Sliding mode controller design for linear systems with unmeasured states

Sliding mode controller design for linear systems with unmeasured states

Journal of the Franklin Institute 349 (2012) 1337–1349 www.elsevier.com/locate/jfranklin Sliding mode controller design for linear systems with unmea...

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Journal of the Franklin Institute 349 (2012) 1337–1349 www.elsevier.com/locate/jfranklin

Sliding mode controller design for linear systems with unmeasured states Michael Basin, Pablo Rodriguez-Ramirez Department of Physical and Mathematical Sciences, Autonomous University of Nuevo Leon, San Nicolas de los Garza, Nuevo Leon, Mexico Received 4 December 2010; received in revised form 5 May 2011; accepted 17 June 2011 Available online 25 June 2011

Abstract This paper addresses the optimal controller problem for a linear system over linear observations with respect to different Bolza–Meyer criteria, where (1) the integral control and state energy terms are quadratic and the non-integral term is of the first degree or (2) the control energy term is quadratic and the state energy terms are of the first degree. The optimal solutions are obtained as sliding mode controllers, each consisting of a sliding mode filter and a sliding mode regulator, whereas the conventional feedback LQG controller fails to provide a causal solution. Performance of the obtained optimal controllers is verified in the illustrative example against the conventional LQG controller that is optimal for the quadratic Bolza–Meyer criterion. The simulation results confirm an advantage in favor of the designed sliding mode controllers. & 2011 The Franklin Institute. Published by Elsevier Ltd. All rights reserved.

1. Introduction Since the sliding mode control was invented in the beginning of 1970s (see a historical review in [1–3]), it has been applied to solve several classes of problems. For instance, the sliding mode technique is actively used in observation and identification methods [4–9]. Application of the sliding mode technique is also extended to stochastic systems [10–12] and stochastic filtering problems [13,14]. However, although it is possible to design a sliding manifold so that an infinite-horizon quadratic cost functional including the system Corresponding author.

E-mail addresses: [email protected] (M. Basin), [email protected] (P. Rodriguez-Ramirez). 0016-0032/$32.00 & 2011 The Franklin Institute. Published by Elsevier Ltd. All rights reserved. doi:10.1016/j.jfranklin.2011.06.019

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state only is minimized [1], it seems, to the best of authors’ knowledge, that no sliding mode controller algorithms, solving the optimal controller problem for a Bolza–Meyer criterion with the quadratic control term [15,16], have been designed. Two of those optimal controller problems are considered in this paper, while the companion paper [17] presents a sliding mode regulator being the solution to an optimal control problem for linear systems. This paper presents the solutions to the optimal controller problems for a linear system over linear observations with respect to different Bolza–Meyer criteria, where (1) the integral control and state energy terms are quadratic and the non-integral term is of the first degree or (2) the control energy term is quadratic and the state energy terms are of the first degree. That type of criteria would be useful in the joint control and parameter identification problems where the objective should be reached for a finite time. The theoretical results are complemented with illustrative examples verifying the performance of the designed controller algorithms. The designed sliding mode controllers are compared to the feedback LQG controller corresponding to the quadratic Bolza–Meyer criterion, which is based on the Kalman–Bucy filter [18] and the conventional LQ regulator [15,16]. The simulation results confirm an advantage in favor of the designed sliding mode controllers. The paper is organized as follows. Section 2 states the optimal controller problems for a linear system over linear observations with non-quadratic Bolza–Meyer criteria and provides the sliding mode solutions. Sections 3 and 4 present mean-square and meanmodule controller design, respectively. Both sections contain theoretical substantiation and illustrative examples. 2. Optimal controller problem 2.1. Problem statement Let ðO,F ,PÞ be a complete probability space with an increasing right-continuous family of s-algebras Ft ,tZt0 , and let ðW1 ðtÞ,Ft ,tZt0 Þ and ðW2 ðtÞ,Ft ,tZt0 Þ be independent Wiener processes. The Ft-measurable random process ðxðtÞ,yðtÞÞ is described by a linear differential equation for the system state dxðtÞ ¼ aðtÞxðtÞ dt þ BðtÞuðtÞ dt þ bðtÞ dW 1 ðtÞ,

xðt0 Þ ¼ x0 ,

ð1Þ

and a linear differential equation for the observation process dyðtÞ ¼ AðtÞxðtÞ dt þ GðtÞ dW 2 ðtÞ:

ð2Þ

Here, xðtÞ 2 Rn is the state vector, uðtÞ 2 Rl is the control input, and yðtÞ 2 Rm is the linear observation vector, mrn. The initial condition x0 2 Rn is a Gaussian vector such that x0, W1 ðtÞ 2 Rp , and W2 ðtÞ 2 Rq are independent. The observation matrix AðtÞ 2 Rmn , mrn, is supposed to be of the full rank. It is assumed that GðtÞG T ðtÞ is a positive definite matrix, therefore, mrq. All coefficients in Eqs. (1) and (2) are deterministic functions of appropriate dimensions. Without loss of generality, the system (1) (pair ðaðtÞ,BðtÞÞ) is assumed to be controllable almost everywhere for tZt0 , i.e., the uncontrollable state components are removed from the consideration. The state and observation equations can also be written in an alternative form _ ¼ aðtÞxðtÞ þ BðtÞuðtÞ þ bðtÞc1 ðtÞ, xðtÞ

xðt0 Þ ¼ x0 ,

ð1Þ

M. Basin, P. Rodriguez-Ramirez / Journal of the Franklin Institute 349 (2012) 1337–1349

yðtÞ ¼ AðtÞxðtÞ þ GðtÞc2 ðtÞ,

1339

ð2Þ

where yðtÞ ¼ Y_ ðtÞ, and c1 ðtÞ and c2 ðtÞ are white Gaussian noises, which are the weak meansquare derivatives of standard Wiener processes W1 ðtÞ and W2 ðtÞ (see [19]). The representations (1), (2) and ð1n Þ, ð2n Þ are equivalent [20]. Eqs. ð1n Þ and ð2n Þ present the conventional form for Eqs. (1) and (2), which is actually used in practice. In the classical linear optimal controller problem [15,16], the criterion to be minimized is defined as a quadratic Bolza–Meyer functional:   Z 1 1 T T T T J3 ¼ E ½xðTÞ c½xðTÞ þ ðu ðsÞRðsÞuðsÞ þ x ðsÞLðsÞxðsÞÞ ds , 2 2 t0 where R(t) is positive and c, L(t) are nonnegative definite symmetric matrix functions, and T4t0 is a certain time moment. The symbol E½f ðxÞ means the expectation (mean) of a function f of a random variable x, and aT denotes transpose to a vector (matrix) a. The solution to this problem is well known [15,16] and considered fundamental for the optimal linear systems theory. In this paper, the criteria to be minimized include a non-quadratic terminal term or both non-quadratic state energy terms and are defined as follows: " # Z n X 1 T T T J1 ¼ E cii jxi ðTÞj þ ðu ðsÞRðsÞuðsÞ þ x ðsÞLðsÞxðsÞÞ ds , ð3Þ 2 t0 i¼1 " J2 ¼ E

n X

Z

T

cii jxi ðTÞj þ

i¼1

t0

! # n X 1 T u ðsÞRðsÞuðsÞ þ Lii ðsÞjxi ðsÞj ds , 2 i¼1

ð4Þ

where R(s) is positive and L(s) is a nonnegative definite continuous symmetric matrix function, c is a diagonal nonnegative definite matrix, and jxi j denotes the absolute value of the component xi of the vector x 2 Rn . The optimal controller problem is to find the control un ðtÞ, t 2 ½t0 ,T, that minimizes the criterion J along with the unobserved trajectory xn ðtÞ, t 2 ½t0 ,T, generated upon substituting un ðtÞ into the state equation (1). Solutions to the stated optimal control problems are given in the next sections. 3. Mean-square controller design 3.1. Separation principle. I Solving the first problem, in accordance with the separation principle for linear stochastic systems (see [15,16]), the unmeasured linear state x(t), satisfying Eq. (1), is replaced with its mean-square estimate m(t) over linear observations y(t) (2), which is obtained using the mean-square sliding mode filter for linear systems (see [21] for the corresponding filtering problem statement and solution): _ ¼ aðtÞmðtÞ þ BðtÞuðtÞ mðtÞ þ KðtÞAT ðtÞðGðtÞGT ðtÞÞ1 AðtÞSign½AT ðtÞðAðtÞAT ðtÞÞ1 yðtÞmðtÞ: mðt0 Þ ¼ m0 ¼ Eðxðt0 ÞjFtY0 Þ,

ð5Þ

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K_ ðtÞ ¼ ðbðtÞbT ðtÞÞnjAT ðtÞðAðtÞAT ðtÞÞ1 yðtÞmðtÞj þ aðtÞKðtÞ,

ð6Þ

Kðt0 Þ ¼ E½ðxðt0 Þmðt0 Þðxðt0 Þmðt0 ÞT jFtY0 n jAT ðt0 ÞðAðt0 ÞAT ðt0 ÞÞ1 yðt0 Þmðt0 Þj: Here, the Signum function of a vector x ¼ ½x1 , . . . ,xn  2 Rn is defined as Sign½x ¼ ½signðx1 Þ, . . . ,signðxn Þ 2 Rn , and the signum function of a scalar x is defined as signðxÞ ¼ 1, if x40, signðxÞ ¼ 0, if x ¼ 0, and signðxÞ ¼ 1, if xo0 [22]. A vector jxj ¼ ½jx1 j, . . . ,jxn j 2 Rn is defined as the vector of absolute values of the components of the vector x 2 Rn , and Anb denotes a product between a matrix A 2 Rnn and a vector b 2 Rn , that results in the matrix defined as follows: all entries of the jth column of the matrix A are multiplied by the jth component of the vector b, j¼ 1,y,n. Recall that m(t) is the mean-square estimate for the state vector x(t), based on the observation process Y ðtÞ ¼ fyðsÞ,t0 rsrtg, that minimizes the mean-square norm H ¼ E½ðxðtÞmðtÞÞT ðxðtÞmðtÞÞjFtY  at every time moment t. Here, E½xðtÞjFtY  means the conditional expectation of a stochastic process xðtÞ ¼ ðxðtÞmðtÞÞT ðxðtÞmðtÞÞ with respect to the s-algebra FY t generated by the observation process Y(t) in the interval ½t0 ,t. As known [19], this optimal estimate is given by the conditional expectation mðtÞ ¼ EðxðtÞjFtY Þ of the system state x(t) with respect to the s-algebra FY t generated by the observation process Y(t) in the interval ½t0 ,t. As usual, the matrix function PðtÞ ¼ E½ðxðtÞmðtÞÞðxðtÞmðtÞÞT jFtY  is the estimation error variance. It is readily verified (see [15,16]) that the optimal control problem for the system state (1) and cost function (3) is equivalent to the optimal control problem for the estimate (5) and the cost function J1 represented as Z n X 1 T T cii jmi ðTÞj þ ðu ðsÞRðsÞuðsÞ þ mT ðsÞLðsÞmðsÞÞ ds J1 ¼ 2 t0 i¼1 Z T 1 tr½PðsÞLðsÞ ds þ tr½P1 ðTÞKðTÞc, ð7Þ þ 2 t0 where tr½A denotes trace of a matrix A. Since the latter part of J1 does not depend on control u(t) or state x(t), the reduced effective cost function M1 to be minimized takes the form Z n X 1 T T M1 ¼ cii jmi ðTÞj þ ðu ðsÞRðsÞuðsÞ þ mT ðsÞLðsÞmðsÞÞ ds: ð8Þ 2 t 0 i¼1 Thus, the solution for the optimal control problem specified by Eqs. (1) and (3) can be found solving the optimal control problem given by Eqs. (4) and (8). Finally, the minimal value of the criterion J1 should be determined using Eq. (7). This conclusion presents the separation principle for linear systems with a non-quadratic criterion (3).

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3.2. Optimal controller problem solution. I The optimal solution to the control problem defined by Eqs. (4) and (8) is given in [23]. Applying the separation principle from the previous subsection to the sliding mode meansquare filter in [21] and the sliding mode optimal regulator in [23], the optimal controller solving the original problem (1)–(3) is given by the following theorem. Theorem 1. The optimal controller for a linear system (1) over linear observations (2) with respect to a non-quadratic criterion (3) is given by the control law uðtÞ ¼ R1 ðtÞBT ðtÞQðtÞSign½mðtÞ, where the matrix function Q(t) is the solution of the matrix equation _ ¼ LðtÞnjmðtÞjaT ðtÞQðtÞ: QðtÞ

ð9Þ ð10Þ

The terminal condition for Eq. (10) is defined as QðTÞ ¼ c, if the state m(t) does not reach the sliding manifold mðtÞ ¼ 0 within the time interval ½t0 ,T, mðtÞa0, t 2 ½t0 ,T. Otherwise, if the state m(t) reaches the sliding manifold mðtÞ ¼ 0 within the time interval ½t0 ,T, mðtÞ ¼ 0 for some t 2 ½t0 ,T, then Q(t) is set equal to a matrix function Q0 ðtÞ that is such a solution of Eq. (10) that m(t) reaches the sliding manifold mðtÞ ¼ 0 under the control law (8) with the matrix Q0 ðtÞ exactly at the final time moment t ¼ T, mðTÞ ¼ 0, but mðtÞa0, toT. Upon substituting the optimal control (9) into Eq. (5), the following optimally controlled state estimate equation is obtained: _ ¼ aðtÞmðtÞ þ BðtÞR1 ðtÞBT ðtÞQðtÞSign½mðtÞ mðtÞ þKðtÞAT ðtÞðGðtÞG T ðtÞÞ1 AðtÞSign½AT ðtÞðAðtÞAT ðtÞÞ1 yðtÞmðtÞ,

ð11Þ

mðt0 Þ ¼ Eðxðt0 ÞjFtY Þ.

with the initial condition Proof readily follows applying the separation principle from the previous subsection to the sliding mode mean-square filter in [21] and the sliding mode optimal regulator in [23]. & Thus, the optimally controlled estimate equation (11), the control gain matrix equation (10), the optimal control law (9), and the filter gain matrix equation (5) give the complete solution to the optimal controller problem for linear systems over linear observations and a non-quadratic cost function (3).

3.3. Example I This section presents an example of designing the optimal sliding mode controller for a linear system (1) over linear observations (2) with a non-quadratic criterion (3), using the controller (6), (9)–(11), and comparing it to the best available LQG controller. Consider a linear state equation _ ¼ xðtÞ þ uðtÞ þ c1 ðtÞ, xðtÞ

xð0Þ ¼ 1,

ð12Þ

and a linear observation process yðtÞ ¼ xðtÞ þ c2 ðtÞ,

ð13Þ

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where c1 ðtÞ and c2 ðtÞ are white Gaussian noises, which are the weak mean-square derivatives of standard Wiener processes (see [19]). Eqs. (12) and (13) correspond to the alternative conventional form ð1n Þ and ð2n Þ for Eqs. (1) and (2). The controller problem is to find the control u(t), t 2 ½0,T, T ¼ 1.2 that minimizes the criterion Z 1 T 2 J1 ¼ 50jxðTÞj þ ðu ðtÞ þ x2 ðtÞÞ dt, ð14Þ 2 0 In other words, the control problem is to minimize the overall energy of the state x using the minimal overall energy of control u. Applying the sliding-mode controller (6), (19)–(11), the control law (9) is given by uðtÞ ¼ QðtÞsign½mðtÞ,

ð15Þ

where m(t) satisfies the equation _ ¼ mðtÞ þ uðtÞ þ KðtÞsign½yðtÞmðtÞ, mðtÞ

ð16Þ

with the initial condition mð0Þ ¼ Eðxð0Þjyð0ÞÞ ¼ m0 , K(t) satisfies the equation K_ ðtÞ ¼ KðtÞ þ jyðtÞmðtÞj,

ð17Þ

with the initial condition Kð0Þ ¼ Eððxð0Þmð0ÞÞðxð0Þmð0ÞÞT jyð0ÞÞnjyð0Þmð0Þj, and Q(t) satisfies the equation _ ¼ jmðtÞjQðtÞ, QðtÞ

ð18Þ

with the terminal condition Qð1:2Þ ¼ 50, if mðtÞa0 for any to5, and Qn ðtn Þ ¼ 0, where tn is the time that the estimate m(t) reaches the sliding manifold m ¼ 0 at the final moment t¼ T, otherwise. Upon substituting the control (15) and the obtained expressions for K(t) and Q(t) into Eq. (16), the optimally controlled state estimate equation takes the form _ ¼ mðtÞ þ QðtÞsign½mðtÞ þ KðtÞsign½yðtÞmðtÞ, mðtÞ

ð19Þ

with the initial condition mð0Þ ¼ Eðxð0Þjyð0ÞÞ ¼ m0 . The obtained system (17)–(19) can be solved using simple numerical methods, such as ‘‘shooting.’’ This method consists in varying initial conditions of Eq. (18) until the given terminal condition is satisfied. For numerical simulation of the system (12), (13) and the controller (15)–(19), the initial values xð0Þ ¼ 1, mð0Þ ¼ 10, and Pð0Þ ¼ 866:25 are assigned. The final time is set to T¼ 1.2. The disturbances c1 ðtÞ in Eq. (12) and c2 ðtÞ in Eq. (13) are realized using the built-in MatLab white noise function. The system (17)–(19) is first simulated with the terminal condition Qn ð1:2Þ ¼ 50. As the simulation shows, the state m(t) reaches zero before the final moment T ¼ 1.2. Accordingly, the terminal condition for Eq. (18) is reset to Qn ð1:2Þ ¼ c0 such that mð1:2Þ ¼ 0, and the system (17)–(19) is simulated again. The results obtained by applying the controller (15)–(19) to the system (12) are shown in Fig. 1, which presents the graphs of the controlled state (12) x(t), the controlled estimate (19) m(t), the control (15) u(t), and the criterion (14) J1 ðtÞ in the interval ½0,1:2. The value of the criterion (14) at the final moment T ¼ 1.2 is J1 ð1:2Þ ¼ 4:985.

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state and estimate

50 10 8

45

6 4 40

2 0 0.2

0.4

0.6 time

0.8

1

1.2

35

10

30

8 criterion

state and estimate

0

6 4 2 0

25

20 0

0.2

0.4

0.6 time

0.8

1

1.2

control

15 5 0 −5 −10 −15 −20 −25

10

5 0

0.2

0.4

0.6 time

0.8

1

1.2

0

0

0.2

0.4

0.6 time

0.8

1

1.2

Fig. 1. Sliding mode controller optimal with respect to criterion J1 vs. linear feedback controller in the entire simulation interval ½0,1:2. 1. Sliding mode controller. Graphs of the controlled state (12) x(t) (thin solid line) and the controlled estimate (19) m(t) (thick solid line). 2. Linear feedback controller. Graphs of the controlled state (12) x(t) (thin solid line) and the controlled estimate (25) m(t) (thick solid line). 3. Control. Graphs of the sliding mode control (15) un ðtÞ (thick solid line) and the linear feedback control (21) u(t) (thin solid line). 4. Criterion. Graphs of the criterion (14) J1 produced by the sliding mode controller (thick solid line) and by the linear feedback controller (thin solid line).

The designed sliding mode controller (6), (9)–(11) is compared to the best linear controller for the criterion J3 with the quadratic non-integral term Z 1 T 2 J3 ¼ 25x2 ðTÞ þ ðu ðtÞ þ x2 ðtÞÞ dt: ð20Þ 2 0 As follows from the optimal LQG theory [15,16], the linear control law is given by uðtÞ ¼ QðtÞmðtÞ,

ð21Þ

where m(t) satisfies the equation _ ¼ mðtÞ þ uðtÞ þ PðtÞ½yðtÞmðtÞ, mðtÞ

ð22Þ

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with the initial condition mð0Þ ¼ Eðxð0Þjyð0ÞÞ ¼ m0 , the variance P(t) satisfies the Riccati equation _ ¼ 1 þ 2PðtÞP2 ðtÞ, PðtÞ ð23Þ with the initial condition Pð0Þ ¼ Eððxð0Þmð0ÞÞðxð0Þmð0ÞÞT jyð0ÞÞ, and Q(t) satisfies the Riccati equation _ ¼ 12QðtÞQ2 ðtÞ, Qð1:2Þ ¼ 50: ð24Þ QðtÞ Upon substituting the control (21) and the obtained expressions for P(t) and Q(t) into Eq. (22), the optimally controlled state estimate equation takes the form _ ¼ mðtÞ þ QðtÞmðtÞ þ PðtÞ½yðtÞmðtÞ, mðtÞ

ð25Þ

with the initial condition mð0Þ ¼ Eðxð0Þjyð0ÞÞ ¼ m0 . Note that the comparison of the designed sliding mode controller (6), (9)–(11) to the best LQG controller (21)–(25) with respect to the criterion (14) is conducted for illustration purposes, since the controller (6), (9)–(11) should theoretically yield a better result, as follows from Theorem 1. The results obtained by applying the controller (6), (9)–(11) to the system (12), (13) are shown in Fig. 1, which presents the graphs of the controlled state (12) x(t), the controlled estimate (25) m(t), the control (21) u(t), and the criterion (14) J1 ðtÞ in the interval ½0,1:2. The value of the criterion (14) at the final moment T ¼ 1.2 is J1 ð1:2Þ ¼ 7:51. It can be observed that the sliding mode controller (6), (9)–(11) yields a certainly better value of the criterion (14) in comparison to the linear feedback LQG controller (21)–(25). Note that the classical linear feedback LQG controller fails to provide a causal optimal control for the criterion (14). 4. Mean-module controller design 4.1. Separation principle. II Solving the second problem, the unmeasured linear state x(t), satisfying (1), is replaced with its mean-module estimate m(t) over linear observations y(t) (2), which is obtained using the mean-module sliding mode filter for linear systems (see [24] for the corresponding filtering problem statement and solution): _ ¼ aðtÞmðtÞ þ BðtÞuðtÞ þ KðtÞAT ðtÞðGðtÞG T ðtÞÞ1 mðtÞ AðtÞSign½AT ðtÞðAðtÞAT ðtÞÞ1 yðtÞmðtÞ,

ð26Þ

mðt0 Þ ¼ m0 ¼ Eðxðt0 ÞjFtY0 Þ, K_ ðtÞ ¼ bðtÞbT ðtÞ þ aðtÞKðtÞ,

ð27Þ

Kðt0 Þ ¼ E½ðxðt0 Þmðt0 ÞðSignðAT ðtÞðAðtÞAT ðtÞÞ1 AðtÞxðt0 Þmðt0 ÞÞÞT jFtY0 : Here, m(t) is the mean-square estimate for the state vector x(t), based on the observation process Y ðtÞ ¼ fyðsÞ,t0 rsrtg that minimizes the mean-module norm Y ^ J ¼ E½ðjxðtÞxðtÞjÞjF t 

at every time moment t.

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It is readily verified (see [15,16]) that the optimal control problem for the system state (1) and cost function (4) is equivalent to the optimal control problem for the estimate (9) and the cost function J2 represented as ! Z T n n X X 1 T u ðsÞRðsÞuðsÞ þ cii jmi ðTÞj þ Lii ðsÞjmi ðsÞj ds J2 ¼ 2 t0 i¼1 i¼1 Z 1 T þ tr½KðsÞLðsÞ ds þ tr½KðTÞc, ð28Þ 2 t0 where tr½A denotes trace of a matrix A. Since the latter part of J2 does not depend on the control u(t) or state x(t), the reduced effective cost function M2 to be minimized takes the form ! Z T n n X X 1 T u ðsÞRðsÞuðsÞ þ M2 ¼ cii jmi ðTÞj þ Lii ðsÞjmi ðsÞj ds: ð29Þ 2 t0 i¼1 i¼1 Thus, the solution for the optimal control problem specified by Eqs. (1) and (4) can be found solving the optimal control problem given by Eqs. (26) and (29). The minimal value of the criterion J2 should be determined using Eq. (28). 4.2. Optimal controller problem solution. II The optimal solution to the control problem defined by Eqs. (26) and (29) is given in [25]. Applying the separation principle from the previous subsection to the sliding mode mean-module filter in [24] and the sliding mode optimal regulator in [25], the optimal controller solving the original problem (1), (2), (4) is given by the following theorem. Theorem 2. The optimal controller for a linear system (1) over linear observations (2) with respect to a non-quadratic criterion (4) is given by the control law uðtÞ ¼ R1 ðtÞBT ðtÞQðtÞSign½mðtÞ, where the matrix function Q(t) is the solution of the matrix equation _ ¼ LðtÞaT ðtÞQðtÞ: QðtÞ

ð30Þ

ð31Þ

The terminal condition for Eq. (31) is defined as QðTÞ ¼ c, if the state x(t) does not reach the sliding manifold mðtÞ ¼ 0 within the time interval ½t0 ,T, mðtÞa0, t 2 ½t0 ,T. Otherwise, if the state m(t) reaches the sliding manifold mðtÞ ¼ 0 within the time interval ½t0 ,T, then the terminal condition for Q(t) is set to zero at the time moment tn, Qðtn Þ ¼ 0, where tn is the maximum possible time of reaching the sliding manifold mðtÞ ¼ 0. In other words, there exists no such solution to the system of equations (1), (30), (31) satisfying the conditions mðt0 Þ ¼ m0 and Qðt1 Þ ¼ 0, t1 4tn that mðtÞa0 for tot1 and mðtÞ ¼ 0 for some tZt1 . Upon substituting the optimal control (30) into Eq. (26), the following optimally controlled state estimate equation is obtained: _ ¼ aðtÞmðtÞ þ BðtÞR1 ðtÞBT ðtÞQðtÞSign½mðtÞ mðtÞ þKðtÞAT ðtÞðGðtÞG T ðtÞÞ1 AðtÞSign½AT ðtÞðAðtÞAT ðtÞÞ1 yðtÞmðtÞ, with the initial condition

mðt0 Þ ¼ Eðxðt0 ÞjFtY Þ.

ð32Þ

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Proof readily follows applying the separation principle from the previous subsection to the sliding mode mean-module filter in [24] and the sliding mode optimal regulator in [25]. Thus, the optimally controlled state estimate equation (32), the control gain matrix equation (31), the optimal control law (30), and the filter gain matrix equation (27) give the complete solution to the optimal controller problem for linear systems over linear observations and a non-quadratic cost function (4). 4.3. Example II This section presents an example of designing the optimal sliding mode controller for a linear system (1) over linear observations (2) with a non-quadratic criterion (4), using the controller (27), (30)–(32), and comparing it to the best available LQG controller. Consider a linear state equation _ ¼ xðtÞ þ uðtÞ þ c1 ðtÞ, xðtÞ

xð0Þ ¼ 1,

ð33Þ

and a linear observation process yðtÞ ¼ xðtÞ þ c2 ðtÞ,

ð34Þ

where c1 ðtÞ and c2 ðtÞ are white Gaussian noises. The controller problem is to find the control u(t), t 2 ½0,T, T ¼ 1.2 that minimizes the criterion:  Z T 1 2 u ðtÞ þ jxðtÞj dt: J2 ¼ 50jxðTÞj þ ð35Þ 2 0 In other words, the control problem is to minimize the overall energy of the state x using the minimal overall energy of control u. Applying the sliding-mode controller (27), (30)–(32), the control law (30) is given by uðtÞ ¼ QðtÞsign½mðtÞ,

ð36Þ

where m(t) satisfies the equation _ ¼ mðtÞ þ uðtÞ þ KðtÞsign½yðtÞmðtÞ, mðtÞ with the initial condition mð0Þ ¼ Eðxð0Þjyð0ÞÞ ¼ m0 , K(t) satisfies the equation K_ ðtÞ ¼ KðtÞ þ 1,

ð37Þ

ð38Þ

with the initial condition Kð0Þ ¼ Eððxð0Þmð0ÞÞðSignðxð0Þmð0ÞÞÞT jyð0ÞÞ, and Q(t) satisfies the equation _ ¼ 1QðtÞ, QðtÞ ð39Þ with the terminal condition Qn ðtn Þ ¼ 0, where tn is the maximum possible time of reaching the sliding manifold mðtÞ ¼ 0 by the state estimate m(t). Upon substituting the control (36) and the obtained expressions for K(t) and Q(t) into Eq. (33), the optimally controlled state estimate equation takes the form _ ¼ mðtÞ þ QðtÞsign½mðtÞ þ KðtÞsign½yðtÞmðtÞ, mðtÞ

ð40Þ

with the initial condition mð0Þ ¼ Eðxð0Þjyð0ÞÞ ¼ m0 . For numerical simulation of the system (33), (34) and the controller (36)–(40), the initial values xð0Þ ¼ 1, mð0Þ ¼ 10, and Pð0Þ ¼ 100 are assigned. The final time is set to T ¼ 1.2. The

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disturbances c1 ðtÞ in Eq. (33) and c2 ðtÞ in Eq. (34) are realized using the built-in MatLab white noise function. The results obtained by applying the controller (36)–(40) to the system (33) are shown in Fig. 2, which presents the graphs of the controlled state (33) x(t), the controlled estimate (40) m(t), the control (36) u(t), and the criterion (35) J2 ðtÞ in the interval ½0,1:2. The value of the criterion (35) at the final moment T ¼ 1.2 is J2 ð1:2Þ ¼ 5:634. The optimal sliding-mode controller (27), (30)–(32) is compared to the best linear regulator (21), (22)–(25) for the criterion (20) J3. Again, the comparison of the designed sliding mode controller (27), (30)–(32) to the best LQG controller (21)–(25) with respect to the criterion (20) is conducted for illustration purposes, since the controller (27), (30)–(32) should theoretically yield a better result, as follows from Theorem 2.

45

5 0 −5

40

0

0.2

0.4

0.6 time

0.8

1

1.2

35

10

30

5

criterion

state and estimate

state and estimate

50 10

0 −5

25

20 0

0.2

0.4

0.6 time

0.8

1

1.2 15

10 control

0

10

−10 −20 −30

5 0

0.2

0.4

0.6 time

0.8

1

1.2

0

0

0.2

0.4

0.6 time

0.8

1

1.2

Fig. 2. Sliding mode controller optimal with respect to criterion J1 vs. linear feedback controller in the entire simulation interval ½0,1:2. 1. Sliding mode controller. Graphs of the controlled state (33) x(t) (thin solid line) and the controlled estimate (40) m(t) (thick solid line). 2. Linear feedback controller. Graphs of the controlled state (33) x(t) (thin solid line) and the controlled estimate (25) m(t) (thick solid line). 3. Control. Graphs of the sliding mode control (36) un ðtÞ (thick solid line) and the linear feedback control (21) u(t) (thin solid line). 4. Criterion. Graphs of the criterion (35) J1 produced by the sliding mode controller (thick solid line) and by the linear feedback controller (thin solid line).

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M. Basin, P. Rodriguez-Ramirez / Journal of the Franklin Institute 349 (2012) 1337–1349

The results obtained after applying the controller (27), (30)–(32) to the system (33), (34) are shown in Fig. 2, which presents the graphs of the controlled state (33) x(t), the controlled estimate (40) m(t), the control (36) u(t), and the criterion (35) J2 ðtÞ in the interval ½0,1:2. The value of the criterion (35) at the final moment T ¼ 1.2 is J2 ð1:2Þ ¼ 7:586. It can be observed that the sliding mode controller (27), (30)–(32) yields a certainly better value of the criterion (35) in comparison to the linear feedback LQG controller (21)–(25). Note again that the classical linear feedback LQG controller fails to provide a causal optimal control for the criterion (35). Acknowledgment The author thanks the Mexican National Science and Technology Council (CONACyT) for financial support under Grant 55584 and joint Mexico-EU FONCICyT Grant 93302. References [1] V.I. Utkin, Sliding Modes in Control and Optimization, Springer, 1992. [2] C. Edwards, S.K. Spurgeon, Sliding Mode Control: Theory and Applications, Taylor and Francis, London, 1998. [3] L. Fridman, A. Levant, Higher order sliding modes, in: W. Perruquetti, J.P. Barbot (Eds.), Sliding Mode Control in Engineering, Marcel Dekker Inc., New York, 2002, pp. 53–101. [4] H. Yang, Y. Xia, P. Shi, Observer-based sliding mode control for a class of discrete systems via delta operator approach, Journal of the Franklin Institute 347 (2010) 1199–1213. [5] N. Orani, A. Pisano, E. Usai, Fault diagnosis for the vertical three-tank system via high-order sliding-mode observation, Journal of the Franklin Institute 347 (2010) 923–939. [6] I. Boiko, Frequency domain precision analysis and design of sliding mode observers, Journal of the Franklin Institute 347 (2010) 899–909. [7] Z. Mao, B. Jiang, P. Shi, Observer based fault-tolerant control for a class of nonlinear networked control systems, Journal of the Franklin Institute 347 (2010) 940–956. [8] H.R. Karimi, M. Zapateiro, N. Luo, A linear matrix inequality approach to robust fault detection filter design of linear systems with mixed time-varying delays and nonlinear perturbations, Journal of the Franklin Institute 347 (2010) 957–973. [9] B. Mirkin, P.-O. Gutman, Y. Shtessel, Robust adaptive tracking with an additional plant identifier for a class of nonlinear systems, Journal of the Franklin Institute 347 (2010) 974–987. [10] Y. Xia, Y. Jia, Robust sliding mode control for uncertain stochastic time-delay systems, IEEE Transactions on Automatic Control 48 (2003) 1086–1092. [11] Y. Niu, D.W.C. Ho, J. Lam, Robust integral sliding mode control for uncertain stochastic systems with timevarying delay, Automatica 41 (2005) 873–880. [12] P. Shi, Y. Xia, G.P. Liu, D. Rees, On designing of sliding mode control for stochastic jump systems, IEEE Transactions on Automatic Control 51 (2006) 97–103. [13] M.V. Basin, L. Fridman, M. Skliar, Optimal and robust sliding mode filter for systems with continuous and delayed measurements, in: Proceedings of the 41st Conference on Decision and Control, Las Vegas, NV, 2002, pp. 2594–2599. [14] M.V. Basin, L. Fridman, J. Rodriguez-Gonzalez, P. Acosta, Integral sliding mode design for robust filtering and control of linear stochastic time-delay systems, International Journal of Robust Nonlinear Control 15 (2005) 407–421. [15] H. Kwakernaak, R. Sivan, Linear Optimal Control Systems, Wiley-Interscience, New York, 1972. [16] W.H. Fleming, R.W. Rishel, Deterministic and Stochastic Optimal Control, Springer, 1975. [17] M.V. Basin, P. Rodriguez-Ramirez, A. Ferrara, D. Calderon-Alvarez, Sliding mode optimal control for linear systems, Journal of the Franklin Institute (this Special Issue on Optimal Sliding Mode Algorithms for Dynamic Systems), doi:10.1016/j.jfranklin.2011.05.010. [18] R.E. Kalman, R.E. Bucy, New results in linear filtering and prediction theory, ASME Transactions, Part D (Journal of Basic Engineering) 83 (1961) 95–108.

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[19] V.S. Pugachev, I.N. Sinitsyn, Stochastic Systems: Theory and Applications, World Scientific, 2001. ˚ om, [20] K.J. Astr Introduction to Stochastic Control Theory, Academic Press, New York, 1970. ¨ [21] M.V. Basin, P. Rodriguez-Ramirez, Sliding mode mean-square filtering for linear stochastic systems, in: Proceedings of the 2010 IEEE International Conference on Industrial Technology, Valparaiso del Mar, Chile, 2010, pp. 1761–1764. [22] A.F. Filippov, Differential Equations with Discontinuous Righthand Sides, Kluwer, 1988. [23] M.V. Basin, A. Ferrara, D. Calderon-Alvarez, Sliding mode regulator as solution to optimal control problem, in: Proceedings of the 47th Conference on Decision and Control, Cancun, Mexico, 2008, pp. 2184–2189. [24] M.V. Basin, P. Rodriguez-Ramirez, Sliding mode mean-module filtering for linear stochastic systems, Proceedings of the 2010 IEEE International Conference on Industrial Technology, Valparaiso del Mar, Chile, 2010, pp. 1757–1760. [25] M.V. Basin, A. Ferrara, D. Calderon-Alvarez, F. Dinuzzo, Sliding mode optimal regulator for a BolzaMeyer criterion with non-quadratic state energy terms, in: Proceedings of the 2009 American Control Conference, St. Louis, MO, 2009, pp. 4951–4955.