Approximate causal output tracking for linear

International Journal of Control

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Approximate causal output tracking for linear perturbed systems via sliding mode control Jorge E. Ruiz-Duarte & Alexander G. Loukianov To cite this article: Jorge E. Ruiz-Duarte & Alexander G. Loukianov (2018): Approximate causal output tracking for linear perturbed systems via sliding mode control, International Journal of Control, DOI: 10.1080/00207179.2018.1500034 To link to this article: https://doi.org/10.1080/00207179.2018.1500034

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INTERNATIONAL JOURNAL OF CONTROL https://doi.org/10.1080/00207179.2018.1500034

Approximate causal output tracking for linear perturbed systems via sliding mode control Jorge E. Ruiz-Duarte

and Alexander G. Loukianov

Automatic Control Laboratory, CINVESTAV-IPN Guadalajara, Zapopan, Jalisco, Mexico ABSTRACT

ARTICLE HISTORY

A new approach to the solution of what is termed causal output tracking problem for linear time-invariant systems in the presence of both matched and unmatched unmodelled disturbances is presented. The proposed solution is addressed through the design of a dynamic steady state estimator based on the desired system structure and an observer, considering the reference as the system output. The unmatched disturbance is estimated and used in the steady state estimator to achieve invariance, while the matched disturbance robustness is provided via sliding mode control using the super-twisting algorithm. A useful application of the presented techniques is output tracking for non-minimum phase systems; thus, the performed simulation results confirm the effectiveness of the proposed control scheme for this kind of systems.

Received 11 September 2017 Accepted 8 July 2018

1. Introduction Output tracking and disturbance rejection for dynamic systems are one of the most important problems in control theory. For linear systems, this problem can be divided into two sub-problems: (A) Steady state calculation. Given the output reference signal yr (t), calculate the steady state values xss for the state and uss for the control input, respectively, ensuring that the tracking error in the steady state is equal to zero, i.e. yss − yr (t) = Cxss − yr (t) = 0. (B) Stabilisation. Defining the control error e = x − xss , design a control law which achieves the asymptotic convergence of the control error to zero. The first problem (A) can, in turn, be considered for two cases: (A1) The output reference signal yr (t) to be tracked is generated by an exogenous system, called exosystem. (A2) The output reference signal yr (t) is an arbitrary function of time, i.e. there is no exosystem. The first case (A1) is called Output Regulation (OR) problem, and its solution is based on the solvability of a set of two linear matrix equations (Francis, 1976), called Francis equation. This solution can be derived off-line using the exosystem state. Solving the Francis equation and defining the control error e = x − xss , the output tracking problem is converted into a stabilisation problem (Francis & Wonham, 1975). In recent years, the robustness issue has become important. To deal with perturbation which can include both plant CONTACT Jorge E. Ruiz-Duarte 45019, Jalisco, Mexico

[email protected]

© 2018 Informa UK Limited, trading as Taylor & Francis Group

KEYWORDS

Sliding mode control; output tracking; non-minimum phase; linear systems; robust control; arbitrary references

parameter variations and external unmodelled disturbances, Sliding Mode (SM) control technique (Utkin, Guldner, & Shijun, 1999) combined with OR theory has been implemented in Gopalswamy and Karl Hedrick (1993), Castillo-Toledo and Castro-Linares (1995), Jeong and Utkin (1999), Loukianov, Castillo-Toledo, and Garcia-Rocha (1999), Bonivento, Marconi, and Zanasi (2001), Utkin and Utkin (2014), Loza, Bejarano, and Fridman (2009), GuiZhi and Kemao (2014), providing that the uncertainties satisfy the matching condition (Draženović, 1969). The second case (A2) is more in-line with real life and theoretical applications. This is known as Causal Output Tracking (COT) problem, since only the information regarding the present and the past are known, but not future, as in the case of the OR problem (A1). For linear Minimum-Phase (MP) systems, the COT problem can be easily achieved through classical control techniques, such as the input-output transformation to the canonical form obtaining the required in this case output reference derivatives by means of a differentiator design. However, in the case of Non-Minimum-Phase (NMP) systems, having just the output reference signal yr (t) on-line, the exact obtaining of the system steady state xss and feedback control part uss (sub-problem A) is a very challenging problem since the zero dynamics are unstable. As result, the exact tracking of arbitrary references in NMP systems is almost unrealisable. Moreover, this problem becomes more difficult for NMP systems subject to perturbations. Therefore, the general problem for linear systems has been converted in some different setups by considering certain constraints. In Wang and Chen (2001), a stable causal inversion was proposed in the event that the reference signal is equal to zero after a

Automatic Control Laboratory, CINVESTAV-IPN Guadalajara, 1145 Avenida del Bosque, Zapopan

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J. E. RUIZ-DUARTE AND A. G. LOUKIANOV

determined time. In Jafari and Mukherjee (2012), considering the COT problem for continuous time NMP SISO systems and using the discretisation technique with two different intercalated sampling times, an intermittent control which ensures the internal dynamics stability, was proposed. Using the first sampling time, the resultant system is NMP, while using the second one, MP system is obtained. The reference matching is then obtained when the MP system discretisation is active. The main disadvantage of that work is that the proposed controller guarantees output matching with reference trajectory only at regular intervals of time. In Ruiz-Duarte and Loukianov (2016), the causal output tracking problem for continuous-time linear systems was solved approximately by using a dynamic steady state estimator based on an observer. However, this work considered only the unperturbed case. In the presence of uncertainties, the COT problem (A2) has been investigated using the SM technique. In Shtessel and Shkolnikov (1999), Shkolnikov and Shtessel (2001), a system centre method, proposed first for MP systems, has been extended for the NMP case, designing a dynamic SM controller. However, in this work, a class of restricted reference signals which includes only functions with a finite number of non-zero time derivatives has been considered. In Baev, Shtessel, Edwards, and Spurgeon (2008), Pisano, Baev, Salimbeni, Shtessel and Usai (2013), the reference signal and the disturbances must be generated by an unknown linear exosystem. To identify the exosystem characteristic polynomial online, a higher order SM observer was used. In the present work, the COT problem is considered for linear (minimum or non-minimum phase) time-invariant (LTI) systems in the presence of both matched and unmatched unmodelled disturbances which are not generated by any exosystem. To solve this problem, a novel robust control scheme based on the SM technique is designed. This can be considered as the main contribution of the paper. First, to solve the problem (A), i.e. to define the required steady state xss in the presence of uncertainty, a dynamic estimator is proposed. This estimator is designed in the form of a state observer which uses the output reference signal as a measured output. To design this observer, the unmatched disturbance is estimated by implementing a robust exact differentiator (Levant, 1998). To solve the stabilisation problem (B) and achieve robustness, an SM controller is designed using the super-twisting algorithm (STA) (Fridman & Levant, 2002). To show the effectiveness of the proposed method, a robust controller is designed for a linear model of the Pendubot non-minimum phase system.

The control objective is to provide approximate output tracking of a smooth arbitrary reference yr (t) ∈ m as well as unknown disturbances rejection. This problem can be solved by designing a control law such that the closed-loop system is stable and the tracking error y − yr (t) converges to a neighbourhood of the origin, i.e.   y − yr (t) < 1 , ∀t > tf (2) for some small nonnegative constant 1 and finite time tf . To solve the considered problem, the following Assumptions which are instrumental for the control design, are introduced: Assumption 2.1: The pair {A, B} is controllable. Assumption 2.2: Matrices A, B and C satisfy   A B rank = n + m. C 0 Assumption 2.3: The number of the system states is greater than or equal to the double of control inputs, i.e. n ≥ 2m (see Remark 2.2). Remark 2.1: Assumption 2.2 implies both, that the realisation (A, B, C) is invertible and that the system (1) has no zeros at λ = 0. Remark 2.2: Assumption 2.3 implies that there is a nonsingular matrix T such that ⎤ ⎡ 0(n−2m)×m ⎦, B1 T −1 B = ⎣ B2 where B1 , B2 ∈ m×m and rank(B1 ) = rank(B2 ) = m. If this Assumption is not satisfied, a 2m−n or greater dynamic extension can be proposed in the form of a dynamic control law. Such control law will not be the applied control, but it is only used to the design of the steady state estimator. As in the case of classical output regulation, in this work, it is argued that the robust causal output tracking problem is solvable if there is a pair of time functions π(t) and c(t), with π ∈ n and c ∈ m , such that the following differential-algebraic equations (DAE) system is satisfied: π˙ (t) = Aπ(t) + B [c(t) + φ(π(t), t)] + D(t), 0 = Cπ(t) − yr (t).

(3)

(1)

Functions π(t) and c(t) represent the required steady state for x and u, respectively, such that the system output tracks the reference yr (t), i.e. xss = π(t) and uss = c(t). The DAE (3) are equivalent to the Francis equation (Francis, 1976) for linear systems and to the Francis-Isidori-Byrnes equation presented in Isidori and Byrnes (1990) for nonlinear systems. However, the solution of (3) depends on time functions instead of the exosystem state.

where x ∈ n is the system state; u ∈ m is the control; y ∈ m is the system output; φ : n × + ∪ {0} → m is a matched disturbance;  : + ∪ {0} → p is an unmatched disturbance; A, B, C and D are known constant matrices of appropriate dimensions, and rank(B) = rank(C) = m.

Remark 2.3: It is easy to see that if there is a solution for Equation (3), then a control law can be proposed such that the robust causal output tracking is achieved even if the internal or zero dynamics (Isidori, 1995) of system (1) are unstable, i.e. even if (1) is an NMP system.

2. Problem statement and proposed control scheme Consider a linear perturbed system presented in the form x˙ = Ax + B [u + φ(x, t)] + D(t), y = Cx,

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3

2.1 Existence and uniqueness of the DAE system solution The DAE system (3) can be rewritten in the form of a semiexplicit descriptor system as follows:         A B π(t) D(t) In 0 π˙ (t) = + , (4) 0 0 c˙ (t) C 0 c (t) −yr (t) where c (t) = c(t) + φ(π(t), t). In general form, the descriptor system (4) can be represented of the form E˙z + Fz = q(t),

(5)

where E and F are square matrices which will be defined later, and det(E) = 0. Definition 2.1: The pair {E, F} is said to be regular if the determinant of the pencil λE + F is regular; i.e. if det(λE + F) ≡ 0. Otherwise, the pair is said to be singular.

On the other hand, if the pair {E, F} is regular, then, there exists a constant λ ∈ C , such that   λIn − A −B = 0. (7) det −C 0 This condition is satisfied under Assumption 2.2, for λ = 0 and therefore, the DAE solution exists and it is unique.  2.2 Proposed control scheme

Proposition 2.1 (Proposition 1.3 in Lamour, März, and Tischendorf (2013)): For any regular pair {E, F}, E, F ∈ Rmd ×md , there exist non-singular matrices L, K ∈ Rmd ×md and integers 0 ≤ l ≤ md , 0 ≤ ν ≤ l, such that     W 0 }md − l I 0 }md − l , LFK = . LEK = }l 0 I }l 0 N (6) Thereby, N is absent if l = 0, and otherwise N is nilpotent of order ν, i.e. N ν = 0, N ν−1 = 0. The integers l and ν as well as the eigenstructure of the blocks N and W are uniquely determined by the pair {E, F}. Theorem 2.1: A solution for the descriptor system (5) exists and it is unique if and only if the pair {E, F} is regular and q(t) is a vector of class Cν−1 , with ν defined in Proposition 2.1. Proof: See Appendix 1.

Figure 1. Proposed control scheme.



Theorem 2.2 (Sufficient conditions): Let the reference signal yr (t) and the unmatched disturbance (t) be smooth enough (Cν−1 ). Then, if the system (1) satisfies Assumption 2.2, the DAE system (3) has a solution and it is unique. Proof: Define the vectors     π(t) D(t) z=  , q(t) = , c (t) −yr (t) both with dimension n + m, and the (n + m) × (n + m) matrices     I 0 A B E= n , F=− . 0 0 C 0 The reference and disturbance smoothness condition is necessary from the class Cυ−1 requirement for the vector q(t) in Theorem 2.1.

The stated robust causal output tracking problem is based on the solution of the DAE system (3). Since the disturbances are unknown and the reference yr (t) is measured on-line, it is not possible to find a solution to this system and, therefore, to achieve the exact tracking. In the foregoing sections, based on the estimation of a solution to the system (3), a robust SM control scheme is proposed to approximately track the arbitrary reference yr (t). Figure 1 shows a schematic diagram of the upcoming proposed control.In the block diagram, using the measured state x, the unmatched disturbance estimator block (see Subsection 3.2) generates the term D(t), ˆ the estimate of D(t). Then, the dynamic steady state estimator block (its design is presented in Section 3) using the output reference yr (t) and the unmatched disturbance estimate D(t), ˆ delivers πˆ (t) which is the estimate of required steady state π(t). Finally, the SM control block (presented in Section 4) produces the control which is in charge of stabilising the control error system rejecting the matched disturbance term φ(x, t).

3. Steady state estimator As it was noted above, the objective of this work is to provide the approximate output tracking of an arbitrary reference for a linear system (even if NMP) in the form (1), despite disturbances. A solution can be obtained by solving the DAE system (3), that is impossible in the presence of the measured on-line reference signal and unknown disturbances. To solve this problem, the design of a dynamic estimator which delivers an on-line approximate of the system (3) solution will be proposed. 3.1 Estimator construction Using a similarity transformation (or a dynamic extension mentioned in Remark 2.2), the system (1) under Assumptions 2.1

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J. E. RUIZ-DUARTE AND A. G. LOUKIANOV

to 2.3, without loss of generality, is reduced to the following form:

where the matrix ⎡ 0 0 ⎣ 0 G := 0 0 B2 B−1 1

x˙ 1 = A11 x1 + A12 x2 + A13 x3 + D1 (t), x˙ 2 = A21 x1 + A22 x2 + A23 x3 + B1 [u + φ(x, t)] + D2 (t), x˙ 3 = A31 x1 + A32 x2 + A33 x3 + B2 [u + φ(x, t)] + D3 (t), y = Cx,

(8)

where x = col(x1 , x2 , x3 ), x1 ∈ n−2m , x2 , x3 , u, y ∈ m , B1 , B2 ∈ m×m are full rank matrices and Aij , Di , i,j = 1,2,3 are constant matrices of appropriate dimensions. Using the system desired structure (8), the DAE system (3) can be represented in the form π˙ 1 (t) = A11 π1 (t) + A12 π2 (t) + A13 π3 (t) + D1 (t),

GB = B.

(9a)

+ B1 [c(t) + φ(π(t), t)] + D2 (t),

(9b)

π˙ 3 (t) = A31 π1 (t) + A32 π2 (t) + A33 π3 (t) + B2 [c(t) + φ(π(t), t)] + D3 (t),

1 1 π˙ e = − πe − 2 Hπ(t), τ τ 1 πpe = πe + Hπ(t), τ

(9d)

where π(t) = col(π1 (t), π2 (t), π3 (t)), π1 ∈ n−2m , π2 , π3 ∈ m . To begin the estimator construction, the term c(t) is calculated from Equations (9b) and (9c), respectively, as follows: c(t) = −φ(π(t), t) + B−1 ˙ 2 (t) − A21 π1 (t) 1 [π − A22 π2 (t) − A23 π3 (t) − D2 (t)] ,

(10)

and −φ(π(t), t) + B−1 ˙ 3 (t) − A31 π1 (t) 2 [π − A32 π2 (t) − A33 π3 (t) − D3 (t)] .

(11)

Replacing Equation (11) into (9b) and (10) into (9c) results in π˙ 1 (t) = A11 π1 (t) + A12 π2 (t) + A13 π3 (t) + D1 (t), 



















π˙ 3 (t) = A31 π1 (t) + A32 π2 (t) + A33 π3 (t) + D3 (t) + B2 π˙ 2 (t), (12) 

(16)

where πe ∈ 2m is the filter state, πpe = col(πpe2 , πpe3 ) ∈ 2m is the filter output or the derivative approximate, and matrix H is defined as  H := 0 I2m ∈ 2m×n . (17) Using the filter output πpe , the system (13) is represented as π˙ (t) = (In − G)Aπ(t) + GH T πpe + (In − G)D(t) + G(t), (18) where (t) := π˙ (t) − H T πe − τ1 π(t) is considered as an unknown disturbance. Thus, the complete system (16) and (18) is represented of the form    (In − G)A + τ1 G GH T π(t) ˙ π(t) = π˙ e πe − τ12 H − τ1 I2m   (I − G)D(t) + G(t) , + n 0



π˙ 2 (t) = A21 π1 (t) + A22 π2 (t) + A23 π3 (t) + D2 (t) + B1 π˙ 3 (t), yr (t) = Cπ(t),

s , τs + 1

where τ is a small positive constant. The filter inputs are the estimates of π2 and π3 , respectively, while their outputs are the estimates of π˙ 2 and π˙ 3 , respectively. The filter dynamics are represented in the state space form as

(9c)

yr (t) = Cπ(t),



−1 with A21 = A21 − B1 B−1 2 A31 , A22 = A22 − B1 B2 A32 , A23 =   −1 −1 A23 − B1 B−1 2 A33 , A31 = A31 − B2 B1 A21 , A32 = A32 − B2 B1     −1 −1 −1 A22 , A33 = A33 − B2 B1 A23 , B1 = B1 B2 , B2 = B2 B1 , D2 =  −1 D2 − B1 B−1 2 D3 and D3 = D3 − B2 B1 D2 . The system (12) can be represented in general form as

yr (t) = Cπ(t).

(19)

The obtained system (19) will be used furthermore to design a steady state estimator for π(t) in the form of a state observer. 3.2 Unmatched disturbance estimation

π˙ (t) = (In − G)Aπ(t) + Gπ˙ (t) + (In − G)D(t), yr (t) = Cπ(t),

(15)

To design an estimator which approximates the system (13) solution on-line, considering the output reference signal yr (t) as the plant (13) output, an observer based on the plant model (13) is proposed. The term π˙ (t) at right-hand side of (13) can be approximated by using a first order filter with transfer function

π˙ 2 (t) = A21 π1 (t) + A22 π2 (t) + A23 π3 (t)



(14)

satisfies

ϒ(s) =

c(t) =

⎤ 0 −1 ⎦ B1 B2 0

(13)

In this subsection, a disturbance estimator is designed to obtain an exact estimate of the disturbance term D(t).

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First, the system (1) is reduced by a similarity transformation x∗ = T1−1 x to Regular form (Utkin, 1978) x˙ 1∗ = A∗11 x1∗ + A∗12 x2∗ + D∗ (t),  x˙ 2∗ = A∗21 x1∗ + A∗22 x2∗ + B∗ u + φ(x∗ , t) ,

(20a) (20b)

where x∗ = col(x1∗ , x2∗ ), x1∗ ∈ n−m , x2∗ ∈ m and  ∗  A∗12 A T1−1 AT1 = 11 . A∗21 A∗22 Using the subsystem (20a), the disturbance term estimate D∗ (t) ˆ can be obtained as ˆ = x˙ˆ 1∗ − A∗11 x1∗ − A∗12 x2∗ , D∗ (t)

(21)

where x˙ˆ 1∗ is a derivative estimate which can be obtained using a robust exact differentiator (Levant, 1998). Then, the disturbance term estimate D(t) ˆ is recovered from Equation (21), using the transformation matrix T1 as  ∗  ˆ D (t) D(t) ˆ = T1 . (22) 0m×p After the exact differentiator converges in finite time, the estimate (22) will be equal to the real disturbance D(t). This estimated unmatched disturbance term will be used in the steady state estimator instead of the real one. 3.3 Main result Define the matrices

  (In − G)A + τ1 G GH T , Cπ := C Aπ := 1 1 − τ2 H − τ I2m

0 .

The observability conditions of the system (19) are presented in the following theorem: Theorem 3.1: There exists a constant τ , possibly after a similarity transformation z = T2−1 x with ⎡ i ⎤ i i B B−1 T11 T12 −T12 1 2 ⎢ i ⎥ i i (23) T2−1 = ⎣T21 T22 T23 ⎦, i T31

i T32

i T33

such that the pair {Aπ , Cπ } (or the transformed pair) is observable if and only if the system (1) satisfies Assumption 2.2. Proof: See Appendix 2.



5

proposed estimator (24) delivers an approximate of the steady state required to achieve the output tracking, even in the presence of unknown disturbance (t). 3.4 Estimator convergence analysis Defining the estimation errors π˜ = π(t) − πˆ and π˜ e = πe − πˆ e and using (19) and (24), results in the following estimation error dynamics:     (In − G)A + τ1 G − L1 C GH T π˙˜ π˜ = π˙˜ e − τ12 H − L2 C − τ1 I2m π˜ e   ˜ G(t) + (In − G)D(t) , (25) + 0 where (t) ˜ = (t) − (t). ˆ Under Theorem 3.1, there are observer matrices L1 and L2 and a filter time constant τ such that the matrix

 (In − G)A + τ1 G − L1 C GH T Aπ ,2 := (26) − τ12 H − L2 C − τ1 I2m is Hurwitz. The matrices L1 and L2 can be designed by using any linear control technique, including pole placement; one advisable way to design L1 and L2 is presented in Appendix 5. To analyse the error dynamics (25) convergence, define the state vector   π˜ πT := , (27) π˜ e and the disturbance vector   ˜ G(t) + (In − G)D(t) . g(t) := 0

(28)

Then, the error system (25) is represented in the general form π˙ T = Aπ ,2 πT + g(t),

(29)

where the vector g(t) is a bounded perturbation, as the following lemma shows. Lemma 3.1: The perturbation g(t) is bounded by   g(t) ≤ γ1 ,

(30)

where γ1 is a non-negative constant. Moreover, γ1 → 0 when the second derivative of reference yr (t) and unmatched disturbance (t) tend to zero.

Then, using the Theorem 3.1 result and the obtained disturbance estimate D(t) ˆ (22), a steady state estimator can be designed in the form of a state observer     (In − G)A + τ1 G − L1 C GH T π˙ˆ πˆ = 1 1 ˙ π ˆe πˆ e − τ 2 H − L2 C − τ I2m   (I − G)D(t) ˆ + L1 yr (t) + n , (24) L2 yr (t)

To analyse the convergence of the estimation error system (29), consider the candidate Lyapunov function

where πˆ (t) is the estimate of π(t), πˆ e (t) is the estimate of πe (t),  T L = LT1 LT2 ∈ (n+2m)×m is the observer gain matrix. The

P1 Aπ ,2 + ATπ ,2 P1 = −Q1 , Q1 > 0.

Proof: See Appendix 3.

V1 (πT ) = πTT P1 πT ,



(31)

where P1 is a symmetric positive definite matrix, the solution of (32)

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Theorem 3.2: Considering the bound (30), the system (29) solution is Globally Ultimately Bounded by  λmax (P1 )

πT < b1 , ∀t > tf 1 , b1 = μ1 , λmin (P1 ) μ1 =

2γ1 λmax (P1 ) , θ1 λmin (Q1 )

for a finite time tf 1 and 0 < θ1 < 1. Moreover, the ultimate bound b1 is proportional to the output reference and unmatched disturbance second derivatives. Proof: See Appendix 4.

SM controller. Having just the estimated steady state π, ˆ define the estimate transformed state error as ζˆ = T1−1 (x − πˆ ), where ζˆ = col(ζˆ1 , ζˆ2 , ζˆ3 ) is equivalent to ˜ ζˆ = T1−1 (e + π˜ ) = ζ + T1−1 π.

4. Tracking control 4.1 Control design Having the estimated steady state, the robust COT problem can be converted into a stabilisation problem. Let us define the control error as e = x − π(t),

(33)

where e = col(e1 , e2 , e3 ), e1 = x1 − π1 (t), e2 = x2 − π2 (t), e3 = x3 − π3 (t). Using the state error (33), the tracking problem (2) can be reduced to design a control law such that

e < 2 ,

∀t > tf

σ = K1 ζˆ1 + K2 ζˆ2 + ζˆ3 ,

with φc = c(t) − φ(x, t) + φ(π(t), t). Using the following similarity transformation: ⎤ ⎡ 0 In−m 0 ⎦, ζ = T1−1 e, T1 = ⎣ 0 Im B1 B−1 2 0 0 Im

(40)

or according to (39) ˜ σ = KT ζ + KT T1−1 π,

(41)

where σ = σ (ζˆ ) = col(σ1 , . . . , σm ) : n → m , K1 and K2 are constant  design control matrices of appropriate dimensions and KT := K1 K2 Im . The projection of the system (37) on the subspace σ is governed by σ˙ = Aσ ζ + B2 (u − φc ) + δσ ,

(42)

˙˜ where Aσ = KT A¯ and δσ = KT T1−1 π. To reduce the chattering effect and achieve robustness with respect to matched unknown disturbances φc , a combination of the super-twisting algorithm and the known part of the equivalent control is used to design the following control law:   u = B−1 −Aσ ζˆ + u1 + u2 , u˙ 2 = −KM sign(σ ), 2

(43)

(34)

for another small positive constant 2 . From the system representation (1) and the steady state Equation (3), the state error (33) dynamics result in e˙ = Ae + B (u − φc )

(39)

Using the estimated error (38), a sliding function is formulated of the form



For the small second derivatives in both the output reference and unmatched disturbance, the ultimate bound b1 becomes also small. Thus, the estimator (24) delivers an on-line approximate of π(t). This approximate will be used to solve the robust COT problem along with the control law that will be designed in Section 4.

(38)

where

(35)

⎤ −k11 |σ1 |1/2 sign(σ1 ) ⎥ ⎢ .. u1 = ⎣ ⎦, . 1/2 −km1 |σm | sign(σm ) ⎡

sign(σ ) = col(sign(σ1 ), . . . , sign(σm )) and matrix KM = diag (k12 , . . . , km2 ). Substituting the control law (43) into Equation (42), yields (36)

σ˙ = φT + u1 + u2 , u˙ 2 = −KM sign(σ ),

(44)

the error system (35) is presented in Regular form ¯ + B¯ (u − φc ) , ζ˙ = Aζ

(37)

where ζ = col(ζ1 , ζ2 , ζ3 ), ζ1 ∈ n−2m , ζ2 ∈ m , ζ3 ∈ m , ⎤ ⎡ ⎡ ⎤ A¯ 11 A¯ 12 A¯ 13 0 A¯ = T1−1 AT1 = ⎣A¯ 21 A¯ 22 A¯ 23 ⎦ , B¯ = T1−1 B = ⎣ 0 ⎦ . B2 A¯ 31 A¯ 32 A¯ 33 It can be noted that presenting the system in Regular form (37) provides a comfortable structure for the design of an

where φT := −Aσ T1−1 π˜ − Bσ φc + δσ . Providing that the derivative φ˙ T is bounded, following Moreno and Osorio (2008) and Polyakov and Poznyak (2009), there are control gains k11 , . . . , km1 , k12 , . . . , km2 such that the closed-loop system (44) state converges to the sliding manifold      := ζˆ  σ = 0 in finite time.

(45)

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4.2 Sliding mode equation stability analysis Once the manifold  (45) is reached, the variable ζ3 can be used as a virtual control for (37). From (40), (41) and (45), it follows ζ3 = −K1 ζ1 − K2 ζ2 − KT T1−1 π. ˜

(46)

SM equation is then expressed as

where ζ1,2

˜ ζ˙1,2 = (As − Bs Ks )ζ1,2 − Bs KT T1−1 π,  = col(ζ1 , ζ2 ), Ks := K1 K2 , 

A¯ As := ¯ 11 A21

Theorem 4.1: Let the system (1) satisfy Assumptions 2.1–2.3. Let the matrices L1 , L2 and Ks be such that Acl is Hurwitz. Then, the tracking error dynamics described by Equation (49) have a Globally Ultimately Bounded solution. Moreover, the solution bound depends on reference and disturbance second derivatives. Proof: Follows from the proof of Theorem 3.2.

5. Simulation results

Under Assumption 2.1, the control matrix Ks can be chosen such that (As − Bs Ks ) is Hurwitz. However, taking into account that system (47) depends on the steady state estimation error π, ˜ it is necessary to analyse the complete SM equation, including the estimation error dynamics (25). Combining tracking error dynamics (47) with steady state estimation error (25), it results in the following error system dynamics: ⎤⎡ ⎤ ζ1,2 ⎥ GH T ⎦ ⎣ π˜ ⎦ π˜ e − τ1 I2m 0

(48)

where AG = (In − G)A + τ1 G. For the sake of simplicity, define ⎤ ⎤ ⎡ ζ1,2 0 χ = ⎣ π˜ ⎦ , g1 (t) = ⎣G(t)⎦ and 0 π˜ e ⎡ ⎤ As − Bs Ks −Bs KT T1−1 0 ⎢ ⎥ GH T ⎦ . 0 AG − L1 C Acl = ⎣ 0 − τ12 H − L2 C − τ1 I2m ⎡

To show the effectiveness of the steady state estimator and the super-twisting controller, the presented technique was applied to a linearised model of the Pendubot system (Figure 2). The Pendubot is an underactuated mechanical NMP system. The Pendubot model and its parameters are taken from Serrano-Heredia, Loukianov, and Bayro-Corrochano (2011). The linearised model with disturbances results in ⎤ 0 0 1 0 ⎢ 0 0 0 1 ⎥ ⎥ x˙ = ⎢ ⎣ 49.4 −15.7 −0.18 0.03 ⎦ x −50.3 70.4 0.34 −0.09 ⎡ ⎤ ⎤ ⎡ 0 0 ⎢0⎥ ⎢ 0 ⎥ ⎢ ⎥ ⎥ +⎢ ⎣ 33.6 ⎦ [u + φ(x, t)] + ⎣1⎦ (t), 0 −62.3  y = 0 1 0 0 x. ⎡

(49)

where Acl is a Hurwitz matrix. From the bound (30), it follows   g1 (t) ≤ γ1 . (50) Consider now the candidate Lyapunov function V2 = χ T P2 χ,

(51)

where P2 is a symmetric positive definite matrix, solution of P2 Acl + ATcl P2 = −Q2 , Q2 > 0.

(52)

The stability property of the system (49) is analysed in the following theorem:

(53)

The matched disturbance was simulated as φ(x, t) = 0.5x1 cos (3t), while the unmatched one was (t) = 0.1 sin(0.4t + 4) + 0.2 cos(0.1t) + 0.1.

Then, the system (48) can be expressed in general form as χ˙ = Acl χ + g1 (t),



(47)

   A¯ A¯ 12 , Bs := ¯ 13 . A¯ 22 A23

⎡ ⎤ ⎡A − B K −Bs KT T1−1 s s s ζ˙1,2 ⎢ ⎣ π˙˜ ⎦ = ⎣ 0 AG − L1 C π˙˜ e 0 − τ12 H − L2 C ⎤ ⎡ 0 + ⎣G(t)⎦ , 0

7

Figure 2. Output tracking.

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J. E. RUIZ-DUARTE AND A. G. LOUKIANOV

Model (53) is actually in the form (8), therefore, no matrix T is necessary. Matrix T1 was chosen as ⎤ ⎡ 1 0 0 0 ⎢0 1 0 0 ⎥ ⎥. T1 = ⎢ ⎣0 0 1 33.6 ⎦ 0 0 0 −62.33

0.2 0.1 0

Constant τ was set to 0.5 and, setting the desired observer eigenvalues at {−30, −31, −32, −33, −34, −35}, the observer  T matrix L, L = LT1 LT2 was obtained as

−0.1

 T L = 1 × 103 74.25 0.155 196.4 9.69 −2437.7 −53.53 .

−0.3

For the control design, the desired SM equation eigenvalues were set to −5 all of them and matrix KT was calculated as KT = −0.724 −0.634 −0.153 1 . Finally, the supertwisting gains were set to k11 = 10 and k12 = 25. The output reference yr (t) was simulated by a signal 0.1 sin(0.5t) added to an arbitrary signal obtained using a random number generator, a Butterworth lowpass filter of order 20 with a cutoff frequency of 0.5 rad/s, and a gain of 100.

−0.2

−0.4 0

0.2

0.4

0.6

0.8

1

Figure 5. Sliding function (σ ).

4 2 0 −2

2

−4 1

−6

5

−8

0

0 −1

−10 0.5

−2 −3 −4 −5 0

−5 0

−12 0

0.5

20

40

1

60

1

2

Table 1. Causal tracking methods advantages and disadvantages. 20

40

60

80

100 •Robustness to matched Approximate tracking. and unmatched disturbances. •Wide variety of references. •No previous knowledge of any reference characteristic needed.

Pisano et al.

•Robustness. •Improved transient features of the error variables. •Exact output tracking.

Works only with references generated by linear exosystems (output regulation problem).

Jafari and Mukherjee

The reference trajectories to be tracked are not constrained to any specific set.

Match with the reference just at periodic time instants.

Shtessel and Shkolnikov

•Insensitivity to matched disturbances and uncertain nonlinearities. •Exact output tracking.

Works only with references with a finite number of non-zero derivatives.

Wang and Chen

Exact output tracking.

Works only with references that are zero after a determined time.

0.02 0.01 0 −0.01 0.5

−0.04 0

0 −0.5 0

1

2

20

40

60

Figure 4. Tracking error (y − yr (t)).

80

100

Disadvantages

Our method Figure 3. Output tracking.

−0.03

100

Figure 6. Control input (u).

−0.5 0

Advantages

−0.02

80

0

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Figure 3 shows the comparison between the reference profile yr (t) and system output y responses. It can be seen that the tracking is appropriately approximated in less than two seconds. Figure 4 shows the tracking error behaviour. This error achieves an ultimate bound with an approximate magnitude of 0.01 rad. In Figure 5, the sliding function σ response is shown. It can be seen that the sliding mode is achieved in approximately 0.3 s with a small overshoot. Finally, Figure 6 shows the applied control input u response. It can be seen that the proposed control in the steady state counteracts the disturbance effect, while the transient response is bounded in an acceptable region less than 6 N − m. Table 1 shows a comparison between the advantages and disadvantages of our method and other four methods.

6. Conclusions The causal output tracking problem for linear time-invariant perturbed systems has been addressed. A dynamic steady state estimator was proposed. Having an arbitrary output reference, this estimator is able to produce an approximated required steady state of the complete system on-line, despite the unmatched disturbances. Based on the SM super-twisting algorithm, a nonlinear controller was designed, providing robustness with respect to the matched disturbances and reducing the chattering effect. The proposed approach can be applied to both, minimum and non-minimum phase systems. Future research will address to the error-feedback causal output tracking problem, for the case when only the system output is available for measurement in the presence of both matched and unmatched uncertainties. The results obtained in this work will be extended to nonlinear affine control perturbed systems.

Disclosure statement No potential conflict of interest was reported by the authors.

Funding This work was supported by Consejo Nacional de Ciencia y Tecnología [grant number 252405].

ORCID Jorge E. Ruiz-Duarte http://orcid.org/0000-0002-0834-0240 Alexander G. Loukianov http://orcid.org/0000-0001-5708-803X

References Baev, S., Shtessel, Y., Edwards, C., & Spurgeon, S. (2008). Output feedback tracking in causal nonminimum-phase nonlinear systems using HOSM techniques. 2008 International Workshop on Variable Structure Systems, Antalya (pp. 209–214). Bonivento C., Marconi L., & Zanasi R. (2001). Output regulation of nonlinear systems by sliding mode. Automatica, 37(4), 535–542. Castillo-Toledo B., & Castro-Linares R. (1995). On robust regulation via sliding mode for nonlinear systems. Systems & Control Letters, 24(5), 361–371. Draženović B. (1969). The invariance conditions in variable structure systems. Automatica, 5(3), 287–295. Francis B. (1976). The linear multivariable regulator problem. In IEEE conference on decision and control including the 15th symposium on adaptive processes, 1976 (pp. 873–878).

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Francis B., & Wonham W. (1975). The internal model principle for linear multivariable regulators. Applied Mathematics and Optimization, 2(2), 170–194. Fridman L., & Levant A. (2002). Higher order sliding modes. Sliding Mode Control in Engineering, 11, 53–102. Gopalswamy S., & Karl Hedrick J. (1993). Tracking nonlinear nonminimum phase systems using sliding control. International Journal of Control, 57(5), 1141–1158. GuiZhi M., & Kemao M. (2014). Output regulation for nonlinear systems via integral sliding mode. In 33rd Chinese Control Conference (CCC), 2014 (pp. 2331–2335). Isidori, A. (1995). Nonlinear control systems. London: Springer Science & Business Media. Isidori A., & Byrnes C. (1990). Output regulation of nonlinear systems. IEEE Transactions on Automatic Control, 35(2), 131–140. Jafari R., & Mukherjee R. (2012). Intermittent output tracking for linear single-input single-output non-minimum-phase systems. In American Control Conference (ACC), 2012 (pp. 5942–5947). Jeong H.-S., & Utkin V. I. (1999). Sliding mode tracking control of systems with unstable zero dynamics. In Variable structure systems, sliding mode and nonlinear control (pp. 303–327). Springer. Khalil, H. K. (2002). Nonlinear systems. Upper Saddle River, NJ: Prentice Hall. Lamour, R., März, R., & Tischendorf, C. (2013). Differential-algebraic equations: A projector based analysis. Heidelberg: Springer Science & Business Media. Levant A. (1998). Robust exact differentiation via sliding mode technique. Automatica, 34(3), 379–384. Loukianov A., Castillo-Toledo B., & Garcia-Rocha R. (1999). Output regulation in sliding mode. In Proceedings of the American Control Conference, 1999 (Vol. 2, pp. 1037–1041). Loza A. Ferreira de, Bejarano F., & Fridman L. (2009). Robust output regulation with exact unmatched uncertainties compensation based on HOSM observation. In Proceedings of the 48th IEEE conference on decision and control, 2009 held jointly with the 2009 28th Chinese Control Conference. CDC/CCC 2009 (pp. 6692–6696). Moreno J., & Osorio M. (2008). A Lyapunov approach to second-order sliding mode controllers and observers. In 47th IEEE conference on decision and control, 2008. CDC 2008 (pp. 2856–2861). Pisano A., Baev S., Salimbeni D., Shtessel Y., & Usai E. (2013). A new approach to causal output tracking for non-minimum phase nonlinear systems via combined first/second order sliding mode control. In European Control Conference (ECC), 2013 (pp. 3234–3239). Polyakov A., & Poznyak A. (2009). Reaching time estimation for supertwisting second order sliding mode controller via Lyapunov function designing. IEEE Transactions on Automatic Control, 54(8), 1951–1955. Riaza, R. (2008). Differential-algebraic systems: Analytical aspects and circuit applications. Toh Tuck Link: World Scientific. Ruiz-Duarte J. E., & Loukianov A. G. (2016). Arbitrary references output tracking in linear systems. In 13th international conference on Electrical Engineering, Computing Science and Automatic Control (CCE), 2016. Serrano-Heredia J., Loukianov A. G., & Bayro-Corrochano E. (2011). Sliding mode block control regulation of the pendubot. In 50th IEEE conference on decision and control and European Control Conference, 2011 (pp. 8249–8254). Shkolnikov I.A., & Shtessel Y.B. (2001). Tracking controller design for a class of nonminimum-phase systems via the method of system center. IEEE Transactions on Automatic Control, 46(10), 1639–1643. Shtessel Y., & Shkolnikov I. (1999). Causal nonminimum phase output tracking in mimo nonlinear systems in sliding mode: Stable system center technique. In Proceedings of the 38th IEEE conference on decision and control, 1999 (Vol. 5, pp. 4790–4795). Utkin V. (1978). Methods for constructing discontinuity planes in multidimensional variable structure systems. Automation and Remote Control, 39, 1466–1470. Utkin, V., Guldner, J., & Shijun, M. (1999). Sliding mode control in electromechanical systems (Vol. 34). Boca Raton: CRC Press. Utkin V. A., & Utkin A. (2014). Problem of tracking in linear systems with parametric uncertainties under unstable zero dynamics. Automation and Remote Control, 75(9), 1577–1592.

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Wang X., & Chen D. (2001). Causal inversion of nonminimum phase systems. In Proceedings of the 40th IEEE conference on decision and control, 2001 (Vol. 1).

Appendices

Consider now that the rank of matrix (A6) is less than n, so, there is a vector q3 such that

 s sIn − (In − G)A − G (A7) q3 = 0. τs + 1 C Since the matrix (In − G) is singular, s = 0 is the matrix Aπ eigenvalue and a solution to (A7) if

Appendix 1. Proof of Theorem 2.1

(In − G)Aq3 = 0,

This proof is based on Riaza (2008, p. 29). Consider the transformation Kϑ = z and transform the system (5) to the form LEK ϑ˙ + LFKϑ = Lq(t).

(A1)

Defining ϑ = col(ϑ1 , ϑ2 ) and Lq(t) = col(˜q1 (t), q˜ 2 (t)), the system (A1) can be written as ϑ˙ 1 + Wϑ1 = q˜ 1 (t),

(A2a)

N ϑ˙ 2 + ϑ2 = q˜ 2 (t).

(A2b)

The Equation (A2a) is an explicit linear constant coefficient ordinary differential equation (ODE) for ϑ1 ∈ md −l not involving the ϑ2 component. An initial value problem is well-defined by any ϑ1 (0) ∈ md −1 and therefore this equation has (md − l) dynamical degrees of freedom. In turn, the Equation (A2b) is decoupled from (A2a). This equation can be rewritten as (ND + I)ϑ2 = q˜ 2 (t), ν−1 

(A8)

Since the null space of the matrix (In − G) is the image of B (Im(B)), the conditions (A8) are satisfied if and only if (A4) fulfils (with q3 = q1 ), that is, the system (1) has zeros at λ = 0, which was supposed to be true. Now, consider a similarity transformation of the system (1) as z = T2−1 x, with the matrix T2−1 defined in (23). The transformed system results in z˙ = T2−1 AT2 z + T2−1 Bu, y = CT2 z.

(A9)

Define the matrix G (14) for the transformed system (A9) as ⎤ ⎡ 0 0 0 ¯ := ⎣0 0 B¯ 1 B¯ −1 G 2 ⎦, −1 ¯ ¯ 0 B2 B1 0

(A10)

i B + T i B ) and B i B + T i B ). Considering a ¯ 2 = (T32 where B¯ 1 = (T22 1 1 23 2 33 2 −1  vector q3 = T2 q3 , conditions (A8) are changed to

whit D := d/dt, and then ϑ2 = (ND + I)−1 q˜ 2 (t) =

Cq3 = 0.

(−1)j (ND)j q˜ 2 (t),

¯ −1 Aq3 = 0, (In − G)T 2

j=0

since N j = 0 for j ≥ ν. This means that ϑ2 ∈ Rl has no degree of freedom since it is completely determined from q˜ 2 (t) via the relation ϑ2 = q˜ 2 (t) − N q˙˜ 2 (t) + . . . + (−1)ν−1 N ν−1 q˜ (ν−1) (t). 2

(A3)

Note that in this framework q˜ (t) (or equivalently q(t)) must be in Cν−1 , and that in higher (ν ≥ 2) index DAEs solutions will depend explicitly on the derivatives of the excitation q(t).

Appendix 2. Proof of Theorem 3.1 Necessity. Suppose that system (1) does not satisfy Assumption 2.2. So, there is a vector q = col(q1 , q2 ) such that

Cq3 = 0.

¯ satisfies GT ¯ −1 B = T −1 B, and as (A4) is considered to be hold, The matrix G 2 2 the conditions (A11) are satisfied by the same vector q3 = q1 that (A8). Therefore, the pair {Aπ , Cπ } is unobservable for any choosing of constant τ and matrix T2 . Sufficiency. Suppose that the system (1) satisfies Assumption 2.2. Consider that there is a vector q3 , such that (A7) holds for all τ . Constant τ is changed to τ1 = τ and also s changes to s1 . Then, the first condition of (A7) becomes   s1 (In − G)Aq3 = s1 In − (A12) G q3 . τ1 s1 + 1 Assume now s = 0 and s1 = 0, since such case involves the conditions (A8). Substituting (A12) into (A7), it yields

Aq1 = Bq2 , Cq1 = 0.

for all s. Considering s = − τ1 , the rank of matrix (A5) remains after left multiplying by matrix ⎡ ⎤ τ In GH T 0 ⎢ ⎥ τs + 1 ⎣0 0 ⎦, I2m 0 0 Im resulting in the following condition:

 s G sIn − (In − G)A − rank = n. τs + 1 C

Gq3 =

(A4)

Following the well-known Popov–Belevitch–Hautus (PBH) observability rank test, the pair {Aπ , Cπ } is observable if and only if ⎤ ⎡ 1 sIn − (In − G)A − G −GH T ⎥ ⎢ τ   ⎥ ⎢ τs + 1 1 ⎥ = n + 2m, (A5) rank ⎢ ⎥ ⎢ H I 2m ⎦ ⎣ τ2 τ C 0

(A6)

(A11)

(τ s + 1)(τ1 s1 + 1)(s − s1 ) q3 , s(τ1 s1 + 1) − s1 (τ s + 1)

Cq3 = 0.

(A13)

Following (A13), the only possibility that (A7) holds for every choosing of constant τ is that vector q3 is a matrix G eigenvector. It is easy to see that the matrix G eigenvalues are 0 with algebraic multiplicity n−2m, −1 and 1, both with algebraic multiplicity m. Therefore, if q is a matrix G eigenvector, it means ⎤ 0 (a) q3 ∈ Im ⎣ B1 ⎦, −B ⎡ 2 ⎤ In−2m (b) q3 ∈ Im ⎣ 0 ⎦, 0 (c) q3 ∈ Im(B), ⎡

where (a), (b) and (c) correspond to the matrix G eigenvalues −1, 0 and 1, respectively. From Equation (A13), it can be seen that the case c) is not possible, since Im(In − G) ∩ Im(B) = ∅. Considering the cases a), b), and the similarity transformation (A9), it results in the matrix (A10) and, using the vector

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q3 = T2−1 q3 , yields

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Appendix 5. Estimation error reduction

¯ −1 q3 = (τ s + 1)(τ1 s1 + 1)(s − s1 ) T −1 q3 , GT 2 s(τ1 s1 + 1) − s1 (τ s + 1) 2 Cq3 = 0.

(A14)

By appropriately choosing matrix T2 , it is possible to avoid that the cases (a) and (b) to be satisfied for the transformed system (A9), i.e. there is T2 such that ⎤ ⎤ ⎡ ⎡ 0 In−2m −1 −1 T2 q3 ∈ / Im ⎣ B¯ 1 ⎦ , T2 q3 ∈ / Im ⎣ 0 ⎦ , 0 −B¯ 2 while, as it can be seen, the case (c) is not possible. The Equation (A14) turns then into an inequality. Therefore, the pair {Aπ , Cπ } can be made observable by choosing appropriate similarity transformation T2 and constant τ .

Appendix 3. Proof of Lemma 3.1 Using the disturbance estimator (22), the term (t) ˜ converges to zero in finite time (Levant, 1998). As the error system (29) is a linear system, it is input-to-state stable. Therefore, this term has no effect on the system stability. Then, the term G(t) must be analysed. Considering the derivative approximation filter dynamics (16), the second time derivative of πe is obtained as 1 1 1 1 1 π¨ e = − π˙ e − 2 H π(t) ˙ = 2 πe + 3 Hπ(t) − 2 H π(t). ˙ (A15) τ τ τ τ τ The term H π(t) ˙ is obtained from (A15) of the form 1 (A16) Hπ(t) − τ 2 π¨ e . τ From the matrices G (14) and H (17) forms, the following equivalences hold: G2 = H T H, G3 = G and GH T H = G3 = G. Left multiplying the Equation (A16) by the term GH T and using the previous equivalences, the term G(t) is obtained as H π˙ (t) = πe +

1 (A17) Gπ(t) = −τ 2 GH T π¨ e . τ From the Equation (28), it is possible to define the constant γ1 from the inequality τ 2 GH T π¨ e ≤ γ1 as a bound for the perturbation vector in (30). Moreover, from the linear systems theory, it is known that term π¨ e is proportional to the second derivatives of the output reference and the unmatched disturbance, then γ1 is as well. Gπ(t) ˙ − GH T πe −

Appendix 4. Proof of Theorem 3.2 The derivative of the Lyapunov function (31) along the trajectories of system (29) is V˙ 1 (πT ) = −πTT Q1 πT + 2πTT P1 g(t) ≤ −λmin (Q1 ) πT 2 + 2γ1 λmax (P1 ) πT . Adding the term (θ1 λmin (Q1 ) πT some 0 < θ1 < 1, it yields

2

− θ1 λmin (Q1 ) πT

(A18)

2 )

to (A1), for

V˙ 1 (πT ) ≤ − (1 − θ1 ) λmin (Q1 ) πT 2 + 2γ1 λmax (P1 ) πT − θ1 λmin (Q1 ) πT 2 ≤ − (1 − θ1 ) λmin (Q1 ) πT 2 < 0,

(A19)

for

2γ1 λmax (P1 ) (A20) = μ1 . θ1 λmin (Q1 ) Thus, the estimation error system solution is bounded by (Khalil, 2002)  λmax (P1 )

πT < b1 , ∀t > tf 1 , b1 = μ1 λmin (P1 ) ∀ πT >

and the bound b1 depends on the constant γ1 . Then, reference and disturbance second derivatives dependence follows from Lemma 3.1.

From the estimation error system (25), it can be seen that the filter time constant τ has a direct impact on the disturbance term. From the disturbance bound (30) it could be thought that due to the direct proportionality of the term τ 2 , the choice of τ as small as possible is going to achieve a better result. However, this time constant has an inverse effect on the term π¨ ess , i.e. the larger the constant τ , the larger the steady state for π¨ e . To find a filter time constant τ which minimises the estimation error, various algorithms could be implemented. In this Appendix, a simple one is proposed. From the disturbance g(t) bound (30), it is also easy to see that, due to the proportionality to π¨ ess , the estimation error is going to be proportional in some sense to the reference and disturbance second derivative. Therefore, consider the case when the reference yr (t) and the unmatched disturbance (t) are arbitrary periodical functions of time, and the maximum (dominant) frequency components (equivalent to the second derivative) contained in yr (t) and (t) are generated by an exosystem, that is ω˙ = Sω, yr (t) = Qω, (t) = Nω,

(A21)

where ω ∈ r is the state of the exosystem, S ∈ r×r is a constant matrix with eigenvalues on the imaginary axis, Q ∈ m×r and N ∈ p×r are constant matrices. Consider the system (1) in the absence of matched disturbance (φ(·) = 0). Define a steady state for x and u as functions of the exosystem (A1) state ω, that is, xss = ω and uss = ω, respectively, where  ∈ n×r and  ∈ m×r . Solving the corresponding Francis equations: S = A + B + DN, C = Q,

(A22)

the solution for the matrices  and  can be found. The estimator system (24) state converges to a steady state since the ˆ matrix Aπ ,2 is Hurwitz. This steady state can be represented as πˆ ss = ω ˆ e ω. In this case, the corresponding Francis equation which and πˆ e,ss =  represents the steady state solution for the estimator (24) is expressed of the form ⎡ ⎤ 1 T     (I − G)A + C GH G − L n 1 ˆ ˆ  ⎢ ⎥  τ S = ⎣ ⎦ ˆ ˆ 1 1 e e − 2 H − L2 C − I2m τ τ   (I − G)DN + L1 Q + n . (A23) L2 Q Propose desired eigenvalues and find the desired characteristic polynomial of matrix Aπ ,2 as α(s) = sn+2m + α1 sn+2m−1 + · · · + αn+2m−1 s + αn+2m . Then, compute the actual characteristic polynomial of matrix Aπ ,2 as a(s) = sn+2m + a1 sn+2m−1 + · · · + an+2m−1 s + an+2m , where the coefficients ai depends on matrix L and constant τ . To achieve that matrix Aπ ,2 characteristic polynomial a(s) equal the  desired one α(s), consider the equality a1 . . . an+2m = [α1 . . . αn+2m ]. Then, it is possible to obtain the terms of matrix L as functions of time constant τ as   L (τ ) . (A24) L= 1 L2 (τ ) Substituting (A4) into the Equation (A3) yields ⎡ ⎤ 1 T     (I − G)A + (τ )C GH G − L n 1 ˆ ˆ  ⎢ ⎥  τ S = ⎣ ⎦ ˆe ˆe 1 1   − 2 H − L2 (τ )C − I2m τ τ   (I − G)DN + L1 (τ )Q + n , (A25) L2 (τ )Q which depends only on the time constant τ . Solve the Equation (A5) and ˆ = (τ ˆ ) and find the steady state solutions as functions of τ , of the form  ˆe = ˆ e (τ ). Then, define the estimation error steady state in terms of the 

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J. E. RUIZ-DUARTE AND A. G. LOUKIANOV

˜ and express it also as a function of τ , of the form exosystem as π˜ ss = ω   ˜ )ω =  − (τ ˆ ) ω. (τ (A26) The estimation error (A6) is minimised for all ω, if the cost functional    ˜ (A27) ) J = (τ is minimised for some real positive constant τ .

Constant τ value which minimises J (A7) is substituted into (A4) to obtain the matrix L. The use of these values for τ and L in the designed steady state estimator (24) guarantees a similar error performance for real references and disturbances similar to the generated by the exosystem (A1).