Sliding Mode Control Part 13 pot

21 A Biomedical Application by Using Optimal Fuzzy Sliding-Mode Control Bor-Jiunn Wen Center for Measurement Standards, Industrial Technology Research Institute Hsinchu, Taiwan, R.O.C Introduction The development of biochips is a major thrust of the rapidly growing biotechnology industry Research on biomedical or biochemical analysis miniaturization and integration has made explosive progress by using biochips recently For example, capillary electrophoresis (CE), sample preconcentration, genomic DNA extraction, and DNA hybridization have been successfully miniaturized and operated in a single-step chip However, there is still a considerable technical challenge in integrating these procedures into a multiple-step system In biometric and biomedical applications, the special transporting mechanism must be designed for the μTAS (micro total analysis system) to move samples and reagents through the microchannels that connect the unit procedure components in the system Therefore, an important issue for this miniaturization and integration is microfluid management technique, i.e., microfluid transportation, metering, and mixing This charter introduced a method to achieve the microfluidic manipulated implementation on biochip system with a pneumatic pumping actuator and a feedbacksignal flowmeter by using an optimal fuzzy sliding-mode control (OFSMC) design based on the 8051 microprocessor However, the relationships of the pumping mechanisms, the operating conditions of the devices, and the transporting behavior of the multi-component fluids in these channels are quite complicated Because the main disadvantages of the mechanical valves utilized moving parts are the complexity and expense of fabrication, and the fragility of the components Therefore, a novel recursively-structured apparatus of valveless microfluid manipulating method based on a pneumatic pumping mechanism has been utilized in this study The working principle of this pumping design on this device should not directly relate to the nature of the fluid components The driving force acting on the microliquid drop in the microchannel of this device is based on the pneumatic pumping which is induced by a blowing airflow Furthermore, the pneumatic pumping actuator should be independent of the actuation responsible for the biochemical analysis on the chip system, so the contamination of pneumatic pumping source can be avoided The total biochip mechanism consists of an external pneumatic actuator and an on-chip planar structure for airflow reception In order to achieve microfluidic manipulation in the microchannel of the biochip system, pneumatic pumping controller plays an important role Therefore, a design of the controller 410 Sliding Mode Control has been investigated numerically and experimentally in the present charter In the control structure of biochip system, at first, the mathematical model of the biochip mechanism is identified by ARX model Second, according to the results of the biochip-mechanism identification, the control-algorithm design is developed By the simulation results of the biochip system with a feedback-signals flowmeter, they show the effectiveness of the developed control algorithm Third, architecture of the control algorithm is integrated on a microprocessor to implement microfluidic manipulation Since the mathematical model of the flow control mechanism in the biochip microchannels is a complicated nonlinear plant, the fuzzy logic control (FLC) design of the controller will be utilized Design of the FLC based on the fuzzy set theory has been widely applied to consumer products or industrial process controls In particular, they are very effective techniques for complicated, nonlinear, and imprecise plants for which either no mathematical model exists or the mathematical model is severely nonlinear The FLC can approximate the human expert’s control behaviors to work fine in such ill-defined environments For some applications, the FLC can be divided into two classes 1) the general-purpose fuzzy processor with specialized fuzzy computations and 2) the dedicated fuzzy hardware for specific applications Because the general-purpose fuzzy processor can be implemented quickly and applied flexibly, and dedicated fuzzy hardware requires long time for development, the general-purpose fuzzy processor-8051 microcontroller can be used Nevertheless, there are also systemic uncertainties and disturbance in FLC controller Because sliding-mode control (SMC) had been known as an effective approach to restrain the systemic uncertainties and disturbance, SMC algorithm was utilized In order to achieve a robust control system, the microcontroller of the biochip system combining FLC and SMC algorithms optimally has been developed Therefore, an OFSMC based on an 8051 microcontroller has been investigated numerically and experimentally in this charter Hence, microfluidic manipulation in the microchannel of the biochip system based on OFSMC has been implemented by using an 8051 microcontroller The microfluidic manipulation based on the microcontroller has successfully been utilized to improve the reaction efficiency of molecular biology First, it was used in DNA hybridization There are two methods to improve the efficiency of the nucleic acid hybridization in this charter The first method is to increase the velocity of the target nucleic acid molecules, which increases the effective collision into the probe molecules as the target molecules flow back and forth The second method is to introduce the strain rates of the target mixture flow on the hybridization surface This hybridization chip was able to increase hybridization signal 6-fold, reduce non-specific target-probe binding and background noises within 30 minutes, as compared to conventional hybridization methods, which may take from hours to overnight Second, it was used in DNA extraction When serum existed in the fluid, the extraction efficiency of immobilized beads with solution flowing back and forth was 88-fold higher than that of free-beads Third, it could be integrated in lab-on-a-chip For the Tee-connected channels, it demonstrated the ability of manipulating the liquid drop from a first channel to a second channel, while simultaneously preventing flow into the third channel Because there is a continuous airflow at the “outlet” during fluid manipulation, it can avoid contamination of the air source similar to the “laminar flow hook” in biological experiments The charter is organized as follows In Section 2, we introduce the structure of the biochip control system In Section 3, the fundamental knowledge of OFSMC and the model of the biochip system are introduced, and we address the OFSMC scheme and the associated 411 A Biomedical Application by Using Optimal Fuzzy Sliding-Mode Control simulations In Section 4, the OFSMC IC based on 8051 microprocessor is designed, and the results of the real-time experiment are presented In Section 5, the efficiency improvement for the molecular biology reaction and DNA extraction by using OFSMC method are presented Finally, the conclusion is given in Section Structure of the biochip control system The structure of the biochip control system (Fig 1) contains six parts: an air compressor, two flow controllers and two flowmeters, a flow-control chip, a biochip, photodiodes system, and a control-chip circuit system One had designed a pneumatic device with planar structures for microfluidic manipulation (Chung, Jen, Lin, Wu & Wu, 2003) Pneumatic devices without any microfabricated electrodes or heaters, which will have a minimal effect on the biochemical properties of the microfluid by not generating electrical current or heat, are most suitable for µTAS A pneumatic structure possessing the ability of bi-directional pumping should be utilized in order to implement a pneumatic device which can control the movement of microfluid without valves or moving parts Biochip Air Compressor Flow controller Buffer Tank PD2 Flow Control Chip flowmeter IR IR Control-input signals Feedback signals PD1 ADC DAC 8051 CONTROL CIRCUIT Feedback-Signal Process of Photodiode System Fig Structure of the biochip control system The schematic diagram of the single pneumatic structure, which provides suction and exclusion by two inlets, is depicted in Fig When the air flows through inlet A only, it causes a low-pressure zone behind the triangular block and suction occurs in the vertical microchannel Furthermore, when the air flows through inlet B only, the airflow is induced into the vertical microchannel to generate exclusion The numerical and experimental results of the pressure and the stream tracers for the condition of the flow-control chip have been demonstrated (Marquardt, 1963) According to the principle of the flow-control chip, the microfluidic manipulation on the biochip is presented in this study by using OFSMC rules 412 Sliding Mode Control with two flow controllers and two flowmeters Since the biochip in the biochip system is a consumer, the photodiodes system should be utilized for sensing the feedback signals of the position of the reagent in the microchannel of the biochip Hence, DNA extraction can be achieved in this study 2.0 QA 3.0 Inlet A Y=4.0 QB Inlet B 2.0 2.0 24.0 20.0 Suction Exclusion Unit: mm Fig Single pneumatic structure Design of the biochip control system 3.1 Design of optimal fuzzy sliding mode control The biochip system of this design is shown in Fig If the biochip is DNA extraction chip, the extraction beads are immobilized on the channels When the bio-fluidics does not flow the place without beads, the time of not extracting DNA can be reduced, and the extraction efficiency will also be improved So the control of bio-fluidics’ location is critical to DNA extraction (or hybridization) efficiency The biochip system depicted in Fig is a nonlinear system Since the mathematical model of the flow-control mechanism and the microchannels in the biochip is a complicated nonlinear model, FLC design of the controller was utilized The basic idea behind FLC is to incorporate the expert experience of a human operator in the design of the controller in controlling a process whose input-output relationship is described by a collection of fuzzy control rules (Altrock, Krause & Zimmermann, 1992) The heart of the fuzzy control rules is a knowledge base consisting of the so-called fuzzy IF-THEN rules involving linguistic variables rather than a complicated dynamic model The typical architecture of a FLC, shown in Fig 3, is comprised of four principal components: a fuzzification interface, a knowledge base, an inference engine, and defuzzification interface The fuzzification interface has the effect of transforming crisp measured data into suitable linguistic values; it was designed first so that further fuzzy inferences could be performed according to the fuzzy rules (Polkinghorne, Roberts, Burns & Winwood, 1994) The heart of the fuzzification interface is the design of membership function There are many kinds of membership A Biomedical Application by Using Optimal Fuzzy Sliding-Mode Control 413 functions - Gaussian, trapezoid, triangular and so on - of the fuzzy set In this paper, a triangular membership function was utilized, as shown in Figs 4-5 Degree of membership Fig Architecture of a fuzzy logic controller NB NM NS -10 ZE PS PM PB +10 d NB: negative big NM: negative medium NS: negative small PB: positive big PM: positive medium PS: positive small ZE: zero Fig Membership function-input variable (d) of photodiode detector 414 Degree of memebership Sliding Mode Control MNB MNM MNS M 0 MPS MPM MPB 10 z MNB: medium negative big MPB: medium positive big MNM: medium negative medium MPM: medium positive medium MNS: medium negative small MPS: medium positive small M: medium Fig Membership function-output variable (z) of photodiode detector The overall fuzzy rules for the biochip system are defined as the following: IF d is NB then z j is MPB IF d is NM then z j is MPM IF d is NS then z j is MPS IF d is ZE then z j is M IF d is PS then z j is MNS IF d is PM then z j is MNM IF d is PB then z j is MNB where d is input variable of the photodiode signal, and z is output variable of the photodiode signal The inference engine is based on the compositional rule of inference with knowledge base for approximate reasoning suggested by Zadeh (Zadeh, 1965; Zadeh, 1968) An inference engine is the kernel of the FLC in modeling human decision making within the conceptual framework of fuzzy logic and reasoning Hence, the fuzzification interface and fuzzy rules are designed completely before fuzzy reasoning In this paper, since there are many structures of inference engine, fuzzy reasoning-Mamdani’s minimum fuzzy implication rule (MMFIR) method (Mamdani, 1977; Lee, 1990; Altrock, Krause & Zimmermann, 1992; Lin and Lee, 1999) was utilized For simplicity, assume two fuzzy rules as follows: R1: IF x is A1 and y is B1, then z is C1, R2: IF x is A2 and y is B2, then z is C2 415 A Biomedical Application by Using Optimal Fuzzy Sliding-Mode Control Then the firing strengths α and α of the first and second rules may be expressed as α = μ A1 ( x0 ) ∧ μB1 ( y0 ) and α = μ A2 ( x0 ) ∧ μB2 ( y0 ) , where μ A1 ( x0 ) and μ B1 ( y0 ) indicate the degrees of partial match between the user-supplied data and the data in the fuzzy rule base In MMFIR fuzzy resoning, the ith fuzzy control rule leads to the control decision μ C ' ( w ) = α i ∧ μCi ( w ) i The final inferred consequent C is given by μ C ( w ) = μC ' ∨ μC ' = [α ∧ μC1 ( w )] ∨ [α ∧ μC2 ( w )] The fuzzy reasoning process is illustrated in Fig μA 1 μB μC A1 B1 1 C1 μC X μA x0 μC B2 X Y μB A2 y0 Z Y C2 Z Z Fig Fuzzy reasoning of MMFIR method Defuzzification is a mapping from a space of fuzzy control actions defined over an output universe of discourse into a space of crisp control actions This process is necessary because fuzzy control actions cannot be utilized in controlling the plant for practical applications Hence, the widely used center of area (COA) method, which generates the center of gravity of the possibility distribution of a control action, was utilized In the case of a discrete universe, this method yields ∑ j =1 μC ( z j )z j n ∑ j =1 μC ( z j ) n zCOA = (1) where n is the number of quantization levels of the output, z j is the amount of control output at the quantization level j, and μC ( z j ) represents its membership degree in the output fuzzy set C 416 Sliding Mode Control The biochip system depicted in Fig is a nonlinear system that has been used as an application to study real world nonlinear control problems by different control techniques (Cheng & Li, 1998; Li & Shieh, 2000) The model of the biochip system is identified by ARX model, as ⎧X ( k + 1) = Az X( k ) + Bzu( k ) ⎨ ⎩ y( k ) = C z X ( k ) (2) where X ( k ) ∈ R n is the state variables of system, u( k ) ∈ Rm is the input voltage of the flow controller and y( k ) ∈ R r is the assumed model output related to the position of the reagent in the microchannel of the biochip The system is controllable and observable Sliding mode control’s robust and disturbance-insensitive characteristics enable the SMC controller to perform well in systems with model uncertainty, disturbances and noises In this paper, in addition to FLC controller, SMC controller was utilized to design the control input voltage of the flow controller To design SMC controllers, a sliding function was designed first, and then enforced a system trajectory to enter sliding surface in a finite time As soon as the system trajectory entered the sliding surface, they moved the sliding surface to a control goal To sum up, there are two procedures of sliding mode, as shown in Fig X (0) A pproaching m ode Sliding m ode C ontrol goal point X ( ∞ )=0 Super space S(X )=0 X (t h ) Touch super space in a finite tim e t h Fig Generation of sliding mode The proposed SMC controller was based on pole placement (Chang, 1999), since the sliding function could be designed by pole placement Some conditions were set for the sliding vector design in the proposed sliding mode control: Re {λi } < , α j ∈ R , α j < , α j ≠ λi Any eigenvalue in {α , ,α m } is not in the spectrum of Az The number of any repeated eigenvalues in {λ1 , , λn − m ,α , ,α m } is not greater than m, the rank of Bz where {λ1 , λ2 , , λn − m } are sliding-mode eigenvalues and {α ,α , ,α m } are virtual eigenvalues A Biomedical Application by Using Optimal Fuzzy Sliding-Mode Control 417 As proved by Sinswat and Fallside (Sinswat & Fallside, 1977), if the condition (3) in the above is established, the control system matrix Az − BzK can be diagonalized as −1 ⎡V ⎤ ⎡Φ Az − BzK = ⎢ ⎥ ⎢ V ⎣F ⎦ ⎣ 0 ⎤ ⎡V ⎤ ΓF ⎥ ⎢ F ⎥ ⎦⎣ ⎦ (3) where ΦV = diag [ λ1 , λ2 , , λn − m ] , ΓF = diag [α ,α , ,α m ] , and V and F are left eigenvectors with respect to ΦV and Γ F , respectively Hence, Eq (3) can be rewritten as ⎧V ( Az − BzK ) = ΦVV ⎨ ⎩ F( Az − BzK ) = Γ F F (4) FAz − Γ F F = (FBz )K (5) rank(FAz − Γ F F ) = rank(F ) (6) Rearrangement of Eq (4) yields According to Chang (Chang, 1999), Since F contains m independent left eigenvectors, one has rank(F ) = m From Eqs (5) and (6), it is also true that rank(FAz − Γ F F ) = rank((Fbz )K ) = rank(F ) = m In other words, FBz is invertible With the designed left eigenvector F above, the sliding function S( k ) is designed as S( k ) = FX ( k ) (7) The second step is the discrete-time switching control design A different and much more expedient approach than that of Gao et al (Gao, Wang & Homaifa, 1995) is adopted here This approach is called the reaching law approach that has been proposed for continuous variable structure control (VSC) systems (Gao, 1990; Hung, Gao & Hung, 1993; Gao & Hung, 1993) This control law is synthesized from the reaching law in conjunction with a plant model and the known bounds of perturbations For a discrete-time system, the reaching law is (Gao, Wang & Homaifa, 1995) S( k + 1) − S( k ) = −qTS( k ) − ε T sgn(S( k )) (8) where T > is the sampling period, q > , ε > and − qT > Therefore, the switching control law for the discrete-time system is derived based on this reaching law From Eq (7) and pole-placement method, S( k ) and S( k + 1) can be obtained in terms of sliding vector F as, ⎧S( k ) = FX ( k ) ⎨ ⎩S( k + 1) = FX( k + 1) = F( Az − BzK )X( k ) + FBzu( k ) (9) where K ∈ Rn is a gain matrix obtained by assigning n desired eigenvalues {λ1 , , λn −m ,α , ,α m } of A − BK It follows that 418 Sliding Mode Control S( k + 1) − S( k ) = F( Az − BzK )X( k ) + FBzu( k ) − FX( k ) (10) From Eqs (8) and (10), S( k + 1) − S( k ) = −qTS( k ) − ε T sgn(S( k )) = F( Az − BzK )X ( k ) + FBzu( k ) − FX ( k ) Solving for u( k ) obtains the switching control law u( k ) = −( FBz )−1[F( Az − BzK )X( k ) + ( qT − 1)FX( k ) + ε T sgn(FX( k ))] (11) In order to achieve the output tracking control, a reference command input r ( k ) is introduced into the system by modifying the state feedback control law up ( k ) = −KX( k ) with pole-placement design method (Franklin, Powell & Workman, 1998) to become up ( k ) = N ur ( k ) − K ( X( k ) − N x r( k )) (12) where ⎡ N u ⎤ ⎡ Az − I ⎢N ⎥ = ⎢ C ⎣ x⎦ ⎣ z −1 Bz ⎤ ⎡0 ⎤ ⎥ ⎢I ⎥ ⎦ ⎣ ⎦ (13) The proposed SMC input, based on Eq (13), is assumed to be us ( k ) = up ( k ) + u = N ur ( k ) − K ( X( k ) − N x r( k )) + u (14) Substituting Eq (11) into (14) gives the proposed SMC input as us ( k ) = N ur ( k ) − K ( X( k ) − N x r( k )) −(FBz )−1[F( Az − BzK )X( k ) + (qT − 1)FX( k ) + ε T sgn(FX( k ))] (15) The pole-placement SMC design method utilizes the feedback of all the state variables to form the desired sliding vector In practice, not all the state variables are available for direct measurement Hence, it is necessary to estimate the state variables that are not directly measurable In practice, a discrete linear time-invariant system sometimes has system disturbances and measurement noise Hence, linear quadratic estimator (LQE) will be applied here to estimate optimal states in having system disturbances and measurement noise According to Eq (2), consider a system model as ⎧X ( k + 1) = Az X( k ) + Bzu( k ) + Gν ( k ) ⎨ ⎩ y( k ) = C z X ( k ) + ω ( k ) (16) where X( k ) ∈ Rn is the state variable, u( k ) ∈ Rm is the control input voltage , y '( k ) ∈ Rr is the assumed plant output related to the XY stage position, and ν ( k ) ∈ Rn and ω ( k ) ∈ Rr are system disturbances and measurement noise with covariances E[ωω T ] = Q , E[νν T ] = R and E[ων T ] = ˆ The objective of LQE is to find a vector X( k ) which is an optimal estimation of the present state X ( k ) Here “optimal” means the cost function Part New Trends in the Theory of Sliding Mode Control 22 Sliding Mode Control of Second Order Dynamic System with State Constraints Aleksandra Nowacka-Leverton and Andrzej Bartoszewicz Technical University of Łódź, Institute of Automatic Control 18/22 Stefanowskiego St 90-924 Łódź, Poland Introduction In recent years much of the research in the area of control theory focused on the design of discontinuous feedback which switches the structure of the system according to the evolution of its state vector This control idea may be illustrated by the following example Example Let us consider the second order system x1 = x x2 = x2 + ui i = 1, , (1) where x1(t) and x2(t) denote the system state variables, with the following two feedback control laws u = f1 ( x1 , x ) = -x2 - x1 (2) u = f2 ( x1 , x ) = -x2 - 4x1 (3) The performance of system (1) controlled according to (2) is shown in Fig 1, and Fig presents the behaviour of the same system with feedback control (3) It can be clearly seen from those two figures that each of the feedback control laws (2) and (3) ensures the system stability only in the sense of Lyapunov However, if the following switching strategy is applied ⎧1 ⎪ i=⎨ ⎪2 ⎩ for for {x1 , x } < {x1 , x } ≥ (4) then the system becomes asymptotically stable This is illustrated in Fig Moreover, it is worth to point out that system (1) with the same feedback control laws may exhibit completely different behaviour (and even become unstable) For example, if the switching strategy (4) is modified as ⎧1 ⎪ i=⎨ ⎪2 ⎩ for for {x1 , x } ≥ {x1 , x } < (5) 432 Sliding Mode Control then the system output increases to infinity The system dynamic behaviour, in this situation, is illustrated in Fig x2 -1 -2 -3 -3 -2 -1 x1 Fig Phase portrait of system (1) with controller (2) x2 -1 -2 -3 -4 -4 -2 x1 Fig Phase portrait of system (1) with controller (3) This example presents the concept of variable structure control (VSC) and stresses that the system dynamics in VSC is determined not only by the applied feedback controllers but also, to a large extent, by the adopted switching strategy VSC is inherently a nonlinear technique and as such, it offers a variety of advantages which cannot be achieved using conventional linear controllers Our next example shows one of those favourable features – namely it demonstrates that VSC may enable finite time error convergence Sliding Mode Control of Second Order Dynamic System with State Constraints 433 1.5 t →∞ x2 0.5 t0 -0.5 -1 -1.5 -1.5 -1 -0.5 0.5 1.5 x1 Fig Phase portrait of system (1) when switching strategy (4) is applied 20 15 10 t0 x2 -5 -10 -15 -20 -20 -10 10 20 x1 Fig Phase portrait of system (1) when switching strategy (5) is applied Example In this example, again we consider system (1), however now we apply the following controller u = -x - a sgn ( x1 ) - b sgn ( x ) (6) where a > b > Closer analysis of the behaviour of system (1) with control law (6) demonstrates that, in this example, the system error converges to zero in finite time which can be expressed as T= a ⎞ ⎛ x 10 ⎜ + ⎟ b a+b ⎠ ⎝ a-b (7) 434 Sliding Mode Control where x10 and x20 = represent initial conditions of system (1) Even though the error converges to zero in finite time, the number of oscillations in the system tends to infinity, with the period of the oscillations decreasing to zero This is illustrated in Figs and In the simulation example presented in the figures, the following parameters are used a = 7, b = 3, x10 = 20 and x20 = Consequently, the system error is nullified at the time instant T = 12.045 and remains equal to zero for any time greater than T Clearly these favourable properties are achieved using finite control signal This controller, due to the way the phase trajectory – shown in Fig – is drawn, is usually called “twisting controller” It also serves as a good, simple example of the second order sliding mode controllers The two examples presented up to now demonstrate the principal properties of VSC systems However, the main advantage of the systems is obtained when the controlled plant exhibits the sliding motion (DeCarlo et al., 1988; Hung et al., 1993; Slotine & Li, 1991; Utkin, 1977) The idea of sliding mode control (SMC) is to employ different feedback controllers acting on the opposite sides of a predetermined surface in the system state space Each of those controllers pushes the system representative point (RP) towards the surface, so that the RP approaches the surface, and once it hits the surface for the first time it stays on it ever after The resulting motion of the system is restricted to the surface, which graphically can be interpreted as “sliding” of the system RP along the surface This idea is illustrated by the following example Example Let us consider another second order plant x1 = x x = b cos ( m x1 ) + u b < 1, (8) where b and m are possibly unknown constants We select the following line in the state space s = x2 + c x1 = (9) u = - c x - sgn ( s ) (10) (c = const.) and apply the controller In this equation sgn(.) function represents the sign of its argument, i.e sgn(s < 0) = –1 and sgn(s > 0) = +1 With this controller the system representative point moves towards line (9) always when it does not belong to the line Then, once it hits the line, the controller switches the plant input (in the ideal case) with infinite frequency Therefore, line (9) is called the switching line Furthermore, since after reaching the line, the system RP slides along it, then the line is also called the sliding line This example is illustrated in Fig The system parameters used in the presented simulation are c = 0.5, b = 0.75, m = 10 and the simulation is performed for the following initial conditions x10 = and x20 = Notice that when the plant remains in the sliding mode, its dynamics is completely determined by the switching line (or in general the switching hypersurface) parameters This implies that neither model uncertainty nor matched external disturbance affects the plant dynamics (Draženović, 1969) which is a highly desirable system property This property can also be justified geometrically, if one notices that in our example the slope of line (9) fully governs the plant motion in the sliding mode Therefore, in SMC systems we usually make the distinction between two phases: the first one – called the reaching phase – lasts until the controlled Sliding Mode Control of Second Order Dynamic System with State Constraints 435 plant RP hits the switching surface, and the second one – the sliding phase – begins when the RP reaches the surface In the latter phase the plant insensitivity to a class of modeling inaccuracies and external disturbances is ensured Let us stress that the system robustness with respect to unmodeled dynamics, parameter uncertainty and external disturbances is guaranteed only in the sliding mode Therefore, shortening or (if possible) even complete elimination of the reaching phase is an important and timely research issue (see for example Bartoszewicz & Nowacka-Leverton, 2009; Pan & Furuta, 2007; Sivert, 2004; Utkin & Shi, 1996) in the field of SMC 15 10 t3 x2 t0 t2 t4 -5 -10 -15 -15 t1 -10 -5 10 15 20 25 x1 Fig Phase portrait of system (1) controlled according to (6) 25 20 x1 x1 x2 15 10 x2 -5 -10 -15 time 10 Fig State variables of system (1) controlled according to (6) 15 436 Sliding Mode Control t0 t→∞ x2 -1 -2 -3 -2 s = x2 + cx1 = -1 x1 Fig Phase trajectory of system (1) controlled according to (10) Another immediate consequence of the fact that in the sliding mode, the system RP is restricted to the switching hypersurface (which is a subset of the state space) is reduction of the system order If the system of the order n has m independent inputs, then the sliding mode takes place on the intersection of m hypersurfaces and the reduced order of the system is equal to the difference n – m To be more precise, in multi-input systems the sliding mode may take place either independently on each switching hypersurface or only on the intersection of the surfaces In the first case the system RP approaches each surface at any time instant and once it hits any of the surfaces it stays on this surface ever after In the latter case, however, the system RP does not necessarily approach each of the surfaces, but it always moves towards their intersection In this case the system RP may hit a surface and move away from it (might possibly cross a switching surface), but once it reaches the intersection of all the surfaces, then the RP never leaves it One of the major tasks in the SMC system design is the selection of an appropriate control law This can be achieved either by assuming a certain kind of the control law (usually motivated by some previous engineering experience) and proving that this control satisfies one of the so-called reaching conditions or by applying the reaching law approach The reaching conditions (Edwards & Spurgeon, 1998) ensure stability of the sliding motion and therefore they are naturally derived using Laypunov stability theory On the other hand, if the reaching law approach is adopted for the purpose of a sliding mode controller construction (Bartoszewicz, 1998; Bartoszewicz, 1996; Gao et al, 1995; Golo & Milosavljević, 2000; Hung et al., 1993), then a totally different design philosophy is employed In this case the desired evolution of the switching variable s is specified first, and then a control law ensuring that s changes according to the specification is determined Sliding mode controllers guarantee system insensitivity with respect to matched disturbance and model uncertainty (Draženović, 1969), and cause reduction of the plant order Moreover, they are computationally efficient, and may be applied to a wide range of various, possibly nonlinear and time-varying plants However, often they also exhibit a serious drawback which essentially hinders their practical applications This drawback – high frequency oscillations which inevitably appear in any real system whose input is Sliding Mode Control of Second Order Dynamic System with State Constraints 437 supposed to switch infinitely fast – is usually called chattering If system (8) exhibits any, even arbitrarily small, delay in the input channel, then control strategy (10), will cause oscillations whose frequency and amplitude depend on the delay With the decreasing of the delay time, the frequency rises and the amplitude is getting smaller This is a highly undesirable phenomenon, because it causes serious wear and tear on the actuator components Therefore, a few methods to eliminate chattering have been proposed The most popular of them uses function ⎧ -1 ⎪ ⎪ sat ( s ) = ⎨ s ⎪ρ ⎪ ⎩ for s < -ρ for s ≤ ρ (11) for s > ρ instead of sgn(s) in the definition of the discontinuous control term With this modification the term becomes continuous and the switching variable does not converge to zero but to the closed interval [–ρ, ρ] Consequently, the system RP after the reaching phase termination, belongs to a layer around the switching hyperplane and therefore this strategy is called boundary layer controller (Slotine & Li, 1991) Other approaches to the chattering elimination include: • introduction of other nonlinear approximations of the discontinuous control term, for example the so called fractional approximation defined as approx ( s ) = s ε+ s (12) where ε is a small positive constant (Ambrosino et al., 1984; Xu et al., 1996); • replacing the boundary layer with a sliding sector (Shyu et al., 1992; Xu et al., 1996); • using dynamic sliding mode controllers (Sira-Ramirez, 1993a; Sira-Ramirez, 1993b; Zlateva, 1996); • using fuzzy sliding mode controllers (Palm, 1994; Palm et al., 1997); • using second (or higher) order sliding mode controllers (Bartolini et al., 1998; Levant, 1993) The phenomenon of chattering has been extensively analyzed in many papers using describing function method and various stability criteria (Shtessel & Lee, 1996) As it has already been mentioned, the switching surface completely determines the plant dynamics in the sliding mode Therefore, selecting this surface is one of the two major tasks in the process of the SMC system design In order to stress this issue let us point out that the same controller which has been considered in the last example may result in a very different system performance, if the sliding line slope c is selected in another way This can be easily noticed if one takes into account any negative c Then, controller (10) still ensures stability of the sliding motion on line (9), i.e the system RP still converges to the line, however the system is unstable since both state variables x1 and x2 tend to (either plus or minus) infinity while the system RP slides away from the origin of the phase plane along line (9) Since sliding mode control is well known to be a robust and computationally efficient regulation technique which may be applied to nonlinear and possibly time-varying plants, then the proper design of the sliding mode controllers has recently become one of the most 438 Sliding Mode Control extensively studied research topics within the field of control engineering This design process usually breaks into two distinct parts: in the first part the switching surface is selected, and in the second one the control signal which always makes the system representative point approach the surface is chosen Once the representative point hits the surface, then under the same control signal, the point remains on the surface Thus, the switching surface fully determines the system dynamics in the sliding mode and should be carefully selected by the system designer In this chapter we consider the second order, nonlinear, time-varying system subject to the acceleration and velocity constraints We introduce a continuously time-varying switching line adaptable to the initial conditions of the system which guarantees the existence of a sliding mode on this line At the time t = t0 the line passes through the representative point, specified by the initial conditions of the system, in the error state space Afterwards, the line moves smoothly, with a constant deceleration and a constant angle of inclination, to the origin of the space and having reached the origin the line remains fixed Thus the proposed control algorithm eliminates the reaching phase and forces the representative point of the system to always stay on the switching line Consequently, our control is robust with respect to the external disturbance and model uncertainty from the very beginning of the control action Furthermore, in order to obtain good dynamic performance of the considered system, the switching line is designed in such a way that the integral absolute error (IAE) over the whole period of the control action is minimised and state constraints are satisfied at the same time The presented method is verified by the simulation example The control algorithm proposed in this chapter may be regarded as an alternative solution to the elegant and currently widely accepted integral sliding mode control technique (Utkin & Shi, 1996) The main advantage of our approach is explicit consideration of state constraints in the controller design process Furthermore, the novelty of our work demonstrates itself also in the IAE optimal performance and error convergence without oscillations or overshoots Problem formulation In this chapter we consider the time-varying and nonlinear, second order system described by the following equations x1 = x2 x = f(x, t) + Δf(x, t) + b(x, t)u + d(t) (13) where x1, x2 are the state variables of the system and x(t) = [x1(t) x2(t)]T is the state vector, t denotes time, u is the input signal, b, f – are a priori known, bounded functions of time and the system state, Δf and d are functions representing the system uncertainty and external disturbances, respectively Further in this chapter, it is assumed that there exists a strictly positive constant δ which is the lower bound of the absolute value of b(x, t), i.e < δ = inf{|b(x, t)|} Furthermore, functions Δf and d are unknown and bounded Therefore, there exists a constant μ which for every pair (x, t) satisfies the following inequality |Δf (x, t) + d(t)| ≤ μ The initial conditions of the system are denoted as x10 , x20 where x10 = x1(t0), x20 = x2(t0) System (13) is supposed to track the desired trajectory given as a function of time xd (t) = [x1d (t) x2d (t)]T where x 2d (t) = x 1d (t) and x2d (t) is a differentiable function of time The trajectory tracking error is defined by the following vector 439 Sliding Mode Control of Second Order Dynamic System with State Constraints e(t) = [e1(t) e2(t)]T = x(t) – xd(t) (14) Hence, we have e1(t) = x1(t) – x1d (t) and e2 (t) = x2(t) – x2d (t) In this chapter it is assumed that at the initial time t = t0, the tracking error and the error derivative can be expressed as e1 ( t ) = e0 ≠ 0, e ( t ) = (15) where e0 is an arbitrary real number different from zero This condition is indeed satisfied in many practical applications such as position control or set point change of second order systems An example of these applications is point to point (PTP) control of robot manipulators, that is moving the manipulator arm from its initial location where it is originally at a halt, to another predefined position at which the arm stops and again is expected to remain at rest Further in this chapter, we present a detailed description of the sliding mode control strategy which ensures optimal performance of the system and its robustness with respect to both the system uncertainty Δf (x, t) and external disturbance d(t) Sliding mode controller In order to effectively control system (13), i.e to eliminate the reaching phase and to obtain system insensitivity with respect to both external disturbance d(t) and the model uncertainty Δf (x, t) from the very beginning of the system motion, we introduce a time-varying switching line The line slope does not change during the control process, which implies that the line moves on the phase plane without rotating In other words, the line is shifted in the state space with a constant angle of inclination At the beginning the line moves with a constant deceleration in the state space and then it stops at a time instant tf > t0 Consequently, the switching line can be described by the following equation ( ) s(e, t) = where s(e, t) = e (t) + ce (t) + Ct + Bt + A δ (16) ⎧1 for t ∈ [ 0, t f ) ⎪ δ=⎨ ⎪0 for t ∈ [ t f , ∞ ) ⎩ (17) where and c, A and B are constants The selection of these constants will be considered further in this chapter In order to ensure system (13) stability in the sliding motion on the line described by equations (16) parameter c in this equation must be strictly positive, i.e c > Furthermore, in order to actually eliminate the reaching phase, and consequently to ensure insensitivity of the considered system from the very beginning of its motion, constants A, B, C and c should be chosen in such a way that the representative point of the system at the initial time t = t0 belongs to the switching line For that purpose, the following condition must be satisfied s ⎡e ( t ) , t ⎤ = e ( t ) + ce1 ( t ) + Ct + Bt + A = ⎣ ⎦ Notice that the input signal (18) 440 Sliding Mode Control { } u = -f(x, t) - ce (t) + x 2d (t) - ( 2Ct + B ) δ - γsgn ⎡s ( e, t ) ⎤ b(x, t) ⎣ ⎦ (19) where γ = η + μ and η is a strictly positive constant, ensures the stability of the sliding motion on the switching line (16) In order to verify this property we consider the product s(e, t)s(e, t) = s(e, t)[e (t) + ce (t) + (2Ct + B)δ ] Taking into account (13) and (19), we obtain s(e, t)s(e, t) = s(e, t)[ f(x, t) + Δf(x, t) + b(x, t)u + d(t) - x (t) + ce (t) + (2Ct + B)δ ] = { } = s(e, t) Δf(x, t) - γsgn ⎡s ( e, t ) ⎤ + d(t) ≤ −η s ( e, t ) ⎣ ⎦ (20) which proves the stability of the sliding motion on the switching line (16) In order to find the system tracking error we solve equation (16) First we consider the following equation e (t) + ce (t) + Ct + Bt + A = (21) which determines the considered switching line for any time t ≤ tf, i.e when the line moves and δ=1 Solving equation (21) with initial condition (15) and assuming for the sake of clarity that t0 = 0, we can calculate the tracking error and its derivative for the time t∈〈0, tf) Furthermore, taking into account condition (18) and the assumption that t0 = we obtain A = - ce0 (22) Then, the tracking error and its derivative can be written as C 2C - cB B 2C ⎛ B 2C ⎞ e1 (t) = ⎜ - + ⎟ e -ct - t + t + e0 + - c c ⎠ c2 c c ⎝ c (23) 2C 2C - cB ⎛ B 2C ⎞ e (t) = ⎜ - ⎟ e -ct t+ c c2 ⎝c c ⎠ (24) e (t) + ce1 (t) = (25) Now we solve equation which determines the considered switching line for any time t > tf i.e for the time when the line does not move which is equivalent to the case δ=0 For this purpose we calculate values of (23) and its derivative (24) for t = tf C 2C - cB B 2C ⎛ B 2C ⎞ e1 (t f ) = ⎜ - + ⎟ e -ct f - t f + t f + e0 + - c c ⎠ c c c ⎝ c (26) 2C 2C - cB ⎛ B 2C ⎞ e (t f ) = ⎜ - ⎟ e -ct f tf + c c ⎠ c c2 ⎝ (27) Then, after some calculations, we obtain the evolution of the tracking error ⎡ B 2C ⎛ C 2C - cB B 2C ⎞ ⎤ e1 (t) = ⎢ - + + ⎜ - t + t f + e + - ⎟ e ct f ⎥ e -ct f c c2 c c ⎠ ⎝ c ⎣ c ⎦ (28) Sliding Mode Control of Second Order Dynamic System with State Constraints ⎡ B 2C ⎛ ⎤ 2C - cB B 2C ⎞ e (t) = ⎢ - + ⎜ Ct t f - ce0 - + ⎟ e ct f ⎥ e -ct f c c c ⎠ ⎝ ⎣c c ⎦ 441 (29) Notice that the error described by (23) and (28) converges to zero monotonically Next, we show the procedure for finding the optimal switching line Switching line design Now we present how to choose the optimal switching line under the assumption that the line moves with a constant deceleration to the origin of the error state space It means that we consider the line defined by (16) where C≠ Notice that for the time t>tf , switching line (16) is fixed and passes through the origin of the error state space This leads to the condition Ct + Bt f + A = f (30) Furthermore, in order to avoid rapid input changes, the velocity of the introduced line should change smoothly Thus, the following condition should hold 2Ct f + B = (31) Using relations (30), (31) and (22), we obtain the formula expressing the time when the line stops moving tf = 2e c B (32) In order to choose the switching line parameters, the integral of the absolute error (IAE) ∞ J = ∫ e (t) dt (33) is minimised subject to the system velocity e (t) ≤ vmax (34) e (t) ≤ amax (35) and the system acceleration constraints, where vmax, amax represent the maximum admissible velocity and maximum admissible acceleration of the considered system, respectively In order to facilitate further minimisation procedure, we define the following positive constant k= e0 c2 B (36) c= Bk e0 (37) From (36), we get We begin the procedure for finding optimal switching line parameters with calculating the IAE criterion Substituting equations (23) and (28) into (33), calculating appropriate integrals and considering relation (37), we obtain 442 Sliding Mode Control J(k,B) = e0 32 ⎛ ⎞ + k⎟ ⎜ B ⎝ k ⎠ (38) This criterion will be minimised with constraints (34) and (35) Since the considered criterion decreases with increasing value of B, the minimisation procedure of two variable function J(k, B) can be replaced by the minimisation of a single variable function This remark will be very useful further in the chapter Considering constraints, firstly we take into account each of the two constraints separately, and then we require both of them to be satisfied simultaneously 4.1 Velocity constraint In this section we will consider system (13) subject to velocity constraint (34) For any time t ≤ tf the system velocity is described by equation (24) and for the time t ≥ tf by relation (29) Calculating the maximum value of e (t) we get max e (t) = B ⎡ ln ( + 2k ) ⎤ - 1⎥ ⎢ c⎣ 2k ⎦ (39) Then using relations (34), (39) and taking into account condition (37), we obtain the following inequality v k ⎡ ln ( + 2k ) ⎤ B ≤ max ⎢ - 1⎥ e0 ⎣ 2k ⎦ -2 (40) As it was mentioned, because criterion (38) decreases with increasing value of |B| the minimisation of criterion J as a function of two variables (k, B) with the velocity constraint may be replaced by the minimisation of the following single variable function J v (k) = e ln ( + 2k ) ⎛ 2⎞ -1⎜ + ⎟ vmax 2k ⎝k 3⎠ (41) This function, for any fixed k expresses the minimum value of criterion J(k, B) which can be achieved when the velocity constraint is satisfied Closer analysis of this criterion as a single variable function shows that (41) reaches its minimum for numerically found argument kv opt ≈ 13.467 Then, the optimal parameter B can be calculated from -2 B= vmaxk ⎡ ln ( + 2k ) ⎤ - 1⎥ sgn ( e0 ) ⎢ e0 ⎣ 2k ⎦ (42) Substituting kv opt into (42), we obtain B v opt ( ) -2 ⎤ vmaxk v opt ⎡ ln + 2k v opt ⎢ = - 1⎥ sgn ( e0 ) ⎢ ⎥ e0 2k v opt ⎣ ⎦ (43) The other switching line parameters can be derived from (22), (31), (32) and (37) , and they are given below Sliding Mode Control of Second Order Dynamic System with State Constraints ( ln + 2k v opt A v opt = vmaxk v opt t f v opt = C v opt = - 2k v opt ( e0 ln + 2k v opt 2k v opt vmax ( vmax k v opt ln + 2k v opt 4e0 c v opt = ) -1 2k v opt −1 sgn(e ) ) -1 ) -1 ( vmaxk v opt ln + 2k v opt e0 443 2k v opt (44) (45) −3 sgn ( e0 ) ) -1 (46) (47) That concludes the analysis of the velocity constraint taken into account separately 4.2 Acceleration constraint Now we consider the system acceleration constraint given by (35) Let us calculate the greatest value of e (t) The maximum absolute value of this signal, achieved at the initial time t0 = is equal to e (0) = B Then, the acceleration constraint can be expressed as follows B ≤ amax (48) Now we will analyse the criterion J minimisation task Notice that for any given value of k, the minimum of criterion (38) is obtained for the greatest value of |B| satisfying constraint (48) Therefore, the solution of the considered minimisation task can be found as a minimum of the following single variable function J J a (k) = e0 32 amax ⎛ ⎞ + k⎟ ⎜ ⎝ k ⎠ (49) In order to analyse the minimisation task we calculate the derivative of expression (49) with respect to k Then, we conclude that function (49) reaches its minimum for ka opt = 1.5 and the optimal parameter B can be calculated from B = amaxsgn ( e0 ) (50) The other optimal switching line parameters can be calculated from relations A a opt = - 3a max e t f a opt = sgn(e0 ) e0 amax (51) (52) ... Fuzzy Sliding- mode Control for Biomicrofluidic Manipulation Control Engineering Practice, Vol 15, 1093–1105 Edwards, C and Tan, C P (2006), Sensor fault tolerant control using sliding mode observers,... Micro Electro Mechanical Systems, 19-24, Travemünde, Germany Part New Trends in the Theory of Sliding Mode Control 22 Sliding Mode Control of Second Order Dynamic System with State Constraints... with a sliding sector (Shyu et al., 1992; Xu et al., 1996); • using dynamic sliding mode controllers (Sira-Ramirez, 1993a; Sira-Ramirez, 1993b; Zlateva, 1996); • using fuzzy sliding mode controllers