Dynamic output feedback sliding-mode control ... - Semantic Scholar
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IEEE TRANSACTIONS ON INDUSTRIAL ELECTRONICS, VOL. 50, NO. 5, OCTOBER 2003
Dynamic Output Feedback Sliding-Mode Control Using Pole Placement and Linear Functional Observers Q. P. Ha, Member, IEEE, H. Trinh, H. T. Nguyen, Senior Member, IEEE, and H. D. Tuan, Member, IEEE
Abstract—This paper presents a methodological approach to design dynamic output feedback sliding-mode control for a class of uncertain dynamical systems. The control action consists of the equivalent control and robust control components. The design of the equivalent control and the sliding function are based on the pole-placement technique. Linear functional observers are developed to implement the sliding function and the equivalent control. Stability of the resulting system under the proposed control scheme is guaranteed. A numerical example is given to demonstrate its efficacy. Index Terms—Dynamic output feedback, linear functional observer, sliding mode.
I. INTRODUCTION
V
ARIABLE-STRUCTURE control with sliding modes is widely recognized in the control research community to be insensitive to parameter variations and external disturbances. Sliding-mode control design generally involves two main steps: firstly, the selection of a sliding surface which induces a stable reduced-order dynamics assigned by the designer, and secondly the synthesis of a switching control law to force the closed-loop system trajectories onto and subsequently remain on the sliding surface. A tutorial and survey on variable-structure control can be found in [1] and [2]. A comprehensive guide on sliding-mode control for control engineers is given in [3]. Most of the design techniques for sliding-mode control assume that all the system states are accessible to the control law. In practice, all of these states are not physically available for feedback. In this case, a full-state feedback sliding-mode controller cannot be implemented unless an observer is used to estimate the unmeasured states, or the design methods must be modified such that only a subset of the states are required to implement the control law.
Manuscript received December 25, 2001; revised January 15, 2003. Abstract published on the Internet July 9, 2003. This work was supported in part by the ARC Centre of Excellence programme, funded by the Australian Research Council (ARC) and the New South Wales State Government. Q. P. Ha is with the Faculty of Engineering, University of Technology, Sydney, NSW 2007, Australia, and also with the ARC Centre of Excellence in Autonomous Systems (CAS) (e-mail: [email protected]). H. T. Nguyen is with the Faculty of Engineering, University of Technology, Sydney, NSW 2007, Australia. H. Trinh is with the School of Engineering and Technology, Deakin University, Geelong, VIC 3217, Australia. H. D. Tuan is with the Toyota Technological Institute, Nagoya 468-8511, Japan. Digital Object Identifier 10.1109/TIE.2003.817697
The problem of sliding-mode control design for uncertain dynamical systems using output information only has been inves˙ et al., for example, proposed the tigated by many authors. Zak use of observers to retain the sliding mode and improve control performance of systems with unmodeled actuator and sensor dynamics [4]. In the context of output feedback sliding mode ˙ and Hui designed the output dependent sliding surcontrol, Zak face, based on eigenvector methods [5]. In [6], it is shown that the design problem of a sliding surface is equivalent to a static output feedback problem. A common design methodology proposed in [7] and [8] provides numerical approaches to synthesizing a static output feedback gain based on nonlinear optimization schemes for which no guarantee of a solution exists a priori. An alternative approach using output information with a dynamic compensator is given in [9], a parameterization of both the sliding surface and the compensator was proposed. As later specified in [10], these numerical schemes are applicable under certain structural conditions and restricted only to a specific class of sliding surface. The problem of finding an explicit solution parameterizing both the sliding surface and the controller for uncertain systems has become of increasing interest in the context of sliding-mode control. Woodham and Zinober (1993) proposed to position the closed-loop system eigenvalues in a specified sector in the left-hand half plane, involving the solution of a complex continuous matrix Riccati equation [11]. An explicit form using Ackermann’s formula for the sliding surface is derived in [12] for single input systems. Motivated by the work in [12] and the estimation of a linear functional of the systems state, this paper presents an approach to dynamic output feedback sliding mode control of multi-input uncertain systems using pole placement and linear functional observers. The design technique is simple and computationally efficient. The method has the following attractive features: 1) the sliding surface and the observers are fully parameterized; 2) overall closed-loop system stability and performance is guaranteed; and 3) the control design is rather straightforward. The paper is organized as follows. The design of sliding-mode control using pole placement is presented in Section II. The design of linear functional observers is given in Section III. The design of dynamical output feedback sliding-mode control is presented in Section IV. A numerical example is included in Section V to illustrate the design procedure and to demonstrate the validity of the proposed approach. Finally, a conclusion is given in Section VI.
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II. SLIDING-MODE CONTROL DESIGN USING POLE PLACEMENT Consider a dynamical system described by (1a) (1b) is the state vector, is the control where is the output vector, and is a input, nonlinear perturbation with known upper bound
margin. Using a pole placement algorithm (see, for example, [14]), a state feedback control law of the form (5) can be found for the equivalent control to assign the desired such that eigenstructure (6) is the state feedback control matrix, is the closed-loop matrix, and is the -dimensional unity matrix. Matrix in (2a) will be chosen such that
where (1c) denotes standard Euclidean norm. , and are real constant matrices of the nominal is absent). For the ensuing system (when perturbation discussion, the following is assumed to be valid. ) of the nominal system is controlA1: The triplet ( lable and observable with the matrices and being of full rank. A2: The number of output channels is greater than the . number of inputs, i.e., The design of sliding-mode control for (1) includes the selection of a sliding function so that the sliding motion when restricted to the sliding surface is stable, and then the derivation of a control law to enforce sliding mode in the sliding surface. The sliding function
where
(2a)
(7a) or (7b) where the form
. A solution to (7b) can be found in (8)
is any basis of the null space of , . Theorem 1: For the system (1) if the control law (3) is employed with the equivalent control (6), obtained from placing for the desired eigenstructure the nominal system, and the robust control given by where
, may be determined such that the where sliding-mode dynamics in the sliding surface (2b) ) desired eigenvalues . The have ( )th order sliding-mode equation defined in (2b) is of the ( and does not depend on the perturbation. The following control law is employed [13]):
(9) where the sliding function (2a) satisfies (7b), then the state vector asymptotically converges to zero and are the sliding eigenvalues. Proof: Using the equivalent control given in (6), the first time derivative of the sliding function can be obtained as
(3) is the equivalent control that may be obtained from where a conventional method of the linear system theory applied to is the robust control, which is the nominal system, and switching in nature, developed to guarantee the reaching condition
From (7) the following dynamics can be obtained in the reaching phase: (10)
(4) Substituting (10) and (9) into the expression of . in the presence of perturbation In the following, a sliding-mode controller will be designed chosen by eigenvalue placewith the sliding matrix ment. If the sliding mode is enforced in the surface (2b) then the ) desired system dynamic properties are determined by ( . As noted in [12], the desliding eigenvalues sign of the sliding surface does not generally imply assigning eigenvalues, which can take any arbitrary value. In the rest this paper we make use of this design freedom to place them at , where is some stable eigenvalue called the sliding
With a negative value of
yields
and using (1c) we obtain (11)
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or the reaching condition (4) is satisfied, and the state vector is asymptotically converges to zero. As ), rank rank . Let us consider of rank ( matrix
Without loss of generality, let us assume that matrix has full , and takes the following canonical row rank, i.e., rank form: (13) Note that any full rank matrix can always be transformed into (13), for example by using the following transformation matrix:
Since transformation defined by
,
can be used in the (14) where control matrix
to transform the system dynamics such states remain unthat the first changed and becomes the last states. Using (10) the transformed system is obtained
is any basis of . Let the feedback be partitioned as (15)
, , and are real where matrices to be determined. Our design effort is to implement the by a dynamical output feedback described by relation (16a)
(12) subject to where
,
, and where
is . Once a sliding mode the spectrum of and . Equation (12) is then reduced incurs, we have with the desired dynamics. to the motion equation This concludes the proof. Remark 1: As can be seen in (11) the eigenvalue provides some stability margin for the system. To accelerate the reaching should be chosen at least about 3–5 times the phase (10), dominant roots of the sliding eigenvalues. ) the specRemark 2: For single-input systems ( consists of ( ) desired sliding eigenvalues trum of and the sliding margin . By using the well-known Ackermann formula for the feedback matrix gain in this case, the sliding matrix can have the following explicit form [12]:
(16b) is the state vector of the observer of where is a real constant matrix to be determined, order , is a stable matrix to be selected according to the and observer desired dynamics. be the estimation error Letting (17) by some simple manipulations, the following error equation is obtained:
(18) (12a) (12b)
III. LINEAR FUNCTIONAL OBSERVER DESIGN In order to implement sliding-mode control when the states are not physically available an observer is required for state estimation or at least for estimation of their subsets. Examining the control law (3) reveals that it could be implemented by using estimates of the equivalent control and the sliding function. In this section we propose a simple, thus, efficient technique to design an observer for estimating asymptotically a linear state , for the system with unknown inputs functional, (1). Relevant research on this subject can be found, e.g., [15], [16], or more recently, in [17] and [18].
Equations (15) and (18) imply that (16b) can act as a linear provided that matrix functional observer for is chosen to be stable, and matrices and fulfill the following constraints: (19) (20) (21) As matrix can be selected according to the desired dynamics of the observer, there are four unknown matrices ( , , , and ) in (19)–(21) to be solved for. Using (13), (19) and (21) can be expressed as (22a)
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(22b)
Augmenting the set of (25a) with the set of (26a) and (27a) gives
and
(28a) (23a)
where (28b)
(23b) can be It is clear from (22a) and (23a) that matrices and easily derived, once matrices and are obtained. It therefore and . remains to solve (20), (22b), and (23b) for matrices To proceed, let matrix be expressed as (24) ( ) is the where th column of . Incorporating (24) into (23b), and after some rearranging, the following matrix-vector equation is obtained (25a) where (25b) (25c) (25d) is the th column of . and Let us now consider (22b). As commented on earlier, can be chosen to be any matrix with a set of stable eigenvalues. By some simple rearranging, (22b) can be cast in a matrix-vector form as (26a) is given by (26b), shown at the where matrix denotes the ( ) element of matrix bottom of the page, and . Similarly, (20) can be rewritten as (27a) where
(27b) and
denotes the (
) element of matrix
.
Lemma 1: [18]: For the dynamical system (1), system (29) is any stable matrix, is selected where is obtained from (28), and arbitrarily with full rank, and are derived, respectively, from (22a) and (23a); can be constructed to generate asymptotically any linear state func, provided that: 1) tional, and 2) matrix defined in (28b) has full row rank. Note that 2) is a necessary and sufficient condition for the solvability of (28). Provided that the above two conditions are satisfied, vector is solved for from linear (28), and thus matrix is obtained from notation (25d). It is then followed that can be derived, respectively, from (22a) and matrices and (23a). As a result, all of the observer parameters are determined , subject to (19)–(21). Consequently, from (18), is defined in (17) as . Since matrix is where asymptotically. Therefore, the estimate stable,
(30) . asymptotically approaches to A proposed algorithm for designing linear functional observers is given below: Design Algorithm 1: 1. Choose the observer order , where denotes the smallest integer greater than or equal . according to (24) 2. Partition matrices according to (25c). and formulate 3. Choose a stable matrix , and arbi) elements of full row rank trarily ( . matrix 4. Formulate (28b) and solve it for vector . Obtain from (25d). and respectively 5. Derive matrices from (22a) and (23a).
(26b)
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Remark 3: To facilitate solving (28b) a singular value decomposition for matrix may be required. In most of the cases, the proposed technique results in functional observers with an order less than . The observer order may however take a large , i.e., when the number of outputs exvalue when ceeds the number of inputs by one. The observer order becomes , which may be greater than . In this then case or if conditions of Lemma 1 are not satisfied, one can still proceed with the design of linear functional observers by using a Moore–Penrose pseudoinverse matrix approach to solve approximately (19)–(21). The observer order can be increased gradually from a smallest value until an approximate solution is obtained.
Substituting (33) into the above equation and using (7a) yield
Conditions of the theorem are now used to obtain
IV. DYNAMIC OUTPUT FEEDBACK SLIDING-MODE CONTROL WITH LINEAR FUNCTIONAL OBSERVERS Let us now apply the results developed in the previous section for estimation of the equivalent control and the sliding function to implement sliding-mode control for system (1). Assume that the following linear functional observers: (31a) (31b) and (32a) (32b) and , rehave been designed to obtain the estimates spectively, of the equivalent control (5) and the sliding function (2a). As expressed in (30), there exist dynamical errors associated with the estimates (33) and (34) and . where Taking this effect into account, the control law may be modified as formulated in the following theorem. Theorem 2: For the system (1) using the sliding-mode control design described in Theorem 1 and the linear functional observers determined in (31) and (32), if the equivalent control (31a) is employed and the robust control is now given by (35) ( is the order where is obtained from (32b) and is the sliding margin), then the state of the observer (32) and vector asymptotically converges to zero. Proof: Let us check the reaching condition (4) for the new . Taking the first time derivative of (34) sliding function gives
As a sliding mode incurs, and since the observer error , according to (34). Thus the state vector asymptotically converges to zero. This concludes the proof. is chosen at least about 3–5 Remark 4: As the eigenvalue times the dominant roots of the sliding eigenvalues according to quickly enough with respect to the sliding Remark 1, mode dynamics. Let us now summarize the development proposed in previous sections in the following design procedure. Design Algorithm 2: 1. Choose the eigenvalues according to the desired sliding-mode dyaccording namics and the sliding margin to Remark 1. 2. Design a suitable state feedback confor the equivalent control (5). troller 3. Select the sliding function(2a) with satisfying condition (7) by matrix using (8). 4. Design linear functional observer (31) using algorithm 1. , design linear func5. Choose tional observer (32) using algorithm 1. 6. Formulate the equivalent control (31b) and the robust control (32b). V. NUMERICAL EXAMPLE Consider a seventh-order aircraft example in [7]. Two inputs are the rudder and the aileron commands. Four measurable outputs are yaw and roll accelerations, bank angle, and the wash out filter state. Note that as mentioned in [10], existing sliding-mode control techniques using output feedback provide no guarantees for a satisfied reaching condition for this system. In the following the method proposed in this paper will be illustrated through the design of a dynamical output feedback sliding-mode controller using linear functional observers. Let us first apply a transformation of the form (14) to obtain the nominal system matrices in the canonical form, as shown in the first entry at the bottom of the next page. The control design procedure is described below.
HA et al.: DYNAMIC OUTPUT FEEDBACK SLIDING-MODE CONTROL
Step 1: Choose the desired sliding eigen, values located at and a double eigenvalue at the sliding . margin Step 2: The feedback gain is found to be the second entry shown at the bottom of the page. can be Step 3: A solution for matrix found from(8) as shown by the third entry at the bottom of the page. Step 4: Design a functional observer to using design algorithm 1 estimate , , , 4.1 For this system, . the observer order is obtained in step 2, vector 4.2 With is formulated as the last entry at the bottom of the page.
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4.3 Choose
and . 4.4 Formulate (28b). Solving this equation yields the first entry at the bottom of the next page. and 4.5 Derive matrices
Step 5: Design a functional observer to using design algorithm 1. estimate , , 5.1 For this system, . the observer order is
,
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5.2 With obtained in step 3, vector is formulated as the second entry at the bottom of the page. and 5.3 Choose . 5.4 Formulate (28b). Solving for this equation yields the third entry at the bottom of the page. and 5.5 Derive matrices
Step 6: Overall, the control law is . The equivalent control is , where is given by subject to the observer dynamics
Fig. 1.
Responses of the states x (t) and x (t).
Fig. 2.
Responses of the control u (t) and u (t).
The robust control is , where is the known perturbation bound and , governed by the following dynamics:
For the sake of simulation it is assumed that and . Fig. 1 shows the responses of the states and of the original system from some nonzero initial and conditions. Fig. 2 shows the two control inputs using the proposed control scheme. Fig. 3 depicts the responses and to an initial conditions (1, ) of the estimation errors of the functional observers, respectively, for and
HA et al.: DYNAMIC OUTPUT FEEDBACK SLIDING-MODE CONTROL
Fig. 3. Responses of the estimation errors e (t) and e
(t).
. As expected, these responses exponentially approach to . zero according to the selection of and VI. CONCLUSION We have presented a methodological approach to design dynamic output feedback sliding-mode control of uncertain dynamical systems using pole placement and linear functional observers. Estimates of the equivalent control and the sliding function are employed to reconstruct the sliding-mode control action. Stability of the closed-loop system under the observerbased sliding-mode controller is guaranteed. Since the sliding surface and the observers parameters are obtained directly from solving linear equations, the control design is rather straightforward and computationally efficient. Step-by-step design algorithms were provided. A numerical example was used to illustrate the design procedure. Simulation results demonstrate the validity of the proposed approach.
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[9] C. Edwards and S. K. Spurgeon, “Compensator based output feedback sliding mode controller design,” Int. J. Control, vol. 71, no. 4, pp. 601–614, 1998. , “On the limitations of some variable structure output feedback [10] controller designs,” Automatica, vol. 36, pp. 743–748, 2000. [11] C. A. Woodham and A. S. I. Zinober, “Eigenvalue placement in a specified sector for variable structure control systems,” Int. J. Control, vol. 57, no. 5, pp. 1021–1037, 1993. [12] J. Ackermann and V. I. Utkin, “Sliding mode control designed based on Ackermann’s formula,” IEEE Trans. Automat. Contr., vol. 43, pp. 234–237, Feb. 1998. [13] Q. P. Ha, D. C. Rye, and H. F. Durrant-Whyte, “Robust sliding mode control with application,” Int. J. Control, vol. 72, pp. 1087–1096, 1999. [14] J. Kautsky, N. K. Nichols, and P. Van Dooren, “Robust pole assignment in linear state feedback,” Int. J. Control, vol. 41, pp. 1129–1155, 1985. [15] T. Mita, “On the synthesis of an unknown input observer,” Int. J. Control, vol. 26, no. 6, pp. 841–851, 1977. [16] F. W. Fairmann, S. S. Mahil, and L. Luk, “Disturbance decoupled observer design via singular value decomposition,” IEEE Trans. Automat. Contr., vol. AC-29, pp. 84–86, Jan. 1984. [17] M. Aldeen and H. Trinh, “Reduced-order linear functional observer for linear systems,” Proc. IEE—Control Theory Applicat., pt. D, vol. 146, pp. 399–405, 1999. [18] H. Trinh and Q. Ha, “Design of linear functional observers for linear systems with unknown inputs,” Int. J. Syst. Sci., vol. 31, pp. 741–749, 2000.
Q. P. Ha (M’97) received the B.E. degree in electrical engineering from Ho Chi Minh City University of Technology, Ho Chi Minh City, Vietnam, in 1983, the Ph.D. degree in engineering science from the Moscow Power Engineering Institute, Moscow, Russia, in 1992, and the Ph.D. degree in Electrical Engineering from the University of Tasmania, Hobart, Australia, in 1997. From 1997 to 2000, he was a Senior Research Associate at the Australian Centre for Field Robotics, University of Sydney, Sydney, Australia. He is currently a Senior Lecturer at the University of Technology, Sydney, Australia. His research interests include robust control and estimation, robotics, and artificial intelligence applications.
H. Trinh received the B.E. (Hons.), M.Eng.Sc., and Ph.D. degrees from the University of Melbourne, Melbourne, Australia, in 1990, 1992, and 1996, respectively. He is currently a Senior Lecturer in the School of Engineering and Technology, Deakin University, Geelong, Australia. His research interests include large-scale systems theory, robust control and estimation, fault diagnosis and fault-tolerant control, robotics, and power systems.
REFERENCES ˙ [1] R. A. DeCarlo, S. H. Zak, and G. P. Matthews, “Variable structure control of nonlinear multivariable systems: A tutorial,” Proc. IEEE, vol. 76, pp. 212–232, Mar. 1988. [2] J. Y. Hung, W. B. Gao, and J. C. Hung, “Variable structure control: A survey,” IEEE Trans. Ind. Electron., vol. 40, pp. 2–22, Feb. 1993. [3] K. D. Young, V. I. Utkin, and U. Ozguner, “A control engineer’s guide to sliding mode control,” IEEE Trans. Contr. Syst. Technol., vol. 7, pp. 328–342, May 1999. ˙ [4] S. H. Zak, J. D. Brehove, and M. J. Corless, “Control of uncertain systems with unmodeled actuator and sensor dynamics and incomplete information,” IEEE Trans. Syst., Man, Cynern., vol. 19, pp. 241–257, Mar./Apr. 1989. ˙ and S. Hui, “Output feedback in variable structure controllers [5] S. H. Zak and state estimators for uncertain/nonlinear dynamical systems,” Proc. IEE—Control Theory Applicat., pt. D, vol. 140, pp. 41–50, 1993. [6] C. Edwards and S. K. Spurgeon, “Sliding mode stabilization of uncertain systems using only output information,” Int. J. Control, vol. 62, pp. 1129–1144, 1995. [7] B. S. Heck, S. V. Yallapragada, and M. K. H. Fan, “Numerical methods to design the reaching phase of output feedback variable structure control,” Automatica, vol. 31, pp. 275–279, 1995. [8] S. K. Bag, S. K. Spurgeon, and C. Edwards, “Output feedback sliding mode design for linear uncertain systems,” Proc. IEE—Control Theory Applicat., pt. D, vol. 144, pp. 209–216, 1997.
H. T. Nguyen (SM’99) received the B.E. (Honors 1), M.E., and Ph.D. degrees from the University of Newcastle, Newcastle, Australia, in 1976, 1977, and 1980, respectively. He is a Professor of Electrical Engineering at the University of Technology, Sydney, Australia, where he is Head of the Mechatronics and Intelligent Systems Group in the Faculty of Engineering and Co-Director of the Key University Research Centre for Health Technologies. He is also Executive Director of AIMEDICS Pty Ltd. His research interests are in biomedical engineering, advanced control and instrumentation, and artificial intelligence. Prof. Nguyen is a Member of the Order of Australia (AM).
H. D. Tuan (M’95) was born in Hanoi, Vietnam, in 1964. He received the diploma and the Ph.D. degree, both in applied mathematics, from Odessa State University, Odessa, Ukraine, in 1987 and 1991, respectively. From 1991 to 1994, he was a Researcher with the Optimization and Systems Division, Vietnam National Centre for Science and Technologies. From 1994 to 1999, he was an Assistant Professor in the Department of Electronic-Mechanical Engineering, Nagoya University, Nagoya, Japan. In 1999, he joined the Toyota Technological Institute, Nagoya, Japan, where he is an Associate Professor in the Department of Electrical and Computer Engineering. His research interests include theoretical developments and applications of optimization-based methods in broad areas of control, signal processing, and communication.
IEEE TRANSACTIONS ON INDUSTRIAL ELECTRONICS, VOL. 50, NO. 5, OCTOBER 2003
Dynamic Output Feedback Sliding-Mode Control Using Pole Placement and Linear Functional Observers Q. P. Ha, Member, IEEE, H. Trinh, H. T. Nguyen, Senior Member, IEEE, and H. D. Tuan, Member, IEEE
Abstract—This paper presents a methodological approach to design dynamic output feedback sliding-mode control for a class of uncertain dynamical systems. The control action consists of the equivalent control and robust control components. The design of the equivalent control and the sliding function are based on the pole-placement technique. Linear functional observers are developed to implement the sliding function and the equivalent control. Stability of the resulting system under the proposed control scheme is guaranteed. A numerical example is given to demonstrate its efficacy. Index Terms—Dynamic output feedback, linear functional observer, sliding mode.
I. INTRODUCTION
V
ARIABLE-STRUCTURE control with sliding modes is widely recognized in the control research community to be insensitive to parameter variations and external disturbances. Sliding-mode control design generally involves two main steps: firstly, the selection of a sliding surface which induces a stable reduced-order dynamics assigned by the designer, and secondly the synthesis of a switching control law to force the closed-loop system trajectories onto and subsequently remain on the sliding surface. A tutorial and survey on variable-structure control can be found in [1] and [2]. A comprehensive guide on sliding-mode control for control engineers is given in [3]. Most of the design techniques for sliding-mode control assume that all the system states are accessible to the control law. In practice, all of these states are not physically available for feedback. In this case, a full-state feedback sliding-mode controller cannot be implemented unless an observer is used to estimate the unmeasured states, or the design methods must be modified such that only a subset of the states are required to implement the control law.
Manuscript received December 25, 2001; revised January 15, 2003. Abstract published on the Internet July 9, 2003. This work was supported in part by the ARC Centre of Excellence programme, funded by the Australian Research Council (ARC) and the New South Wales State Government. Q. P. Ha is with the Faculty of Engineering, University of Technology, Sydney, NSW 2007, Australia, and also with the ARC Centre of Excellence in Autonomous Systems (CAS) (e-mail: [email protected]). H. T. Nguyen is with the Faculty of Engineering, University of Technology, Sydney, NSW 2007, Australia. H. Trinh is with the School of Engineering and Technology, Deakin University, Geelong, VIC 3217, Australia. H. D. Tuan is with the Toyota Technological Institute, Nagoya 468-8511, Japan. Digital Object Identifier 10.1109/TIE.2003.817697
The problem of sliding-mode control design for uncertain dynamical systems using output information only has been inves˙ et al., for example, proposed the tigated by many authors. Zak use of observers to retain the sliding mode and improve control performance of systems with unmodeled actuator and sensor dynamics [4]. In the context of output feedback sliding mode ˙ and Hui designed the output dependent sliding surcontrol, Zak face, based on eigenvector methods [5]. In [6], it is shown that the design problem of a sliding surface is equivalent to a static output feedback problem. A common design methodology proposed in [7] and [8] provides numerical approaches to synthesizing a static output feedback gain based on nonlinear optimization schemes for which no guarantee of a solution exists a priori. An alternative approach using output information with a dynamic compensator is given in [9], a parameterization of both the sliding surface and the compensator was proposed. As later specified in [10], these numerical schemes are applicable under certain structural conditions and restricted only to a specific class of sliding surface. The problem of finding an explicit solution parameterizing both the sliding surface and the controller for uncertain systems has become of increasing interest in the context of sliding-mode control. Woodham and Zinober (1993) proposed to position the closed-loop system eigenvalues in a specified sector in the left-hand half plane, involving the solution of a complex continuous matrix Riccati equation [11]. An explicit form using Ackermann’s formula for the sliding surface is derived in [12] for single input systems. Motivated by the work in [12] and the estimation of a linear functional of the systems state, this paper presents an approach to dynamic output feedback sliding mode control of multi-input uncertain systems using pole placement and linear functional observers. The design technique is simple and computationally efficient. The method has the following attractive features: 1) the sliding surface and the observers are fully parameterized; 2) overall closed-loop system stability and performance is guaranteed; and 3) the control design is rather straightforward. The paper is organized as follows. The design of sliding-mode control using pole placement is presented in Section II. The design of linear functional observers is given in Section III. The design of dynamical output feedback sliding-mode control is presented in Section IV. A numerical example is included in Section V to illustrate the design procedure and to demonstrate the validity of the proposed approach. Finally, a conclusion is given in Section VI.
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II. SLIDING-MODE CONTROL DESIGN USING POLE PLACEMENT Consider a dynamical system described by (1a) (1b) is the state vector, is the control where is the output vector, and is a input, nonlinear perturbation with known upper bound
margin. Using a pole placement algorithm (see, for example, [14]), a state feedback control law of the form (5) can be found for the equivalent control to assign the desired such that eigenstructure (6) is the state feedback control matrix, is the closed-loop matrix, and is the -dimensional unity matrix. Matrix in (2a) will be chosen such that
where (1c) denotes standard Euclidean norm. , and are real constant matrices of the nominal is absent). For the ensuing system (when perturbation discussion, the following is assumed to be valid. ) of the nominal system is controlA1: The triplet ( lable and observable with the matrices and being of full rank. A2: The number of output channels is greater than the . number of inputs, i.e., The design of sliding-mode control for (1) includes the selection of a sliding function so that the sliding motion when restricted to the sliding surface is stable, and then the derivation of a control law to enforce sliding mode in the sliding surface. The sliding function
where
(2a)
(7a) or (7b) where the form
. A solution to (7b) can be found in (8)
is any basis of the null space of , . Theorem 1: For the system (1) if the control law (3) is employed with the equivalent control (6), obtained from placing for the desired eigenstructure the nominal system, and the robust control given by where
, may be determined such that the where sliding-mode dynamics in the sliding surface (2b) ) desired eigenvalues . The have ( )th order sliding-mode equation defined in (2b) is of the ( and does not depend on the perturbation. The following control law is employed [13]):
(9) where the sliding function (2a) satisfies (7b), then the state vector asymptotically converges to zero and are the sliding eigenvalues. Proof: Using the equivalent control given in (6), the first time derivative of the sliding function can be obtained as
(3) is the equivalent control that may be obtained from where a conventional method of the linear system theory applied to is the robust control, which is the nominal system, and switching in nature, developed to guarantee the reaching condition
From (7) the following dynamics can be obtained in the reaching phase: (10)
(4) Substituting (10) and (9) into the expression of . in the presence of perturbation In the following, a sliding-mode controller will be designed chosen by eigenvalue placewith the sliding matrix ment. If the sliding mode is enforced in the surface (2b) then the ) desired system dynamic properties are determined by ( . As noted in [12], the desliding eigenvalues sign of the sliding surface does not generally imply assigning eigenvalues, which can take any arbitrary value. In the rest this paper we make use of this design freedom to place them at , where is some stable eigenvalue called the sliding
With a negative value of
yields
and using (1c) we obtain (11)
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or the reaching condition (4) is satisfied, and the state vector is asymptotically converges to zero. As ), rank rank . Let us consider of rank ( matrix
Without loss of generality, let us assume that matrix has full , and takes the following canonical row rank, i.e., rank form: (13) Note that any full rank matrix can always be transformed into (13), for example by using the following transformation matrix:
Since transformation defined by
,
can be used in the (14) where control matrix
to transform the system dynamics such states remain unthat the first changed and becomes the last states. Using (10) the transformed system is obtained
is any basis of . Let the feedback be partitioned as (15)
, , and are real where matrices to be determined. Our design effort is to implement the by a dynamical output feedback described by relation (16a)
(12) subject to where
,
, and where
is . Once a sliding mode the spectrum of and . Equation (12) is then reduced incurs, we have with the desired dynamics. to the motion equation This concludes the proof. Remark 1: As can be seen in (11) the eigenvalue provides some stability margin for the system. To accelerate the reaching should be chosen at least about 3–5 times the phase (10), dominant roots of the sliding eigenvalues. ) the specRemark 2: For single-input systems ( consists of ( ) desired sliding eigenvalues trum of and the sliding margin . By using the well-known Ackermann formula for the feedback matrix gain in this case, the sliding matrix can have the following explicit form [12]:
(16b) is the state vector of the observer of where is a real constant matrix to be determined, order , is a stable matrix to be selected according to the and observer desired dynamics. be the estimation error Letting (17) by some simple manipulations, the following error equation is obtained:
(18) (12a) (12b)
III. LINEAR FUNCTIONAL OBSERVER DESIGN In order to implement sliding-mode control when the states are not physically available an observer is required for state estimation or at least for estimation of their subsets. Examining the control law (3) reveals that it could be implemented by using estimates of the equivalent control and the sliding function. In this section we propose a simple, thus, efficient technique to design an observer for estimating asymptotically a linear state , for the system with unknown inputs functional, (1). Relevant research on this subject can be found, e.g., [15], [16], or more recently, in [17] and [18].
Equations (15) and (18) imply that (16b) can act as a linear provided that matrix functional observer for is chosen to be stable, and matrices and fulfill the following constraints: (19) (20) (21) As matrix can be selected according to the desired dynamics of the observer, there are four unknown matrices ( , , , and ) in (19)–(21) to be solved for. Using (13), (19) and (21) can be expressed as (22a)
HA et al.: DYNAMIC OUTPUT FEEDBACK SLIDING-MODE CONTROL
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(22b)
Augmenting the set of (25a) with the set of (26a) and (27a) gives
and
(28a) (23a)
where (28b)
(23b) can be It is clear from (22a) and (23a) that matrices and easily derived, once matrices and are obtained. It therefore and . remains to solve (20), (22b), and (23b) for matrices To proceed, let matrix be expressed as (24) ( ) is the where th column of . Incorporating (24) into (23b), and after some rearranging, the following matrix-vector equation is obtained (25a) where (25b) (25c) (25d) is the th column of . and Let us now consider (22b). As commented on earlier, can be chosen to be any matrix with a set of stable eigenvalues. By some simple rearranging, (22b) can be cast in a matrix-vector form as (26a) is given by (26b), shown at the where matrix denotes the ( ) element of matrix bottom of the page, and . Similarly, (20) can be rewritten as (27a) where
(27b) and
denotes the (
) element of matrix
.
Lemma 1: [18]: For the dynamical system (1), system (29) is any stable matrix, is selected where is obtained from (28), and arbitrarily with full rank, and are derived, respectively, from (22a) and (23a); can be constructed to generate asymptotically any linear state func, provided that: 1) tional, and 2) matrix defined in (28b) has full row rank. Note that 2) is a necessary and sufficient condition for the solvability of (28). Provided that the above two conditions are satisfied, vector is solved for from linear (28), and thus matrix is obtained from notation (25d). It is then followed that can be derived, respectively, from (22a) and matrices and (23a). As a result, all of the observer parameters are determined , subject to (19)–(21). Consequently, from (18), is defined in (17) as . Since matrix is where asymptotically. Therefore, the estimate stable,
(30) . asymptotically approaches to A proposed algorithm for designing linear functional observers is given below: Design Algorithm 1: 1. Choose the observer order , where denotes the smallest integer greater than or equal . according to (24) 2. Partition matrices according to (25c). and formulate 3. Choose a stable matrix , and arbi) elements of full row rank trarily ( . matrix 4. Formulate (28b) and solve it for vector . Obtain from (25d). and respectively 5. Derive matrices from (22a) and (23a).
(26b)
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Remark 3: To facilitate solving (28b) a singular value decomposition for matrix may be required. In most of the cases, the proposed technique results in functional observers with an order less than . The observer order may however take a large , i.e., when the number of outputs exvalue when ceeds the number of inputs by one. The observer order becomes , which may be greater than . In this then case or if conditions of Lemma 1 are not satisfied, one can still proceed with the design of linear functional observers by using a Moore–Penrose pseudoinverse matrix approach to solve approximately (19)–(21). The observer order can be increased gradually from a smallest value until an approximate solution is obtained.
Substituting (33) into the above equation and using (7a) yield
Conditions of the theorem are now used to obtain
IV. DYNAMIC OUTPUT FEEDBACK SLIDING-MODE CONTROL WITH LINEAR FUNCTIONAL OBSERVERS Let us now apply the results developed in the previous section for estimation of the equivalent control and the sliding function to implement sliding-mode control for system (1). Assume that the following linear functional observers: (31a) (31b) and (32a) (32b) and , rehave been designed to obtain the estimates spectively, of the equivalent control (5) and the sliding function (2a). As expressed in (30), there exist dynamical errors associated with the estimates (33) and (34) and . where Taking this effect into account, the control law may be modified as formulated in the following theorem. Theorem 2: For the system (1) using the sliding-mode control design described in Theorem 1 and the linear functional observers determined in (31) and (32), if the equivalent control (31a) is employed and the robust control is now given by (35) ( is the order where is obtained from (32b) and is the sliding margin), then the state of the observer (32) and vector asymptotically converges to zero. Proof: Let us check the reaching condition (4) for the new . Taking the first time derivative of (34) sliding function gives
As a sliding mode incurs, and since the observer error , according to (34). Thus the state vector asymptotically converges to zero. This concludes the proof. is chosen at least about 3–5 Remark 4: As the eigenvalue times the dominant roots of the sliding eigenvalues according to quickly enough with respect to the sliding Remark 1, mode dynamics. Let us now summarize the development proposed in previous sections in the following design procedure. Design Algorithm 2: 1. Choose the eigenvalues according to the desired sliding-mode dyaccording namics and the sliding margin to Remark 1. 2. Design a suitable state feedback confor the equivalent control (5). troller 3. Select the sliding function(2a) with satisfying condition (7) by matrix using (8). 4. Design linear functional observer (31) using algorithm 1. , design linear func5. Choose tional observer (32) using algorithm 1. 6. Formulate the equivalent control (31b) and the robust control (32b). V. NUMERICAL EXAMPLE Consider a seventh-order aircraft example in [7]. Two inputs are the rudder and the aileron commands. Four measurable outputs are yaw and roll accelerations, bank angle, and the wash out filter state. Note that as mentioned in [10], existing sliding-mode control techniques using output feedback provide no guarantees for a satisfied reaching condition for this system. In the following the method proposed in this paper will be illustrated through the design of a dynamical output feedback sliding-mode controller using linear functional observers. Let us first apply a transformation of the form (14) to obtain the nominal system matrices in the canonical form, as shown in the first entry at the bottom of the next page. The control design procedure is described below.
HA et al.: DYNAMIC OUTPUT FEEDBACK SLIDING-MODE CONTROL
Step 1: Choose the desired sliding eigen, values located at and a double eigenvalue at the sliding . margin Step 2: The feedback gain is found to be the second entry shown at the bottom of the page. can be Step 3: A solution for matrix found from(8) as shown by the third entry at the bottom of the page. Step 4: Design a functional observer to using design algorithm 1 estimate , , , 4.1 For this system, . the observer order is obtained in step 2, vector 4.2 With is formulated as the last entry at the bottom of the page.
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4.3 Choose
and . 4.4 Formulate (28b). Solving this equation yields the first entry at the bottom of the next page. and 4.5 Derive matrices
Step 5: Design a functional observer to using design algorithm 1. estimate , , 5.1 For this system, . the observer order is
,
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5.2 With obtained in step 3, vector is formulated as the second entry at the bottom of the page. and 5.3 Choose . 5.4 Formulate (28b). Solving for this equation yields the third entry at the bottom of the page. and 5.5 Derive matrices
Step 6: Overall, the control law is . The equivalent control is , where is given by subject to the observer dynamics
Fig. 1.
Responses of the states x (t) and x (t).
Fig. 2.
Responses of the control u (t) and u (t).
The robust control is , where is the known perturbation bound and , governed by the following dynamics:
For the sake of simulation it is assumed that and . Fig. 1 shows the responses of the states and of the original system from some nonzero initial and conditions. Fig. 2 shows the two control inputs using the proposed control scheme. Fig. 3 depicts the responses and to an initial conditions (1, ) of the estimation errors of the functional observers, respectively, for and
HA et al.: DYNAMIC OUTPUT FEEDBACK SLIDING-MODE CONTROL
Fig. 3. Responses of the estimation errors e (t) and e
(t).
. As expected, these responses exponentially approach to . zero according to the selection of and VI. CONCLUSION We have presented a methodological approach to design dynamic output feedback sliding-mode control of uncertain dynamical systems using pole placement and linear functional observers. Estimates of the equivalent control and the sliding function are employed to reconstruct the sliding-mode control action. Stability of the closed-loop system under the observerbased sliding-mode controller is guaranteed. Since the sliding surface and the observers parameters are obtained directly from solving linear equations, the control design is rather straightforward and computationally efficient. Step-by-step design algorithms were provided. A numerical example was used to illustrate the design procedure. Simulation results demonstrate the validity of the proposed approach.
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[9] C. Edwards and S. K. Spurgeon, “Compensator based output feedback sliding mode controller design,” Int. J. Control, vol. 71, no. 4, pp. 601–614, 1998. , “On the limitations of some variable structure output feedback [10] controller designs,” Automatica, vol. 36, pp. 743–748, 2000. [11] C. A. Woodham and A. S. I. Zinober, “Eigenvalue placement in a specified sector for variable structure control systems,” Int. J. Control, vol. 57, no. 5, pp. 1021–1037, 1993. [12] J. Ackermann and V. I. Utkin, “Sliding mode control designed based on Ackermann’s formula,” IEEE Trans. Automat. Contr., vol. 43, pp. 234–237, Feb. 1998. [13] Q. P. Ha, D. C. Rye, and H. F. Durrant-Whyte, “Robust sliding mode control with application,” Int. J. Control, vol. 72, pp. 1087–1096, 1999. [14] J. Kautsky, N. K. Nichols, and P. Van Dooren, “Robust pole assignment in linear state feedback,” Int. J. Control, vol. 41, pp. 1129–1155, 1985. [15] T. Mita, “On the synthesis of an unknown input observer,” Int. J. Control, vol. 26, no. 6, pp. 841–851, 1977. [16] F. W. Fairmann, S. S. Mahil, and L. Luk, “Disturbance decoupled observer design via singular value decomposition,” IEEE Trans. Automat. Contr., vol. AC-29, pp. 84–86, Jan. 1984. [17] M. Aldeen and H. Trinh, “Reduced-order linear functional observer for linear systems,” Proc. IEE—Control Theory Applicat., pt. D, vol. 146, pp. 399–405, 1999. [18] H. Trinh and Q. Ha, “Design of linear functional observers for linear systems with unknown inputs,” Int. J. Syst. Sci., vol. 31, pp. 741–749, 2000.
Q. P. Ha (M’97) received the B.E. degree in electrical engineering from Ho Chi Minh City University of Technology, Ho Chi Minh City, Vietnam, in 1983, the Ph.D. degree in engineering science from the Moscow Power Engineering Institute, Moscow, Russia, in 1992, and the Ph.D. degree in Electrical Engineering from the University of Tasmania, Hobart, Australia, in 1997. From 1997 to 2000, he was a Senior Research Associate at the Australian Centre for Field Robotics, University of Sydney, Sydney, Australia. He is currently a Senior Lecturer at the University of Technology, Sydney, Australia. His research interests include robust control and estimation, robotics, and artificial intelligence applications.
H. Trinh received the B.E. (Hons.), M.Eng.Sc., and Ph.D. degrees from the University of Melbourne, Melbourne, Australia, in 1990, 1992, and 1996, respectively. He is currently a Senior Lecturer in the School of Engineering and Technology, Deakin University, Geelong, Australia. His research interests include large-scale systems theory, robust control and estimation, fault diagnosis and fault-tolerant control, robotics, and power systems.
REFERENCES ˙ [1] R. A. DeCarlo, S. H. Zak, and G. P. Matthews, “Variable structure control of nonlinear multivariable systems: A tutorial,” Proc. IEEE, vol. 76, pp. 212–232, Mar. 1988. [2] J. Y. Hung, W. B. Gao, and J. C. Hung, “Variable structure control: A survey,” IEEE Trans. Ind. Electron., vol. 40, pp. 2–22, Feb. 1993. [3] K. D. Young, V. I. Utkin, and U. Ozguner, “A control engineer’s guide to sliding mode control,” IEEE Trans. Contr. Syst. Technol., vol. 7, pp. 328–342, May 1999. ˙ [4] S. H. Zak, J. D. Brehove, and M. J. Corless, “Control of uncertain systems with unmodeled actuator and sensor dynamics and incomplete information,” IEEE Trans. Syst., Man, Cynern., vol. 19, pp. 241–257, Mar./Apr. 1989. ˙ and S. Hui, “Output feedback in variable structure controllers [5] S. H. Zak and state estimators for uncertain/nonlinear dynamical systems,” Proc. IEE—Control Theory Applicat., pt. D, vol. 140, pp. 41–50, 1993. [6] C. Edwards and S. K. Spurgeon, “Sliding mode stabilization of uncertain systems using only output information,” Int. J. Control, vol. 62, pp. 1129–1144, 1995. [7] B. S. Heck, S. V. Yallapragada, and M. K. H. Fan, “Numerical methods to design the reaching phase of output feedback variable structure control,” Automatica, vol. 31, pp. 275–279, 1995. [8] S. K. Bag, S. K. Spurgeon, and C. Edwards, “Output feedback sliding mode design for linear uncertain systems,” Proc. IEE—Control Theory Applicat., pt. D, vol. 144, pp. 209–216, 1997.
H. T. Nguyen (SM’99) received the B.E. (Honors 1), M.E., and Ph.D. degrees from the University of Newcastle, Newcastle, Australia, in 1976, 1977, and 1980, respectively. He is a Professor of Electrical Engineering at the University of Technology, Sydney, Australia, where he is Head of the Mechatronics and Intelligent Systems Group in the Faculty of Engineering and Co-Director of the Key University Research Centre for Health Technologies. He is also Executive Director of AIMEDICS Pty Ltd. His research interests are in biomedical engineering, advanced control and instrumentation, and artificial intelligence. Prof. Nguyen is a Member of the Order of Australia (AM).
H. D. Tuan (M’95) was born in Hanoi, Vietnam, in 1964. He received the diploma and the Ph.D. degree, both in applied mathematics, from Odessa State University, Odessa, Ukraine, in 1987 and 1991, respectively. From 1991 to 1994, he was a Researcher with the Optimization and Systems Division, Vietnam National Centre for Science and Technologies. From 1994 to 1999, he was an Assistant Professor in the Department of Electronic-Mechanical Engineering, Nagoya University, Nagoya, Japan. In 1999, he joined the Toyota Technological Institute, Nagoya, Japan, where he is an Associate Professor in the Department of Electrical and Computer Engineering. His research interests include theoretical developments and applications of optimization-based methods in broad areas of control, signal processing, and communication.
