Nonlinear Control of Bioprocess Using Feedback Linearization, Backstepping, and Luenberger Observers

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1 International Journal of Modern Nonlinear Theory and Application 5-6 Published Online September in SciRes. Nonlinear Control of Bioprocess Using Feedback Linearization Backstepping and Luenberger Oervers Muhammad Rizwan S. Khan Robert N. K. Loh Department of Electrical and Computer Engineering Oakland University Rochester Michigan USA Received July ; revised 5 August ; accepted 9 August Copyright by authors and Scientific Research Publishing Inc. This work is licensed under the Creative Commons Attribution International License (CC BY). Atract This paper addresses the analysis design and application of oerver-based nonlinear controls by combining feedback linearization (FBL) and backstepping (BS) techniues with Luenberger oervers. Complete development of oerver-based controls is presented for a bioprocess. Controllers using input-output feedback linearization and backstepping techniues are designed first assuming that all states are available for feedback. Next the construction of oerver in the transformed domain is presented based on input-output feedback linearization. This approach is then extended to oerver design based on backstepping approach using the error euation resulted from the backstepping design procedure. Simulation results demonstrating the effectiveness of the techniues developed are presented and compared. Keywords Bioprocess Feedback Linearization Backstepping Luenberger Oervers Oerver-Based Control. Introduction In process control a major difficulty is to provide direct real-time measurements of the state variables reuired to implement advanced monitoring and control methods on bioreactors []-[5]. Dissolved oxygen concentration in bioreactors temperature in non-isothermal reactors and gaseous flow rates are available for on-line measurement while the values of concentration of products reactants and/or biomass are often available only via on-line analysis []-[] which means that these variables are not available for real-time feedback control. An alternative is to use state oervers which in conjunction with the process model and available measurements How to cite this paper: Khan M.R.S. and Loh R.N.K. () Nonlinear Control of Bioprocess Using Feedback Linearization Backstepping and Luenberger Oervers. International Journal of Modern Nonlinear Theory and Application

2 can generate accurate estimates of the unmeasured and/or inaccessible states effectively. Exponential and asymptotic oervers and their variants to estimate unmeasured states in bioprocess systems have appeared in []-[5]. In [6] Dochain and Perrier applied backstepping [7]-[9] techniues to the nonlinear control of microbial growth problem in a CSTR (continuously stirred tank reactor) and two controllers were proposed. The first one was a non-adaptive version while the second one was an adaptive version in which the maximum specific growth rate was estimated on-line. However backstepping-based oerver design was not considered in [6]. In this paper a complete development of oerver-based control is presented that includes feedback linearization [7] [8] [] [] backstepping [7]-[9] Luenberger oerver [] with feedback linearization and Luenberger oerver with backstepping. The paper is organized as follows. Section presents the bioprocess model for control design. Theoretical foundation of input-output feedback linearization (FBL) and controller design are outlined in Section with simulation results. Section addresses the formulation and application of backstepping (BS) control with simulation results. In Section 5 simulation results are compared for both approaches i.e. FBL and BS. Section 6 addresses the design of Luenberger oervers for FBL and BS controls with simulations. The conclusions are presented in Section 7.. Bioprocess Model The model dynamics in a CSTR (continuous stirred tank reactor) with a simple microbial growth reaction with one sutrate S and biomass X are given by the following euations []: dx dt = µ X DX () ds = kµ X + DSin DS () dt where k µ D S Sin represent the yield coefficient specific growth rate (h ) dilution rate (h ) and sutrate concentration (grams/lit) in the influent and reactor respectively. The biomass concentration X ( t ) (grams/lit) is the variable which is to be controlled. Defining the parameter θ as θ = Dko and expressing specific growth rate µ as µ = kr o ( S) the dynamical Euations () and () above can be written as [6] x = θ r x x x x = kθ r x x x+ u () y = x can be measured with a sensor i.e. the output is given by is the control input. The bioprocess model given by () can be written compactly in an alternate state-space form as: where it is assumed that the biomass concentration x ( t) X ( t) y = x while x ( t) S( t) denotes the sutrate concentration and u( t) Sin µ max µ where C and C k D D tion kinetics which can expressed as Cxx x = + u f x + g x y = x = [ ] x Cx x Ks x + x Cxx x Ks + x max. Note that in () ( ) max = r x o u () r x has been written using Monod form for reac- µ x (5) k K + x S 5

3 r µ max KS = x k K + x o S. (6) We will use () for back stepping control and oerver design and () will be utilized for developing the control law and oerver design using the feedback linearization approach. Typical values of the model parameters needed for the simulation studies are given in Table [6].. Feedback Linearization (FBL) Control Design The main intent of this section is to investigate control design using the input-output feedback linearization (FBL) techniue. Consider a general nonlinear control-affine SISO system described by [7] [8] [] [] u : n n = + D (7) x f x g x f g n y = h( x) h: D (8) n where x is the state vector u y are the control and output signals respectively; h is a smooth function and f g are smooth vector fields on D where D is an open set. Given the nonlinear system (7) and the measurement (8) our goal is to find a diffeomorphism or nonlinear transformation of the form z = T ( x ) n z with T = that transforms the nonlinear system in the x -coordinates to a linear system in the y t with respect to t yields z -coordinates. Differentiating the output h h ( x) ( x) y = Lh + Lh u (9) f where Lh f ( x) = f and Lh g ( x) = g denote the Lie derivatives of h( x ) with respect to f x x g( x ) respectively. If Lh g ( x ) = then y ( t) is independent of ρ times until u( t ) appears explicitly we obtain ( ρ ) ρ ρ f g f g ( x) ( x) α( x) β( x) x and u t. Continuing successive differentiation y = Lh + LL h u + u () where Lh ρ ρ α x f ( x) is a nonlinearity cancellation factor and β ( x) LL g f h( x) ρ smallest integer ρ for which u( t ) appears is referred to as the relative degree i.e. when LL h is a scalar function. The g f x. The nonlinear system (7) - (8) is said to have a well-defined relative degree ρ in a region D D if for all x D. Note that ρ n ( x) h( x) k LLh g f = k k< ρ ; ρ LL g f. From () define ( ρ ) ( x) β( x) () v y = α + u () where vt is a one-dimensional transformed input created by the feedback linearization process. Euation () yields the linearizing feedback control law [7] [8] [] []: Table. Parameter data. u α = β + x x v () Symbol Parameter Numerical Value k Yield coefficient µ max Specific growth rate. h D Dilution rate.5 h K s Saturation constant 5 grams/lit 5

4 provided β ( x ) is invertible. If ρ = n then ( x) ( x) h Lh f z = T x =. n Lf h( x) M. R. S. Khan R. N. K. Loh If ρ < n the diffeomorphism T (x) comprises of both external and internal dynamics i.e. [ ] T T x = ξ η ρ n ρ where ξ represents the external dynamics state vector and η the internal dynamics state vector respectively; furthermore the differential euation for ξ is linear while that for η is typically nonlinear. For the bioprocess model given by () ρ = n = so the system is fully linearizable. We obtain from () () and () where ( Cx Ks x)( Cxx Ksxxx ) CC KSxx CKSxx ( K + x ) ( K + x ) ( K + x ) α x = + (5) s s s S s ( K + x ) () CK x β x = (6) x y z = T x = = Cxx (7) y x Ks + x T x is a local diffeomorphism for the system. Using (7) with () for ρ = the original nonlinear system described by () and () is transformed into a linear system of the form where and [ Ac B c] and [ c c] for the transformed input z = Az c + Bcv z = zo y = Cz c Ac = c c [ ] B = = C y = Cz c A C are respectively controllable and oervable pairs. A suitable tracking control law vt in the linear system (8) for ρ = can be formulated as with (7) r r r r (8) v = K z+ K Y + y = K T x + K Y + y () where z = T ( x ) yr ( t ) is a bounded reference with bounded derivatives y r ( t) and yr ( t) [ ] T r = yr yr and the constant feedback gain matrix K K K is determined such that ( c c ) (9) Y = A BK is Hurwitz. Furthermore the gain matrix K can be determined by various design methods such as pole placement (PP) and linear uadratic regulator (LQR). We shall focus on the PP design in this paper. Sutituting () into (8) yields the closed-loop system ( y ) z = Ac BK c z+ Bc K Yr + r z = zo y = Cz c. The linearizing feedback control law in the x -domain can be written by setting which yields the closed-loop system u = u in () as u = β x α x + v = β x α x K T x + K Yr + yr () () 5

5 x = f x + g x u x = x o y = x. () The design of a PP control law () for the nd -order system (8) is achieved by choosing a damping ratio ξ =. that prohibits overshoot and an undamped natural freuency ω = 8. (rad/sec). The resulting closed- loop poles are given by = [ ξωn ± jωd] = [ ] [ ] p where n d = n = and the resulting gain ω ω ξ K = is computed with Matlab s ACKER command. Simulation studies for the closed-loop system with FBL control were conducted using (). The controller performance was evaluated for a suare-wave set-point reference yr ( t ) that alternates every hours between grams/lit and grams/lit as shown in Figure (dotted line). The initial conditions were chosen as X = x = grams/lit and S = x =.9 [6]. The simulation results are depicted in Figure which shows that the responses are satisfactory.. Backstepping (BS) Control Design We shall address the design of back stepping (BS) [7]-[9] control in this section where the parameter θ is assumed known. The objective here is to design a BS control law u such that the output y = x tracks the reference y r. We will also compare the performance of the closed-loop bioprocess under FBL control u given by () and the BS control u to be developed below. The formulation presented here considers a general bounded differentiable reference signal yr ( t ) instead of the constant set-point regulation in [6]. Consider the nonlinear system in the form of () reproduced below for ease of reference: x = θ r x x x x = kθ r x x x+ u () y = x where r x x as the virtual control of the first suystem x = θ r( x) x x in () and let α be the stabilizing function such that y = x racks y r. Define the tracking errors as θ is a known constant parameter. We treat = y y = x y () r r = θ r x x α (5) x y r (a) x f bl x 6 8 y r 6 8 (b) x f bl 6 (c) u f bl u Figure. Responses of closed-loop bioprocess () under FBL control u t with PP design

6 where is the error between r( x ) Consider the Lyapunov function candidate M. R. S. Khan R. N. K. Loh θ x and α. Taking the derivative of yields with (5) = x y = θ r x x x y = + α x y. (6) r r r which yields the derivative with (6) V = (7) ( α ) V = + x y. (8) r Choosing the stabilizing function α to make αx y r = c in (8) yields α = c + x + y c >. (9) r Sutituting (9) into (6) and (8) yields respectively = c + () V = c c >. () From () if = then V = c < and the origin = is globally asymptotically stable whereby achieving global tracking with y = x yr. The term will be addressed in the next step. The next step is to develop a BS control law for u. The derivative of = θ r( x) x α given by (5) satisfies with () and (9) r r x x x x () = θ + θ α x where r φ α θ = x + x u () x r φ( x) θ r ( θ r ) x + θ x ( kθ rxx) () x and r x is given by (6). To stabilize the ( ) -system the Lyapunov function candidate as The derivative of V is given by with () and () V = V+. (5) Defining u r V c φ α θ = + + x + x u. (6) x u to be the BS control and choosing u to make the term [ ] = c in (6) yields Sutituting (7) into () and (6) we obtain r u = θ x c φ + α x x. (7) c (8) = 55

7 V = c c c > c >. (9) Since V < for all c c = is globally asymptotically stable. Additionally the stability result can also be established by combing the error euations from () and (8) as > > it follows that c = = c A. () Since A is a skew-symmetric Hurwitz matrix for all c > and c > it follows that the euilibrium ( ) = ( ) is globally asymptotically stable. Moreover A C is an oervable pair where C = [ ] (see (5)). Since () is in the form of a standard linear time-invariant (LTI) system a Luenberger oerver [] for state estimation can be constructed for the system and will be investigated in Section 6. Meanwhile the closed-loop bioprocess under BS control is given by ( x) x = f x + g x u x = x o y = h = x where u is given by (7). Simulation studies were conducted using () with the backstepping gains c = c = 8. The reference signal yr ( t ) and the initial condition x = x o were same as those used for the FBL control in Section. The simulation is depicted in Figure which shows that the responses were satisfactory. 5. Comparison of FBL and BS Designs The simulation results for the FBL versus BS designs using the gains reported in Sections and are shown in Figure and Figure for comparison purposes. It can be seen that both x ( t) yr ( t) and x ( t) yr ( t) asymptotically with no overshoot. It can also be seen that the magnitudes of u ( t ) are slightly larger than those of u ( t ). However the reverse can also be obtained by tuning c c. K and { } 6. Oerver-Based FBL and BS Controls As mentioned before that not all state variables are measured in the bioreactor systems; therefore suitable oervers are needed for realizing the full-state feedback control designs proposed in Sections and. We shall () (a) (b) x x y r x x 6 8 y r 6 8 (c) u u Figure. Responses of closed-loop bioprocess () under BS u t for c = c = 8. control 56

8 x (a) x f bl x 6 8 x 6 8 (b) x f bl x u 6 (c) u f bl u Figure. Comparing FBL responses in Figure and BS responses in Figure. (a) x u x f bl x (b) u f bl u x t and Figure. Zoomed-in view of x ( t ) and u ( t ) and u ( t ) in Figure. investigate the constructions of Luenberger oervers for the FBL-based and BS-based control approaches in this section 6.. Luenberger Oerver for FBL Control t is measured in () a Luenberger oerver [] can be constructed for full-state estimation needed for full-state control of the bioprocess system. Using () a full-state oerver can be constructed as Since only x ( c c ) c ( r r ) o c ( r r ) z ˆ = A B K zˆ + B K Y + y + L y yˆ = A zˆ + B K Y + y + Ly y = Cz ˆ ˆ ˆ c = z = x y = Cz c = zˆ = x where Ao Ac Bc K LCc is the oerver system matrix and L the oerver gain matrix to be determined such that A o is Hurwitz provided that ( Ac BK c ) C c is an oervable pair (which is the case in the present problem). The gain matrix L in () can be computed using a Luenberger oerver [] with pole placement (PP) and/or Kalman-Bucy filter [] design techniues. We shall focus on the PP design method; () 57

9 henceforth L and Ao = Ac Bc K LC c in () will be denoted by L= L pp and Aopp = Ac BK c LppC c respectively. It should be noted that for a general LTI system characterized by [ ABC ] where [ AC ] is an oervable pair the pair ( A BK ) C may not be oervable because full-state feedback can destroy oervability; furthermore ( A BK LC ) may be unstable even though ( A LC ) is designed to be stable [] [5]. Now using () and () it can readily be shown that the estimation error z = z zˆ satisfies z = A z z () = z () where the initial condition z is arbitrary. Since A opp is Hurwitz it follows that for all opp ( t) ( t) ˆ( t) o lim z = lim z z = () t t z. Using the transformation defined by (7) the oerver described by () in the z -coordinates can be transformed back to the x -coordinates as where Q ( xˆ ) ( xˆ ) and Q ( ˆ ) T x = = + + (5) ˆ uˆ ( y xˆ ) xˆ Q ˆ ˆ ˆ ˆ x z f x g x Q x Lpp x is the Jacobian matrix associated with (7) given by ( ˆ ) Cxˆ CK ˆ s x Q x =. (6) K ˆ s + x ( K ˆ s + x ) In summary the oerver-based control system with feedback linearization for the bioprocess under consideration has the form where x ˆ is the initial estimate of ( ˆ ) x = f x + g x ˆ x = x (7) u o xˆ = f xˆ + g xˆ + Q xˆ L xˆ = xˆ (8) uˆ ( y xˆ ) x and uˆ pp o α = β x ˆ x ˆ + vˆ (9) ( ˆ ) vˆ = + r + yr K T x K Y (5) ( Cxˆ K ˆ )( ˆ ˆ ˆ ˆ ˆ s x Cxx Ksxxx ) CC ˆ ˆ ˆ ˆ KSxx CKSxx ( K + xˆ ) ( K + xˆ ) ( K + xˆ ) α x = + (5) 6.. Luenberger Oerver for BS Control s s s CK ˆ S x β ( x ) =. (5) ( K + xˆ ) In this section we pursue our final objective i.e. to design a Luenberger oerver based on the BS formulation using the error Euation (). To construct an oerver for () we need an output euation which can be defined as s [ ] where y = = x yr is known and A C is an oervable pair. y = = C (5) 58

10 We present the following proposition. Proposition Consider the bioprocess system described by () and (). A Luenberger oerver for the associated error system () with measurement given by (5) can be constructed as ( y ) ˆ = Aˆ + L Cˆ ˆ = ˆ (5) o where y Cˆ ˆ = yx and L is the oerver gain matrix to be determined by for example poleplacement such that the oerver matrix Ao = A LC is Hurwitz. Since A is already Hurwitz and A C is an oervable pair L can be determined such that the real parts of the eigenvalues of Ao = A LC lie on the left-side of those of A on the open left-half plane if desirable. Furthermore (5) can be expressed in the x-coordinates as x ˆ = f xˆ + g xˆ ˆ + Q L ˆ xˆ = xˆ. (55) u ( y x ) o Proof: First we need to show that the estimate ˆ converges to its true value. Define the estimation error as = ˆ. From () and (5) it follows that satisfies = ˆ = A L C ˆ A = (56) o o where is the initial condition. Since A o is Hurwitz it follows that for arbitrary obtained as [ ] [ ˆ ] lim = lim = (57) t t Next using () (5) and (9) the coordinates transformation for the error-system can be x y x y = T x =. (58) Euation (58) yields the Jacobian matrix where Q is nonsingular for ( y y ) r r r r θ r( x) x α θ r( x) x+ ( c) xcyr y r Q ( x) θ x tions (58) and (59) yield = T = r (59) x θ r x + c θ x x r x so that ( y y ) T x is a local diffeomorphism for (). Eua- r r ˆ Q xˆ xˆ and from (5) and (5) x ˆ = Q ( x ˆ ) ˆ ( ˆ ) ˆ ( ˆ) ( ˆ ) ( ˆ ) ˆ ( ˆ ) ( ˆ = Q x A + L y x = f x + g x u + Q x L yx) (6) where Q ( x ) is the inverse of Q ( ˆ ) ˆ x and r uˆ ˆ ˆ ˆ ( ˆ ) ˆ = θ x c φ + α xˆ x (6) ˆ α = c ˆ + x ˆ + y = c cˆ + ˆ + r xˆ xˆ xˆ + y c > (6) θ r r which complete the proof. The oerver design techniue developed here is interesting and attractive and is different from the two-filter approach in [9]. The techniue can be applied to a wide class of BS-based error systems. In summary the oerver-based control system with the BS formulation for the bioprocess is described by ˆ x = f x + g x x = x (6) u o 59

11 where x ˆ is the initial estimate of ˆ x = f xˆ + g xˆ ˆ + Q xˆ L ˆ xˆ = x ˆ (6) u ( y x ) o x. Simulation studies for the proposed oerver-based FBL and BS controls were conducted and compared. The initial conditions were chosen as x = [.9] T and x ˆ = [.8.] T. The set-point reference yr ( t ) was the same as before. The model parameters were given in Table and Table. In Figure 5 results for oerver-based FBL control scheme described by (7) and (8) are shown. It can be seen that the estimates xˆ ( t) x( t) and xˆ ( t) x( t) converged to the true states around t = 7 h. In Figure 6 results for the oerver-based BS control scheme are presented. Convergence of the estimated states to the actual states can also be seen from this figure and are similar to those presented in Figure. In Figure 7 the behavior of the error variables and defined by () and (5) which satisfy () in the backstepping scheme is shown. It is evident that ˆ( t) ( t) and ˆ( t) ( t) smoothly after the transients are over around t = 8 h. 7. Conclusion Oervers are critical to control system analysis and designs that employ full-state feedback where not all the state variables are accessible for on-line real-time measurements and/or where the measurements are corrupted by noise. Indeed the design of suitable linear or nonlinear oervers or filters leading to oerver-based control technology is an integral part of real world control system applications. In this paper oerver-based control strategies were developed for a nonlinear bioprocess system using feedback linearization and backstepping control techniues; in particular a Luenberger oerver for backstepping control was formulated using the error euation resulted from the backstepping design procedure. The oerver design techniue developed here is interesting and attractive and is different from the two-filter approach known in the literature. Simulation results with and without oervers for both the FBL and BS schemes are presented and compared. The results were excellent and demonstrated the feasibility and effectiveness of the proposed approaches. Table. Controllers and oerver gains. Oerver-Based Control Controller Gain/Coefficients Oerver Gain L FBL K = [ 9 ] [ ] T L = BS [ c c ] = [.5.5] [ ] T L =. 5.8 x y r u v y r (a) Figure 5. Responses of oerver-based FBL control scheme (7) x t x t x t x t smoothly. ˆ x f bl ^x f bl x ˆ and (8): and (c) 6 (b) x f bl ^x f bl ^u f bl ^v f bl 6

12 x y r y r 6 (a) x ^x x (c).5.5 (b) x ^x ^u u Figure 6. Responses of oerver-based BS control scheme (6) x t x t x t x t smoothly. ˆ ˆ and (6): and (a) (b) (t) ^ (t) Acknowledgements Figure 7. Evolution of the backstepping error variables: ˆ t t ˆ t t smoothly. and The authors would like to thank all the reviewers for their feedbacks and constructive criticisms. References [] Bastin G. and Dochain D. (99) On-line Estimation and Adaptive Control of Bioreactors. Elsevier Amsterdam. [] Dochain D. () State and Parameter Estimation in Chemical and Biochemical Processes: A Tutorial. Journal of Process Control [] Dochain D. and Rapaport A. (5) Internal Oervers for Biochemical Processes with Uncertain Kinetics and Inputs. Mathematical Biosciences [] Dochain D. () State Oervers for Tubular Reactors with Unknown Kinetics. Journal of Process Control [5] Hulhoven X. Vande Wouwer A. and Bogaerts Ph. (6) Hybrid Extended Luenberger-Asymptotic Oerver for Bioprocess State Estimation. Chemical Engineering Science (t) ^ (t) 6

13 [6] Dochain D. and Perrier M. () Adaptive Backstepping Nonlinear Control of Bioprocesses. International Federation of Automatic Control Proceedings ADCHEM [7] Khalil H.K. () Nonlinear Systems. rd Edition Prentice Hall Upper Saddle River. [8] Maruez H.J. () Nonlinear Control Systems: Analysis and Design. John Wiley & Sons Hoboken. [9] Krstic M. Kanellakopoulos I. and Kokotovic P. (995) Nonlinear and Adaptive Control Design. John Wiley New York. [] Isidori A. (995) Nonlinear Control Systems. Springer-Verlag New York. [] Krener A.J. (999) Feedback Linearization. In: Baillieul J. and Willems J.C. Eds. Mathematical Control Theory Chapter Springer-Verlag New York [] Luenberger D.G. (96) Oerving the State of a Linear System. IEEE Transactions on Military Electronics [] Kalman R.E. and Bucy R.S. (96) New Results in Linear Filtering and Prediction Theory. Transactions of the ASME 8D 9-8. [] Ogata K. () Modern Control Systems. 5th Edition Prentice Hall Upper Saddle River. [5] Dorf R. and Bishop R. (8) Modern Control Systems. th Edition Pearson Education Upper Saddle River. 6

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