Robust error-feedback arbitrary references tracking for discrete-time linear perturbed systems

Robust Error-Feedback Arbitrary References Tracking for Discrete-Time Linear Perturbed Systems

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Robust Error-Feedback Arbitrary References Tracking for Discrete-Time Linear Perturbed Systems Jorge E. Ruiz-Duarte, Alexander G. Loukianov PII: DOI: Reference:

S0016-0032(19)30741-0 https://doi.org/10.1016/j.jfranklin.2019.10.011 FI 4209

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Journal of the Franklin Institute

Received date: Revised date: Accepted date:

25 January 2018 13 May 2019 10 October 2019

Please cite this article as: Jorge E. Ruiz-Duarte, Alexander G. Loukianov, Robust Error-Feedback Arbitrary References Tracking for Discrete-Time Linear Perturbed Systems, Journal of the Franklin Institute (2019), doi: https://doi.org/10.1016/j.jfranklin.2019.10.011

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Robust Error-Feedback Arbitrary References Tracking for Discrete-Time Linear Perturbed Systems✩ Jorge E. Ruiz-Duartea,∗, Alexander G. Loukianova a CINVESTAV-IPN Unidad Guadalajara, Av. del Bosque 1145, CP 45019, Zapopan, Jalisco, Mexico

Abstract In this paper, the error-feedback output tracking problem for discrete-time linear perturbed systems with arbitrary references and unknown disturbances is approximately solved. To obtain the approximated system state error, a dynamic estimator is designed using a state observer, based on the plant model and considering the tracking error as the system output. The proposed solution is based on the approximation of a discrete-time function, using various previous steps of itself. This approximation is used in the proposed observer to estimate the reference profile of the control input. Based on the estimated state error, a discrete-time sliding mode controller is then designed to achieve the approximate tracking of the arbitrary reference. The simulations show the effectiveness of the proposed control scheme. Keywords: Sliding mode control, causal output tracking, non-minimum phase, discrete-time linear perturbed systems, arbitrary references.

1. Introduction The problem of tracking a reference while rejecting disturbances for discretetime systems using only the system output information is a challenging problem in control theory. Such problem is called the discrete-time error-feedback (or ✩ This

work was supported by CONACYT, Mexico, under grant 252405. author Email addresses: [email protected] (Jorge E. Ruiz-Duarte), [email protected] (Alexander G. Loukianov) ∗ Corresponding

Preprint submitted to Journal of the Franklin Institute

October 17, 2019

output-feedback) robust output tracking (DTEF-ROT) problem. In the case of non-minimum phase (NMP) systems, i.e., when the system internal dynamics are unstable, the DTEF-ROT problem turns into an even more difficult task, since the feedback linearization techniques are not applicable in this case. In the classical setup, it is assumed that both, reference and disturbance are generated by an exogenous and autonomous system, called exosystem. This setup is called the Error-Feedback Output Regulation (EFOR) problem. The EFOR problem was first solved by Francis [1] in the case of linear systems. In that solution, a required reference profile for the system state and for the control input are found based on the internal model principle [2]. Then, a dynamic compensator is designed. After these two steps, the EFOR problem turns into a stabilization problem. The solution conditions are derived in terms of the exosystem state through solving a set of algebraic equations. On the other hand, many real life and theoretical applications of the DTEFROT problem consider an arbitrary reference of time and an unmodelled unknown disturbance, i.e., there is no exosystem which generates neither, the reference to be tracked and the disturbance to be rejected. This problem is called the discrete-time error-feedback robust causal output tracking (DTEF-RCOT) problem, since there is knowledge of the present and past signals information, but there is no information about the future signals, as in the case of the EFOR problem. Because of this, the DTEF-RCOT problem in the case of NMP systems is a very difficult task. Since the exact tracking of arbitrary references in NMP systems is almost unrealizable, in previous works, this general problem has been converted in some different setups through considering certain constraints. In the case of continuous-time systems, Wang [3] proposed a stable causal inversion to find the required state reference profile. The disadvantage of this technique is that it deals only with references which are equal to zero after a determined time. In [4], an extension of the method of system center for NMP systems is used along with a dynamic sliding mode (SM) technique. However, such work considers only references and disturbances with a finite number of non-zero derivatives. In [5], 2

the reference and the disturbance are supposed to be generated by an unknown linear exosystem; then, using a higher order sliding mode (HOSM) parameter estimator, the exosystem characteristic polynomial is identified online. Under the mentioned constraints, the application of this technique is very limited. In [6], it is proposed an intermittent control which ensures the tracking error to zero at regular intervals of time, keeping the internal dynamics stable. However, it only guarantees output matching with the reference trajectory just for an instant. In [7], the causal output tracking problem for continuous-time linear systems is approximately solved by using a dynamic reference profile estimator based on an observer. For discrete-time systems, the robust causal output tracking problem has not been extensively studied. Papers as [8] and [9] have proposed stable inversion techniques for discrete-time systems, similar to the proposed by Wang [3] in the case of continuous-time systems. However, the DT versions have the same disadvantages that the continuous-time. Iskrenovic [10] extended the method of stable system center and SM control [4] for discrete-time systems. The disadvantage is that this work considers only a class of SISO systems and that the reference is generated by an exosystem with known characteristic polynomial. In [11], similar to the continuous-time approach [7], to estimate the system state reference profile, a dynamic estimator was designed in the form of an observer. Using the estimated state reference profile, the output tracking problem was reduced to a stabilization one. However, this approach was proposed just for linear unperturbed systems. The output-feedback case was addressed in [12] for linear systems. In that work, the authors proposed an extended error system, including an approximate of the control input required reference profile. Having the proposed extended system, an observer was designed. Using the obtained error, a DT super-twisting controller was implemented to stabilize the error system. The present paper addresses the DTEF-RCOT problem for linear MultipleInput-Multiple-Output (MIMO) perturbed systems, which is a still open problem. To solve this problem, first, a set of difference-algebraic equations is pro3

posed in order to describe the required system reference profile. Then, a system state error is defined as the difference between the system state and its required reference profile, described by the aforementioned equations. The reference profile for the control input is approximated by using several previous steps of itself, which can be considered as an extension of discrete-time SM control works as [13], [14], [15] and [16]. Using this approximation, the state error system is extended. Based on this extended system and considering the output tracking error as the system output, a dynamic estimator in the form of a state observer is designed. It is shown that the proposed estimator state converges to an approximate solution of the system state error with an ultimate bound of exponential order w.r.t. the sampling time. Finally, a discrete-time control law, using a SM algorithm [17], is designed to stabilize the closed-loop system and achieve the approximate output tracking of the arbitrary reference. The main contribution of this work is the development of a novel technique to achieve an approximate robust output tracking of an arbitrary reference signal in DT linear perturbed, minimum or nonminimum phase systems and by measuring just the system output. In the current literature, there is not a technique that solves this problem. The paper is organized as follows: Section 2 introduces the problem statement and some Assumptions which are necessary to the proposed solution. That section presents also a general solution to the DTEF-RCOT problem, showing the problems for finding an exact solution and introducing the proposed solution general structure. In Section 3, a dynamic system state error estimator, in the form of a state observer, is proposed to obtain an online approximated solution of the DTEF-RCOT problem. Then, Section 4 presents the SM control design and the closed-loop system stability analysis. In Section 5, some simulation results are presented, where the proposed tracking control scheme is applied to a non-minimum phase perturbed system. Finally, Section 6 presents the concluding remarks and future works.

4

2. Problem Statement Consider the discrete-time linear MIMO system subject to perturbation xk+1 = Axk + Buk + %k

(1)

yk = Cxk , where x ∈ Rn is the system state, u ∈ Rm is the control input, y ∈ Rm is the

system output, %k ∈ Rn is a bounded external unknown disturbance, A, B and C are constant matrices of appropriate dimensions and rank(B) = m. Consider that the system (1) fulfills the following assumptions: Assumption 1. The pair {A, B} is controllable. Assumption 2. The pair {C, A} is observable. Assumption 3. Matrices A, B and C satisfy   In − A B  = n + m. rank  C 0m×m

Remark 1. Assumption 3 implies that the realization (A, B, C) is invertible, i.e.,

zIn − A B 6≡ 0, C 0m×m

and that the system (1) has no zeros z = 1.

Let yr,k ∈ Rm represent an arbitrary bounded reference to be tracked by the system (1). Defining the tracking error as εk = yk − yr,k ,

(2)

a general solution for the DTEF-RCOT problem can be formulated as to find an error-feedback control law, which satisfies the following conditions: (Sef ) The closed-loop system in absence of output reference and disturbance is asymptotically stable;

5

(Tef ) The tracking error (2) asymptotically tends to zero, i.e., lim εk = 0.

k→∞

(3)

Definition 1. A time-function vector πk ∈ Rn is said to be a reference profile of the system (1) state xk if, for a control input uk = −Kxk + ϑk , the system solution satisfies lim (xk − πk ) = 0,

k→∞

where ϑk ∈ Rm is a time-function vector and K is a constant matrix of appropriate dimensions, such that (A − BK) is a Schur matrix. A time-function

vector ck ∈ Rm is said to be a reference profile of the control input uk if, for

xk = πk , uk = ck . In the mentioned case ck = −Kπk + ϑk . Remark 2. Note that the control input proposed in Definition 1 can be generalized to any control law, static or dynamic, which imposes stability at the closed-loop system. It is worth to note that the reference yr,k arbitrariness in (2) does not permit to compute the required reference profiles for the system state and the control input by using a classic method as output regulation [1]. Therefore, in this work, based on the output regulation classical approach the following lemma is presented: Lemma 1. Let the system (1) satisfies Assumption 1. Then, the output tracking and disturbance rejection problem represented by the conditions (Sef ) and (Tef ) is solvable if there exists a stable reference profile for state xk given by πk = xrp,k and a bounded control input reference profile ck = urp,k , such that the following difference-algebraic equations (DAE) system is satisfied: πk+1 = Aπk + Bck + %k 0 = Cπk − yr,k . Proof. see Appendix A

6

(4)

It can be seen in (4) that the arbitrary output reference yr,k and the unknown disturbance %k have a direct influence on the solution of the equations (called reference profile equations). Therefore, it is not possible to find offline the reference profile solutions πk and ck , which are necessary for the control law design, even if the disturbance is known. For minimum phase systems, the tracking problem can be easily solved by using some of the robust input-output control strategies as feedback-linearization, optimal control, etc without solving the reference profile equations (4). However, the NMP systems require to find a reference profile by solving the DAE (4). The NMP systems are the main application of the method presented in this work. Of course, the proposed method is also applicable for minimum phase systems. Definition 2 (Definition 1.2 [18]). Given any ordered pair {E, F } of matrices

E, F ∈ Rn×n , the matrix pencil λE + F is said to be regular if the polynomial p(λ) = det(λE + F ) does not vanish identically. Otherwise the matrix pencil is said to be singular. The pair {E, F } is said to be regular if the accompanying matrix pencil is regular, and otherwise nonregular. Remark 3. Existence of the DAE (4) solution depends on the regularity of the pair {E, F } (see [19]), with    In 0n×m A  and F = −  E= 0m×n 0m×m C

B 0m×m



.

Under Assumption 3, the regularity condition for the equation (4) is satisfied; therefore, its solution exists. A general solution for the DTEF-RCOT problem is presented in the following theorem: Theorem 1. Let the system (1) satisfies Assumptions 1-3. Define πk and ck as a stable solution of (4). Then, the DTEF-RCOT problem can be solved by

7

the following dynamic control law uk = N η k

(5)

ηk+1 = Dηk + Gεk + λk , where ηk ∈ Rs , and λk ∈ Rs is used to consider the causal nature of the reference and the disturbance which are not generated by any exosystem, N , D and G are constant matrices of appropriate dimensions. Proof. See Appendix B. Since the term λk in (5) is unknown, the DTEF-RCOT problem cannot be solved exactly, as mentioned above. Then, the control objective of this paper is to provide the system (1) with an approximate output tracking of an arbitrary reference yr,k ∈ Rm by just measuring the tracking error εk , and implementing discrete-time sliding mode control (DTSMC) techniques to achieve robustness. The proposed solution to achieve the control objective is to design a dynamic control of the form uk = Sat(N ηk )

(6)

ηk+1 = D1 ηk + D2 uk + Gεk , such that the sliding variable σk = Φek satisfies kσk k < , ∀k > ks ,

(7)

and, inside the boundary layer (7), the following conditions are satisfied: (Ssm ) the closed-loop system (1) and (6) in absence of output reference yr,k and disturbance %k is asymptotically stable (Tsm ) the tracking error εk tends to an ultimate bound kεk k < 1 , ∀k > ks + kf ,

(8)

where η ∈ Rs , σk ∈ Rm , Sat(·) represents a saturation on the control law, according with its physical constraints,  and 1 are small nonnegative constants, ks and kf are positive integers and N , D1 , D2 , G and Φ, are constant design matrices of appropriate dimensions. 8

3. System State Error Observer In this section an observer for the system state error will be designed. First, a novel method to estimate DT functions is presented. Using the mentioned method, the required reference profile of the control input ck is estimated by a dynamic system, extending the error system dimension. Then, the observer is designed for the extended system. 3.1. Discrete-Time Functions Approximation In several works such as [13], [14] and [15], discrete-time functions of the type fk = f (k∆t) have been approximated using one previous step of itself, i.e., fk ≈ fk−1 ; obtaining an error of the order O(∆t), where ∆t is the sampling time of the discretization. In [16], the discrete-time function is approximated using two previous steps of itself, of the form fk ≈ 2fk−1 − fk−2 , resulting in an approximation error of the order O(∆t2 ). Based on these works, a discrete-time function can be approximated by using h previous steps of itself, of the form fk ≈ −a1 fk−1 − . . . − ah fk−h ,

(9)

where constants ai results from (1−z −1 )h = 1+a1 z −1 +a2 z −2 +. . .+ah z −h and h is the order of the desired approximation. This fact is shown in the following lemma: Lemma 2. Let f (t) be a continuous-time smooth function. Let fk = f (k∆t) be the discretization of f (t) with a sampling time ∆t. Define the binomial expansion (1 − z −1 )h = 1 + a1 z −1 + a2 z −2 + . . . + ah z −h , for some positive integer h. Then, ∆hf,k := fk + a1 fk−1 + . . . + ah fk−h = O(∆th ).

(10)

Proof. See Appendix C. 3.2. Extended Error System Construction Based on the solvability of the reference profile equations (4), the robust causal output tracking problem can be converted into a state stabilization problem. Define the system state error as ek = xk − πk and consider the tracking 9

error definition (2). Then, using the equations (1) and (4), the error dynamics are expressed as ek+1 = Aek + B(uk − ck )

(11)

εk = Cek . From the theory of difference-algebraic equations ([20] and [18]) and descriptor systems ([19] and [21]), it is known that the solution of ck in (11) depends on the functions yr,k and %k . Using the function estimation (9), the term ck can be approximated using h previous steps of itself as ck ≈ −a1 ck−1 − a2 ck−2 − . . . − ah ck−h ,

(12)

where the constants ai are defined as the coefficients of (1 − z −1 )h = 1+ a1 z −1 + a2 z −2 + . . . + ah z −h . In fact, the equation (12) is an h-order dynamic equation h i h i which by defining the variables ς1,k , . . . , ςh,k = ck , . . . , ck−h+1 , is represented by the dynamic system

ς1,k+1 = −a1 ς1,k − a2 ς2,k − . . . − ah ςh,k + ∆hc,k+1 ς2,k+1 = ς1,k .. .

(13)

ςh,k+1 = ςh−1,k ck = ς1,k , which consists of h scalar equations, with ∆hc,k+1 = ck+1 +a1 ck +. . .+ah ck−h+1 . Extending the error system (11) with the ck approximate (13), results in the following extended system: ¯ k + Bu ¯ k + δk ξk+1 = Aξ ¯ k, εk = Cξ

10

(14)

where ξk = col(ek , ςk ) ∈ Rn+mh , ςk = col(ς1,k , . . . , ςh,k ) ∈ Rmh ,  A     0  0n×1    0   δk =  ∆hc,k+1  , A¯ =     0  0m(h−1)×1 )  .. . 

−B

0

···

0

−a1 Im

−a2 Im

···

−ah−1 Im

Im

0

···

0

0 .. .

Im .. .

··· .. .

0 .. .

0

0

···

Im

0





h ¯=  and C¯ = C B 0mh×m B

0



  −ah Im    0  ,  0   ..  .   0

i 0m×mh .

To obtain an approximated estimate of the system state error (11), a state observer for the extended system (14) will be designed. 3.3. State Error Observer Design The observability of the system (14) is analyzed in the following lemma: ¯ A} ¯ Lemma 3. Let the system (1) satisfies Assumption 3. Then, the pair {C, is observable if and only if the pair {C, A} is observable. Proof. See Appendix D. Then, under Assumptions 2 and 3, a state observer is designed for the system (14), of the form ¯ k + L(εk − C¯ ξˆk ), ξˆk+1 = A¯ξˆk + Bu

(15)

where ξˆk is the estimate of ξk and L ∈ R(n+mh)×m is the observer gain matrix.

¯ is Schur with desired eigenvalues. Matrix L must be chosen such that (A¯ − LC)

This can be achieved by using the Bass-Gura or the Ackerman approaches in ¯ A} ¯ to the observer form in the the scalar case or transforming the pair {C, multivariable case. The next step is to analyze the observer (15) convergence.

11

3.4. Observer Convergence Analysis Define the observation error as ξ˜k = ξk − ξˆk . Then, using (14) and (15), the observation error dynamics are obtained as ¯ ξ˜k + δk , ξ˜k+1 = (A¯ − LC)

(16)

where the term δk is considered as a bounded disturbance. Lemma 4. Let the output reference yr,k and disturbance %k derivatives greater or equal than h be bounded of the form

i

i

d yr,k



≤ ci , d %k ≤ di , i = h, h + 1, ...,

dti

dti

where ci , di are nonnegative constants. Then, the disturbance term δk in (16) is bounded by kδk k ≤ γ1 ,

(17)

where γ1 is a small nonnegative constant of the order O(∆th ). Proof. Follows from Lemma 2. The stability of the system (16) is analyzed in the following theorem: Theorem 2. Let the system (1) satisfies Assumptions 2 and 3. Consider that the matrix L is chosen such that A¯2 := A¯ − LC¯ is a Schur matrix. Then, the estimation error system (16) has a Globally Ultimately Bounded solution, with an ultimate bound b1 of the order O(∆th ). Proof. See Appendix E. Since the constant b1 is of the order O(∆th ), by decreasing the sampling time ∆t or increasing the approximation order h it is possible to decrease the observer error ultimate bound and therefore, to achieve a better estimation of the system state error.

12

4. Tracking Control 4.1. Sliding Mode Control Design After the system state error estimator (15) was designed, to solve the EFDTRCOT problem, a robust control law, which achieves that conditions (Ssm ) and (Tsm ) be satisfied, will be designed. This problem solution is equivalent to achieve that the state error ek = xk − πk , described by the system (11), converges to a neighborhood of the origin ek = 0, i.e., kek k < 2 , ∀k > kf ,

(18)

for another small nonnegative constant 2 . Consider that matrix B is represented as   B1 B =   , B1 ∈ R(n−m)×m , B2 ∈ Rm×m , rank(B2 ) = m; B2

then by using the similarity transformation ζk = T ek , with   In−m −B1 B2−1  T = 0 Im

the error system (11) is represented into Regular form [22] as ζ1,k+1 = A11 ζ1,k + A12 ζ2,k ζ2,k+1 = A21 ζ1,k + A22 ζ2,k + B2 (uk − ck ),

(19)

where ζk = col(ζ1,k , ζ2,k ), ζ1,k ∈ Rn−m , ζ2,k ∈ Rm ,     A A 0 0 11 12  and T B =  (n−m)×m  . T AT −1 = A =  A21 A22 B2 Using (19), a sliding variable σk is formulated as σk = Φζk , h where σk = col(σ1,k , . . . , σm,k ) ∈ Rm , Φ = K constant design matrix.

13

(20) i Im and K ∈ Rm×(n−m) is a

From the equation (19), the sliding variable (20) dynamics are obtained of the form σk+1 = ΦA0 ζk + B2 (uk − ck ).

(21)

The next step is to design a control law such that system (19) reaches a boundary layer of the sliding manifold  Σ := ζk σk = 0 ,

(22)

for all k > ks , and some finite positive integer ks . Define an estimated transformed error as

h where H1 = In

ζˆk = T −1 eˆk = T −1 H1 ξˆk = ζk − T −1 H1 ξ˜k , i 0n×mh ∈ Rn×(n+mh) .

(23)

Then, using (15) and (23), an estimated equivalent control is obtained from

the equation σk+1 = 0 (21) as

h

where H2 = 0m×n

Im

u ˆeq,k = H2 ξˆk − B2−1 ΦA0 ζˆk , i 0m×m(h−1) .

(24)

Considering the system actuators physical constraints, the control law uk is implemented as the saturation of the estimated equivalent control (24) as [17]    u ˆeq,k if kˆ ueq,k k ≤ umax  , (25) uk = u ˆeq,k   if kˆ ueq,k k > umax umax kˆ ueq,k k

where umax is a real-life bound for the control uk , i.e., kuk k ≤ umax .

It can be shown that, in the case kˆ ueq,k k > umax , under the condition kB2−1 k·

k(ΦA0 − Φ)ζˆk − B2 H2 ξˆk k < umax , the control (25) produces a monotonous decreasing in the sliding variable norm kσk k and, after a finite number of steps, the condition kˆ ueq,k k ≤ umax is achieved [17]. After these steps, the closed-loop system state (21) and (25) converges to σk+1 = ΦA0 ζ˜k − B2 H2 ξ˜k = H ξ˜k , where H = [ΦA0 T −1

−B2

0m×m(h−1) ] ∈ Rm×(n+mh) . 14

(26)

4.2. Closed-Loop System Stability Analysis After the sliding variable dynamics (21) and (25) converge to (26), it is possible to analyze the complete closed-loop system, formed by the observer error (ξ˜k ) system (16), the sliding variable (σk ) dynamics (26) and, using the change of variable ζ2,k = −Kζ1,k + σk , the system state error (ζ1,k ). This closed-loop system results in    A − A12 K ζ1,k+1   11     ˜  ξk+1  =  0    0 σk+1

0 A¯2 H

    0 ζ1,k      ˜    0   ξk  + δk  .     0 σk 0

A12

For the sake of simplicity define      A − A12 K ζ1,k 0  11       ˜    χk =  ξk  , gk = δk  and Acl =  0      0 0 σk

Then, the system (27) is represented in general form as

0 A¯2 H

(27)

A12



  0 .  0

χk+1 = Acl χk + gk .

(28)

From Lemma 4, it follows that kgk k ≤ γ1 .

(29)

The stability of the system (28) is studied in the following theorem: Lemma 5. The system state error dynamics, described by the equation (28), have a Globally Ultimately Bounded solution with an ultimate bound of the order O(∆th ). Proof. Under Assumption 1 it is easy to verify that the pair {A11 , A12 } is controllable [22]. Then, it is possible to find a matrix K such that (A11 − A12 K) is

a Schur matrix. Besides, under Lemma 3, matrix A¯2 can be Schur as well. The rest m eigenvalues of the system (27) correspond to the σk dynamics, and are equal to zero. Therefore, the matrix Acl is Schur. 15

Consider the candidate Lyapunov function V2 = α2 kχk k,

(30)

1/2

where α2 = (1 − λmax (ATcl Acl ))−1 is a positive constant. Similar to the proof of Theorem 2, the system (28) solution ultimate bound is calculated as kχk k < b2 , ∀k > kf 2 , b2 =

α2 γ1 , θ2

for some finite step kf 2 and constant 0 < θ2 < 1. The order of the ultimate bound follows from (29). Lemma 5 focuses on the error stability analysis, satisfying the condition (Tsm ). However, in general, the error stability does not ensure the original system (1) stability. Therefore, the closed-loop system (1) and (15) stability to satisfy the condition (Ssm ) is analyzed in the following Lemma: Lemma 6. Under Assumptions 1-3 the closed-loop system including the controlled system (1), the designed observer (15) and the control input (25), in absence of output reference and disturbance, is exponentially stable. Proof. See Appendix F. 4.3. Final Results Using the system state error observer (15), the sliding mode control law (25) and defining 2 = b2 , the control objective is satisfied, as the following theorem shows: Theorem 3. Let the system (1) satisfies Assumptions 1-3. Assume that the output reference yr,k and the disturbance %k are the discretization of smooth signals with appropriate sampling time ∆t. Let matrices L and K be chosen ¯ and (A11 − A12 K) are Schur matrices. Then, the dynamic such that (A¯ − LC) control law (15) and (25) satisfies the conditions (Ssm ) and (Tsm ) and therefore, the control objective (7)-(8), with ε and ε1 of the order O(∆th ). Proof. Follows from Lemmas 2, 3, 5 and 6. 16

yr,k

State error observer

ξˆk

Sliding mode controller

uk

yk

Plant

Figure 1: Block Diagram of the Proposed Control Scheme.

The proposed control scheme is presented in Figure 1. It is worth to note that the proposed control law (15) and (25) can be ¯ D2 = B, ¯ represented into the general form (6) with ηk = ξˆk , D1 = A¯ − LC, G = L and N defined in (F.1). Besides, with the knowledge of the term λk = δk , such control law solves exactly the DTEF-RCOT problem.

5. Simulation Results To show the effectiveness of the proposed control scheme, a simulation was performed. Consider the non-minimum phase linear discrete-time perturbed system       1.0607 −0.03383 0.0206 0  xk +   uk +   xk+1 =  0.0789 1.196 0.00076 0.01 cos(0.002k) h i yk = 1 0 xk .

(31)

The output reference was proposed as the function yr,k = 2sin(0.003k) + −7

2sin(0.0015k) + 2cos(0.003k) + 70tan−1 (0.0008k) + 70e−5×10

(k−15000)2

− 2,

which cannot be generated by any exosystem. The approximations order was proposed as h = 4. With desired eigenvalues

17

at 0.9, the matrices A¯ and L resulted in     0.856 1.0607 −0.03383 −0.0206 0 0 0         −13.35 0.0789 1.196 −0.00076 0 0 0           7.2   0 0 4 −6 4 −1 ¯ .    A=   and L =   6.77   0 0 1 0 0 0          6.35   0 0 0 1 0 0     5.94 0 0 0 0 1 0 For the control law, the transformation matrix was chosen as   0 0.0206 , T = 0.01 0.00076

and the sliding variable was designed as σk = 1.718ζ1,k + ζ2,k . Finally, the control input was saturated with umax = 250.

The following simulations show the performance of the proposed control scheme. Figure 2 shows the comparison between the arbitrary reference profile yr,k and the actual system output yk . 20

15

10

5

20 0

0 -20 -5

-10

0

100

0.5

1

yr,k yk

200 1.5

Step (k) Figure 2: Output tracking.

18

2

2.5 ×10 4

The system state error response is shown in Figure 3. It can be seen that the steady state response is reached in less than 250 steps. 5

×10 -4

0

-5

20 0

-10 -20

-15

0

100

0.5

1

200 1.5

Step (k)

2

2.5 ×10 4

Figure 3: Tracking error (εk ).

In Figure 4, the estimated system state error response for both system states, is presented. Figure 5 shows the estimated sliding variable (ˆ σk = Φζˆk ) behavior. It can be seen that it reaches a small boundary layer in less than 200 steps. To show that the proposed technique achieves the output tracking while stabilizing the controlled system, the system state response is presented in Figure 6. Finally, the applied control input response is shown in Figure 7. 6. Conclusions The present paper proposed an approximate solution to the causal output tracking problem for discrete-time linear perturbed systems using error feedback. Taking the tracking error as the measured system output, a state observer was designed to estimate the system state error and the control input reference profile, despite the disturbance. The proposed estimator is based on 19

×10 -4 2 0 -2 -4

1

eˆ1,k 0

0.5

1

1.5

2

2.5 ×10 4

×10 -4

0

-1

eˆ2,k 0

0.5

1

1.5

2

Step (k)

2.5 ×10 4

Figure 4: Estimated reference profile error.

the approximation of a discrete-time function by using various previous steps of itself. Then, a discrete-time sliding mode controller was designed to ensure the closed-loop system stability and that the system state error is ultimately bounded by a small constant of the order O(∆th ). Future works will extend these results for the case of nonlinear discrete-time systems.

Appendix A. Proof of Lemma 1 Define the state error as ek = xk − πk with dynamics ek+1 = Aek + B(uk − ck )

(A.1)

εk = Cek . Under Assumption 1, if there exist πk and ck as a stable solution of (4), a control law can be proposed such that the error system (A.1) is stabilized as uk = ck − K(xk − πk ),

20

(A.2)

2

×10 -3

1 0 -1 -2 -3 -4

500

0

-500

0

100

-5

0.5

200 1

1.5

2

Step(k)

2.5 ×10 4

Figure 5: Estimated sliding variable (ˆ σk ).

where K is a constant matrix of appropriate dimensions, such that (A − BK) is a Schur matrix. The closed-loop system (A.1) and (A.2) becomes ek+1 = (A − BK)ek

(A.3)

εk = Cek , which satisfies the condition (Tef ). In absence of reference and disturbance, the control (A.2) becomes uk = −Kxk and, under Assumption 1, the closed-loop system results exponentially stable, satisfying the condition (Sef ). Appendix B. Proof of Theorem 1 As it was stated in Remark 3, Assumption 3 implies that there exist πk and ck as solutions to (4). Define a reference profile for the variable ηk as πη,k = ηss,k , satisfying πη,k+1 = Dπη,k + λk .

(B.1)

From (5), it follows that the reference profile of the control input uk satisfies ck = N πη,k . Note that the term λk in (B.1) ensures that ck can be generated 21

20 0 -20 -40

x1,k 0

0.5

1

1.5

2

2.5 ×10 4

5

x2,k

0 -5 -10

0

0.5

1

1.5

2

Step (k)

2.5 ×10 4

Figure 6: System state (xk ).

despite there is no an exosystem. Then, the extended reference profile equations (4) and (B.1) become πk+1 = Aπk + BN πη,k + %k πη,k+1 = Dπη,k + λk

(B.2)

0 = Cπk − yr,k . Define the errors for the system state xk and for the dynamic control state ηk as ek = xk − πk and eη,k = ηk − πη,k , respectively. The error system results in ek+1 = Aek + BN eη,k eη,k+1 = Deη,k + GCek

(B.3)

εk = Cek . From linear control theory it is known that, under Assumptions 1 and 2, it is possible to find matrices D, N and G such that the matrix   A BN   GC D

(B.4)

has desired eigenvalues inside the unitary circle, obtaining a Schur matrix. Having a Schur matrix, the system (B.3) will be exponentially stable and the output tracking will be achieved asymptotically, satisfying the condition (Tsm ). 22

60 40

200

20

0

0

-200

-20

0

100

200

-40 -60 -80 -100

0.5

1

1.5

2

Step(k)

2.5 ×10 4

Figure 7: Control law (uk ).

To ensure the system stability presented in the condition (Ssm ), consider the closed-loop system (1) and (5), represented in the absence of reference and disturbance, as xk+1 = Axk + BN ηk

(B.5)

ηk+1 = GCxk + Dηk . Since the matrix (B.4) is Schur, the closed-loop system (B.5) is exponentially stable. Appendix C. Proof of Lemma 2 To prove this lemma, it will be demonstrated that ∆hf,k = where

dh fk dth

=



dh f (t) , dth t=k∆t

Step h = 1.

dh fk h ∆t + O(∆th+1 ) = O(∆th ), dth

(C.1)

using mathematical induction.

Consider the Taylor series of the continuous time function f (t) at the instant (k − 1)∆t, of the form (see [23]) fk−1 = fk −

dfk ∆t + O(∆t2 ). dt

23

Then, ∆1f,k results in ∆1f,k = fk − fk−1 =

dfk ∆t + O(∆t2 ) = O(∆t). dt

(C.2)

Step h = i + 1. Assume that the expression (C.1) is satisfied for h = i, i.e., ∆if,k =

di fk i ∆t + O(∆ti+1 ) = O(∆ti ). dti

(C.3)

Step h = i + 1. Using (i + 1) previous steps to approximate fk , the approximation error is defined as i i ∆i+1 f,k = fk + b1 fk−1 + . . . + bi+1 fk−i−1 = ∆f,k − ∆f,k−1 ,

(C.4)

where the coefficients bi obtained from (1 − z −1 )i+1 = 1 + b1 z −1 + . . . + bi+1 z −i−1 . Using now (C.2) and (C.3), (C.4) becomes ∆if,k − ∆if,k−1 =

d∆if,k ∆t + O(∆if,k )O(∆t2 ). dt

Substituting ∆if,k (C.3) into (C.5), results in ∆i+1 f,k =

di+1 fk i+1 ∆t + O(∆ti+2 ) = O(∆ti+1 ). dti+1

Therefore, (C.1) is satisfied for any h. Appendix D. Proof of Lemma 3 Represent the matrix A¯ (14) of the form  h i A −B 0 n×m(h−1)  A¯ =  , 0mh×n A1

where

 −a1 Im    Im   A1 =  0   ..  .  0

−a2 Im

···

−ah−1 Im

−ah Im

0

···

0

0

Im .. .

··· .. .

0 .. .

0 .. .

0

···

Im

0

24



     .    

(C.5)

Using the well-known Popov-Belevitch-Hautus (PBH) observability test, the ¯ A} ¯ is observable if and only if pair {C,   zIn+mh − A¯  = n + mh, ∀z ∈ C. rank  C¯

(D.1)

It is easy to verify that the submatrix A1 has all its eigenvalues at 1. Therefore, there are two important cases to analyze the condition (D.1): (1) z = 1. In this case, rank(Imh − A1 ) = m(h − 1) and the condition (D.1) becomes    I −A In+mh − A¯  = m(h − 1) + rank  n rank  C C¯

B 0m×m

¯ A} ¯ can be observable only if From (D.2), the pair {C,   In − A B  = n + m, rank  C 0m×m



 . (D.2)

which is fulfilled under Assumption 3.

(2) z 6= 1. In this case, the condition (D.1) becomes     zIn − A zIn+mh − A¯  ,  = mh + rank  rank  C C¯

(D.3)

and it is satisfied if and only if   zIn − A  = n, rank  C

that is, if and only if the pair {C, A} is observable. Appendix E. Proob of Theorem 2 Consider the candidate Lyapunov function V1,k = α1 kξ˜k k 25

(E.1)

1/2 where the constant α1 = (1 − λmax (A¯T2 A¯2 ))−1 and λmax (A¯T2 A¯2 ) is the matrix

A¯T2 A¯2 greater eigenvalue. Therefore, the constant α1 is positive if and only if the matrix A¯2 is Schur. The Lyapunov function difference ∆V1 = V1,k+1 − V1,k is ∆V1 = α1 kA¯2 ξ˜k + δk k − α1 kξ˜k k. Using the norm properties [24], it results in ∆V1 ≤ α1 kA¯2 kkξ˜k k + α1 kδk k − α1 kξ˜k k ≤ −kξ˜k k + α1 γ1 ≤ −(1 − θ1 )kξ˜k k − θ1 kξ˜k k + α1 γ1 ≤ 0,

(E.2)

for kξ˜k k ≥

α 1 γ1 = b1 , θ1

(E.3)

where 0 < θ1 < 1. From the equations (E.2) and (E.3), there exists a finite step kf 1 such that kξ˜k k < b1 , ∀k > kf 1 .

(E.4)

Therefore, the estimation error is Globally Ultimately Bounded with a bound dependent on the constant γ1 (17).

Appendix F. Proof of Lemma 6 After a finite time, the control input (25) will satisfy kuk k ≤ umax . When this happens, the control input can be represented as uk = N ξˆk ,

(F.1)

where N = H2 − B2−1 ΦT −1 AH1 . The matrix N design generates that the

matrix (A + BN H1T ) is Schur, with m eigenvalues at 0 and the rest given by

the choosing of the matrix K in (20). Then, the closed-loop system (1), (15) and (F.1) results in xk+1 = Axk + BN ξˆk + %k ¯ )ξˆk − Lyr,k . ξˆk+1 = LCxk + (A¯ − LC¯ + BN 26

(F.2)

Since the output reference yr,k and the disturbances vector %k are bounded time functions, the closed-loop system stability is analyzed in the absence of these, i.e., considering yr,k = 0 and %k = 0. Then, the closed-loop stability is achieved if and only if the matrix 

A

F = LC

BN ¯ A¯ − LC¯ + BN

is Schur. Using the nonsingular matrix 

In

M = M −1 =  H1T

 

0 −In+mh



,

the matrix F has the same eigenvalues that the matrix   A + BN H1T −BN . MFM =  0 A¯ − LC¯

Therefore, under Assumptions 1-3, F is a Schur matrix and the closed-loop system (F.2), in the absence of yr,k and %k , is exponentially stable. References [1] B. Francis, The linear multivariable regulator problem, in: Decision and Control including the 15th Symposium on Adaptive Processes, 1976 IEEE Conference on, 1976. [2] B. A. Francis, W. M. Wonham, The internal model principle of control theory, Automatica 12 (5). [3] X. Wang, D. Chen, Causal inversion of nonminimum phase systems, in: Decision and Control, 2001. Proceedings of the 40th IEEE Conference on, Vol. 1, 2001. [4] Y. Shtessel, I. Shkolnikov, Causal nonminimum phase output tracking in mimo nonlinear systems in sliding mode: stable system center technique, in: Decision and Control, 1999. Proceedings of the 38th IEEE Conference on, Vol. 5, 1999. 27

[5] S. Baev, Y. Shtessel, J. Shkolnikov, Hosm driven output tracking in the nonminimum-phase causal nonlinear systems, in: Decision and Control, 2007 46th IEEE Conference on, 2007. [6] R. Jafari, R. Mukherjee, Intermittent output tracking for linear singleinput single-output non-minimum-phase systems, in: American Control Conference (ACC), 2012, 2012. [7] J. E. Ruiz-Duarte, A. G. Loukianov, Arbitrary references output tracking in linear systems, in: Electrical Engineering, Computing Science and Automatic Control (CCE), 2016 13th International Conference on, IEEE, 2016. [8] T. Sogo, Stable inversion for nonminimum phase sampled-data systems and its relation with the continuous-time counterpart, in: Decision and Control, 2002, Proceedings of the 41st IEEE Conference on, Vol. 4, 2002. [9] L. Marconi, G. Marro, C. Melchiorri, A solution technique for almost perfect tracking of non-minimum-phase, discrete-time linear systems, International Journal of Control 74 (5) (2001) 496–506. [10] O. Iskrenovic-Momcilovic, C. Milosavljevic, Y. Shtessel, Discrete-time variable structure control for casusal nonminimum phase systems using stable system center, in: Variable Structure Systems, 2006. VSS’06. International Workshop on, 2006. [11] J. E. Ruiz-Duarte, A. G. Loukianov, Causal output tracking for discretetime linear systems, in: American Control Conference (ACC), 2017, IEEE, 2017. [12] J. E. Ruiz-Duarte, A. G. Loukianov, Error-feedback robust causal output tracking for discrete-time linear systems, IFAC-PapersOnLine 50 (1) (2017) 14507 – 14512, 20th IFAC World Congress.

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