.0 0 0 0⎥ , −0.0224 ⎦ 0.3513
−0.2730
(25)
Now, the next step is to design F. Before designing an ISMRO, first of all we need to ensure that, with this F, when u = F x˜ is applied to the base system, (8) with E, A, B, C matrices as given by (22) will provide a NO response for e˜, from an initial condition, E x˜0 = E (x0 − w0 ). To design F, first we need to find the number of stable invariant zeros present in the base system, which will be same as the number of invariant zeros present in the actual system as E, A, B, C matrices for both are the same. From (22), the number of invariant zeros can be found to be zero. i.e. we say we have (q − 3 ∗ p) invariant zeros present in the system (remember, q = rank(E ) = 3 and no. of outputs, p = 1) because q − 3 p = 3 − 3 ∗ 1 = 0. Hence, for the design we can use the methodology
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Fig. 1. (a) Plot of e˜ response of base system (b) Tracking responses with ISMRO (c) Plot of sliding surface variable for ISMRO (d) Plot of tracking error under various conditions.
given in [2,14] for a q − l p case, where l = 3 here. As the system has no invariant zeros, we can arbitrarily place 3 poles which are distinct, real and stable and let this be at λ1 = −10, λ2 = −8 and λ3 = −6. Now, using Moore’s algorithm for generalized system [14], F can be obtained as
F = −5.5306
1.0696
2.0852
0
(26)
The design of a state feedback controller with this F will give an impulse free response as it satisfies deg(det (λE − (A + BF ))) = 3 = rank(E ) [7]. Fig. 1(a) shows that we will get an NO response for e˜, when we apply u = F x˜ to b with F as in (26). ISMRO Design. For the ISMRO design we need to design both uSMC and uNOISM . Here, we have a matched disturbance, d (t ) = [0.5sin(π t )]T . Now, for uSMC in (16), choose k11 = 10, k21 = 2 and ω (0 ) = 0. For designing uNOISM = F x(t ) + Gw(t ), F can be chosen as in (26). Now, with this F and , as in (25), by solving G = − F gives G as follows:
G = 2.0669
1.5153
−14.0577
0.8433
(27)
Fig. 1(b) shows the responses of y(t ) = q(t ) tracking the given reference profile using ISMRO design after rejecting the matched disturbance without any overshoot, which can be confirmed from the error response, e(t ) = y(t ) − r (t ) plot. Thus robust output regulation with NO is achieved for the given system (22). Fig. 1(c) gives the response of sliding surface variable and it assures that the sliding mode is happening from the very beginning itself. Fig. 1(d) shows the response of : (1) tracking error for the closed loop system with ISMRO in presence of disturbance, d(t) where the control is given by (16) (2) tracking error for the closed loop system with NO controller acting alone where the control is given by u = uNOISM = F x(t ) + Gw(t ) but with no d(t) in the system (3)tracking error for the closed loop system with u = uNOISM = F x(t ) + Gw(t ) alone, but with d(t) in the system (22). Hence, it is very evident from the plot that tracking performance of the system
(22) with the proposed ISMRO in presence of d(t) and that of NO controller without d(t) are one and the same. But in the absence of sliding mode controller, when the system (22) is controlled with NO controller alone in presence of d(t) the tracking response is becoming poor and hence robust output regulation with NO cannot be guaranteed. 5. Conclusion This paper presented the design of a robust output regulation control strategy with NO for the MIMO descriptor systems. The design strategy helped to track a time-varying reference generated by an exo-system by rejecting the matched disturbances even if their frequencies are unknown by employing an ISM based technique. Simulation results confirmed the effectiveness of the proposed method. As we have designed a sliding mode based control which is continuous, this will alleviate the harmful chattering effects of the actuator while implementation. Moreover in our simulation to obtain the feedback matrix F we have placed the stable and distinct poles arbitrarily on the real axis. By placing these stable and distinct poles farther from the imaginary axis we may be able to get a faster decay of error response to zero but the price we need to pay may be higher control effort. The control scheme discussed here is based on state feedback based technique. Hence, as a scope for future work one can try designing a robust NO controller using an output feedback technique. One can also try extending the idea of implementing the robust NO output regulation scheme for discrete-time systems. References [1] P.S. Babu, B. Bandyopadhyay, M. Thomas, Robust composite non-linear feedback control for descriptor systems with general reference tracking, in: Proceedings of the 44th Annual Conference of the IEEE Industrial Electronics Society, IECON, 2018.
Please cite this article as: P.S. Babu, N. Xavier and B. Bandyopadhyay, Robust output regulation for state feedback descriptor systems with nonovershooting behavior, European Journal of Control, https://doi.org/10.1016/j.ejcon.2019.08.008
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Please cite this article as: P.S. Babu, N. Xavier and B. Bandyopadhyay, Robust output regulation for state feedback descriptor systems with nonovershooting behavior, European Journal of Control, https://doi.org/10.1016/j.ejcon.2019.08.008