Model predictive control (MPC) algorithms require full knowledge of the system states while using state feedback framework. However, it is not possible to control a process directly using state feedback technique when the states of the process are not available for measurement. In such cases, the controller has to be designed with output feedback framework, i.e., to control the process using the output measurements of the process. Output feedback control is accomplished using either direct outputtoinput mapping or using the state feedback control by estimating the states from the process output measurement.
Linear systems, not affected by any disturbance, can be controlled by linear control techniques that employ an observer/estimator. In such cases separation principle holds good and stability can be ensured. However, stability of the closedloop cannot be ensured, in general, by simply combining a stable estimator with a stable state feedback controller, when the system or its controller is nonlinear (as is the case with MPC for constrained systems) and the state and output disturbances are present^{[6]}. To overcome this drawback some special care has to be taken in the robust controller design to deal with the estimation error along with modeling error.
In this paper, the statefeedback Nash game based mixed MPC (NGMMPC)^{[1, 2]} design has been extended for output feedback case. However, the extension is not straightforward, as it is an observer based controller.
2 Problem formulationConsider the dynamic system
$ x_{k+1} =Ax_{k}+B_{0}w_{k}^{0}+B_{1}w_{k}+B_{2}u_{k} $  (1) 
$ \tilde{z}_{k} =C_{0}x_{k}+D_{01}w_{k}+D_{02}u_{k} $  (2) 
$ z_{k} =C_{1}x_{k}+D_{10}w_{k}^{0}+D_{11}w_{k}+D_{12}u_{k} $  (3) 
$ y_{k} =C_{2}x_{k}+D_{20}w_{k}^{0}+D_{21}w_{k} $  (4) 
where
Assumption 1.
Assumption 2.
^{1} The set of sequences
Assumption 3. The initial condition
^{2}
In
$ \begin{align} \Vert T_{z_{k}, w_{k}}\Vert _{\infty }^{2}=\sup\limits_{w_{k}\in {P}}\frac{ \Vert z_{k}\Vert _{{P}}^{2}}{\Vert w_{k}\Vert _{{P}}^{2}} <\gamma ^{2} \end{align} $  (5) 
whose equivalent form is
$ \begin{align} \sup\limits_{w_{k}\in {P}}\{\Vert z_{k}\Vert _{{P}}^{2}\gamma ^{2}\Vert w_{k}\Vert _{{P}}^{2}\}. \end{align} $  (6) 
Being a multiobjective control problem, where the two performance functions, namely,
$ J_{2}(w_{k}, u_{k})= \arg \min\limits_{u\in {U}}\Vert T_{\tilde{z} _{k}, w_{k}^{0}}\Vert _{2}^{2} $  (7) 
$ J_{\infty }(w_{k}, u_{k})= \sup\limits_{w_{k}\in {P}}\{\Vert z_{k}\Vert _{ {P}}^{2}\gamma ^{2}\Vert w_{k}\Vert _{{P}}^{2}\} $  (8) 
where
As the state feedback problem is solvable, the output feedback problem is also solvable once the states are estimated by an observer. Then the resulting controller will be given as
$ \begin{align} u^*_k=K_{k}\hat{x}_k \end{align} $  (9) 
where
Let
$ x_{k+1} = (A+B_{1}H_{k})x_{k}+B_{0}w_{k}^{0}+B_{2}u_{k} $  (10) 
$ \tilde{z}_{k} =(C_{0}+D_{01}H_{k})x_{k}+D_{02}u_{k} $  (11) 
$ y_{k} = (C_{2}+D_{21}H_{k})x_{k}+D_{20}w_{k}^{0}. $  (12) 
Now the problem to be solved is
$ \begin{align} u_{k}^{\ast }=\arg \min\limits_{u}\parallel T_{\tilde{z}_{k}, w_{k}^{0}}\parallel _{2}^{2}. \end{align} $  (13) 
Generally, for a discrete time linear system given by
$ x_{k+1} = \bar{A}x_{k}+\bar{B}u_{k}+\bar{E}w_{k}^{0} $  (14) 
$ y_{k} = \bar{C}_{1}x_{k}+\bar{D}_{1}w_{k}^{0} $  (15) 
$ z_{k} = \bar{C}_{2}x_{k}+\bar{D}_{2}u_{k} $  (16) 
the solution of discretetime
$ \begin{align} P =&\bar{A}^{\rm T}P\bar{A}+\bar{C}_{2}^{\rm T}\bar{C}_{2}(\bar{C}_{2}^{\rm T}\bar{D} _{2}+\bar{A}^{\rm T}P\bar{B})\times \notag \\ &(\bar{D}_{2}^{\rm T}\bar{D}_{2}+\bar{B}^{\rm T}P\bar{B})^{1}(\bar{D}_{2}^{\rm T}\bar{C }_{2}+\bar{B}^{\rm T}P\bar{A}) \end{align} $  (17) 
$ \begin{align} Q =&\bar{\bar{A}}Q\bar{\bar{A}}^{\rm T}+\bar{E}\bar{E}^{\rm T}(\bar{\bar{A}}Q\bar{C }_{1}^{\rm T}+\bar{E}\bar{D}_{1}^{\rm T})\times \notag \\ &(\bar{D}_{1}\bar{D}_{1}^{\rm T}+\bar{C}_{1}Q\bar{C}_{1}^{\rm T})^{1}(\bar{D}_{1} \bar{E}^{\rm T}+\bar{C}_{1}Q\bar{\bar{A}}^{\rm T}) \end{align} $  (18) 
whose brief derivation is given in Appendix A. The above pair of DAREs could also be rewritten using matrix inversion identity^{3} as
^{3}
$ \begin{align} P =&\bar{A}^{\rm T}P\left\{ I+\bar{B}(\bar{D}_{2}^{\rm T}\bar{D}_{2})^{1}\bar{B} ^{\rm T}P\right\} ^{1}\bar{A} + \nonumber \\ &\bar{C}_{2}^{\rm T}\left\{ I\bar{D}_{2}^{\rm T}(\bar{D}_{2}\bar{D}_{2}^{\rm T})^{1} \bar{D}_{2}^{\rm T}\right\} \bar{C}_{2} \end{align} $  (19) 
$ \begin{align} Q =&\bar{\bar{A}}Q\left\{ I\bar{C}_{1}^{\rm T}(\bar{D}_{1}\bar{D}_{1}^{\rm T})^{1} \bar{C}_{1}Q\right\} ^{1}\bar{\bar{A}} + \nonumber \\ &\bar{E}\left\{ I\bar{D}_{1}^{\rm T}(\bar{D}_{1}\bar{D}_{1}^{\rm T})^{1}\bar{D} _{1}\right\} \bar{E}^{\rm T}. \end{align} $  (20) 
Using the analogy of the above general
$ \bar{A} =A_{C}+(B_{1}B_{2}R_{02}^{1}D_{02}D_{01})H_{k} $  (21) 
$ \bar{\bar{A}} =A_{O}+(B_{1}B_{0}D_{20}^{\rm T}R_{20}^{1}D_{21})H_{k} $  (22) 
$ \bar{B} =B_{2} $  (23) 
$ \bar{C} =C_{0}+D_{21}H_{k} $  (24) 
$ \bar{E} =B_{0} $  (25) 
$ \tilde{R}_{02}= I\bar{D}_{2}^{\rm T}(\bar{D}_{2}\bar{D}_{2}^{\rm T})^{1}\bar{D}_{1} $  (26) 
$ \tilde{R}_{20}= I\bar{D}_{1}^{\rm T}(\bar{D}_{1}\bar{D}_{1}^{\rm T})^{1}\bar{D}_{1}. $  (27) 
we can get the solution for the
$ \begin{align} Ric1 :&0=\left\{ \begin{array}{l} \left[ A_{C}+\left( B_{1}B_{2}R_{02}^{1}D_{02}D_{01}\right) H_{k}\right] ^{\rm T}\times \\ P_{k}^{1}\left( I+B_{2}R_{02}^{1}B_{2}^{\rm T}P_{k}^{1}\right) ^{1}\times \\ \left[ A_{C}+\left( B_{1}B_{2}R_{02}^{1}D_{02}D_{01}\right) H_{k}\right] \\ P_{k}^{1}+\left( C_{0}+D_{21}H_{k}\right) ^{\rm T}\tilde{R}_{02}\times \\ \left( C_{0}+D_{21}H_{k}\right) \end{array} \right. \label{Ric1Out} \end{align} $  (28) 
$ \begin{align} Ric2 :&0=\left\{ \begin{array}{l} \left[ A_{0}+\left( B_{1}B_{0}D_{20}^{\rm T}R_{20}^{1}D_{21}\right) H_{k} \right] \times \\ P_{k}^{2}\left( IC_{2}^{\rm T}R_{20}^{1}C_{2}P_{k}^{2}\right) ^{1}\times \\ \left[ A_{0}+\left( B_{1}B_{0}D_{20}^{\rm T}R_{20}^{1}D_{21}\right) H_{k} \right] ^{\rm T} \\ P_{k}^{2}+B_{0}\tilde{R}_{20}B_{0}^{\rm T} \end{array} \right. \label{Ric2Out} \end{align} $  (29) 
where
$ A_{C} =AB_{2}R_{02}^{1}D_{02}^{\rm T}C_{0} $  (30) 
$ A_{O} =AB_{0}D_{02}^{\rm T}R_{20}^{\rm T}C_{2} $  (31) 
$ R_{02} =D_{02}^{\rm T}D_{02} $  (32) 
$ \tilde{R}_{02} =ID_{02}R_{02}^{1}D_{02}^{\rm T} $  (33) 
$ R_{20} =D_{20}D_{20}^{\rm T} $  (34) 
$ \tilde{R}_{20} =ID_{20}R_{20}^{1}D_{20} $  (35) 
and the state feedback controller and observer gains are
$ \begin{align} K_{k} =&\left[ D_{02}^{\rm T}D_{02}+B_{2}^{\rm T}P_{k}^{1}B_{2}\right] ^{1}\times \nonumber \\ & \left[ \begin{array}{l} B_{2}^{\rm T}P_{k}^{1}(A+B_{1}H_{k})+ \\ D_{02}^{\rm T}(C_{0}+D_{01}H_{k}) \end{array} \right] \end{align} $  (36) 
$ \begin{align} L_{k} =&[(A+B_{1}H_{k})P_{k}^{2}(C_{2}+D_{21}H_{k})^{\rm T}+B_{0}D_{20}^{\rm T}] \times\nonumber \\ & \left[ \begin{array}{l} D_{20}D_{20}^{\rm T}+(C_{2}+D_{21}H_{k})P_{k}^{2}\times \\ (C_{2}+D_{21}H_{k})^{\rm T} \end{array} \right] ^{1} \end{align} $  (37) 
where
This part of the control problem finds out the optimal controller such that
$ \begin{align} \tilde{x}_{k+1} =&\left( \begin{array}{c} A+B_{1}H_{k}+B_{2}K_{k}+ \\ L_{k}C_{2}+L_{k}D_{21}H_{k} \end{array} \right) \tilde{x}_{k}L_{k}y_{k} \end{align} $  (38) 
$ \begin{align} u_{k}^{\ast } =&K_{k}\tilde{x}_{k}. \end{align} $  (39) 
The above two equations represent the outputfeedback controller, the former is an observer and the latter is a state feedback controller.
2.2 Solving theNow, with the optimal control law
$ x_{k+1} =(A+B_{2}K_{k})x_{k}+B_{0}w_{k}^{0}+B_{1}w_{k} $  (40) 
$ z_{k} =(C_{1}+D_{12}K_{k})x_{k}+D_{10}w_{k}^{0}+D_{11}w_{k}. $  (41) 
Using the Stochastic Bounded Real Lemma^{[8]} for a general discretetime system
$ x_{k+1} =\bar{A}x_{k}+\bar{B}_{0}w_{k}^{0}+\bar{B}_{1}w_{k} $  (42) 
$ z_{k} =\bar{C}_{1}x_{k}+\bar{D}_{10}w_{k}^{0}+\bar{D}_{11}w_{k} $  (43) 
the corresponding general discretetime
$ \begin{align} 0 =&\bar{A}^{\rm T}X\bar{A}X+(\bar{B}_{1}^{\rm T}X\bar{A}+\bar{D}_{11}^{\rm T}\bar{C} _{1})^{\rm T} \times \notag \\ & (\gamma ^{2}I\bar{D}_{11}^{\rm T}\bar{D}_{11}\bar{B}_{1}^{\rm T}X\bar{B} _{1})^{1}\times \notag \\ & (\bar{B}_{1}^{\rm T}X\bar{A}+\bar{D}_{11}^{\rm T}\bar{C}_{1})+\bar{C} _{1}^{\rm T}\bar{C}_{1} \label{Ric_22} \end{align} $  (44) 
where
$ \begin{align} w_{k}^{\ast }=(\gamma ^{2}I\bar{D}_{11}^{\rm T}\bar{D}_{11}\bar{B}_{1}^{\rm T}X \bar{B}_{1})^{1}(\bar{B}_{1}^{\rm T}X\bar{A}+\bar{D}_{11}^{\rm T}\bar{C}_{1})x_{k}. \label{uCons} \end{align} $  (45) 
In the light of the analogy of structure of the above given general discretetime system with the system given by (40)(41), one can get
$ \bar{A} =A+B_{2}K_{k} $  (46) 
$ \bar{B}_{0} =B_{0} $  (47) 
$ \bar{B}_{1} =B_{1} $  (48) 
$ \bar{C}_{1} =C_{1}+D_{12}K_k $  (49) 
$ \bar{D}_{10} =D_{10} $  (50) 
$ \bar{D}_{11} =D_{11} $  (51) 
and the corresponding discretetime
$ \begin{align} Ric3:\, 0=\left\{ \begin{array}{l} \left( A+B_{2}K_{k}\right) ^{\rm T}P_{k}^{3}\left( A+B_{2}K_{k}\right) P_{k}^{3}+ \\ \left[ B_{1}^{\rm T}P_{k}^{3}\left( A+B_{2}K_{k}\right) +D_{11}^{\rm T}\left( C_{1}+D_{12}K_{k}\right) \right] ^{\rm T} \\ \left( \gamma ^{2}ID_{11}^{\rm T}D_{11}B_{1}^{\rm T}P_{k}^{3}B_{1}\right) ^{1} \\ \left[ B_{1}^{\rm T}P_{k}^{3}\left( A+B_{2}K_{k}\right) +D_{11}^{\rm T}\left( C_{1}+D_{12}K_{k}\right) \right]+ \\ \left( C_{1}+D_{12}K_{k}\right) ^{\rm T}\left( C_{1}+D_{12}K_{k}\right) \end{array} \right. \label{Ric3Out} \end{align} $  (52) 
where
$ \begin{align} H_{k}=(\gamma ^{2}ID_{11}^{\rm T}D_{11}B_{1}^{\rm T}P_{k}^{3}B_{1})^{1}(B_{1}^{\rm T}P_{k}^{3}A+D_{11}^{\rm T}C_{1}) \end{align} $  (53) 
and
$ \begin{align} w_{k}^{\ast }=H_{k}\hat{x}_{k}. \end{align} $  (54) 
Thus (28), (29) and (53) are the three coupled algebraic Riccati equations (3cAREs) of the optimal output feedback control problem. The coupling is attributed due to the presence of the
The saddle point solution, viz. Nash equilibria, for the output feedback case is similar to that of the state feedback case, except that the state information is been estimated from the observer:
$ \frac{1}{2}\hat{x}_{k}^{\rm T}P_{k}^{1}\hat{x}_{k} \geq \beta _{k} $  (55) 
$ \frac{1}{2}\hat{x}_{k}^{\rm T}P_{k}^{3}\hat{x}_{k} \leq \beta _{k}. $  (56) 
With this bound, updating the control law using the current state of the system, the optimal values of the stabilizing matrices viz.
This is again similar to the case of state feedback control problem. However, it involves 3cAREs in the case of output feedback optimal control problem.
Theorem 1. For a system defined by
$ Ric1+\epsilon _{k}^{1} =0 $  (57) 
$ Ric2+\epsilon _{k}^{2} =0 $  (58) 
$ Ric3+\epsilon _{k}^{3} =0 $  (59) 
and along with the saddle point optimal condition bounds
$ \hat{x}_{k}^{\rm T}P_{k}^{1}\hat{x}_{k} \geq \beta _{k} $  (60) 
$ \hat{x}_{k}^{\rm T}P_{k}^{3}\hat{x}_{k} \leq \beta _{k} $  (61) 
if it exists, such that (
Proof. Let
Likewise, there may exist a different solution set for a different choice of
$ \begin{align} \lim\limits_{\delta _{j}\rightarrow 0}\epsilon _{k}^{j^{\ast }}=\tilde{\epsilon} _{k}^{j^{\ast }}+\delta _{j}=\tilde{\epsilon}_{k}^{j^{\ast }}, \;\;j=1, 2, 3 \end{align} $  (62) 
which in turn gives
One of the main objectives of this paper is to extend the mixed
Model predictive control relies explicitly on the process model to make an optimal decision on the control input/law to be implemented at every time instant. Hence the correctness of the process model at the prevailing operating condition, is a major factor for efficient performance of MPC. The Wiener type Laguerrewavelet network model^{[3]} is a datadriven nonlinear model. Usually the inputoutput data collected for identifying nominal stable systems are obtained by exciting the process about its nominal equilibrium point. Hence the efficient control of the process by linearising the model at the nominal operating point and controlling the process using linear controller is possible only in the close vicinity of the nominal operating point of the process. However, in situations where the process operating point is shifted far away from the previous value, the above mentioned model (linearised at the nominal operating point)based control strategy can fail miserably.
Successive linearisation approach^{[9]} could be adopted to circumvent the above shortcoming. The successive linearisation of the static nonlinear gain by finding the Jacobian linearisation of the nonlinearity around the nominal state trajectory is made to obtain a local linear equivalent model of the statetooutput mapping. So linearising the wavelet network about the nominal value of the current Laguerre states gives a local linear statetooutput gain. The necessary condition to ensure a smooth linearized representation is that the static nonlinear gain
From the standard reference for the subject^{[11]} it is stated that:
Algorithmic or automatic differentiation (AD) is concerned with the accurate and efficient evaluation of derivatives for functions defined by computer programs.
AD is a numerical differentiation technique that uses chainrule of calculus (differentiation) for the floating point evaluation of a function and/or its derivatives^{[13]}. The principal difference between AD and the usual numerical finite difference methods is that in AD there is no discretisation or cancellation errors; moreover, the results (derivative values) are accurate to the machine precision or within the roundoff used. Symbolic differentiation methods makes a function derivative into a single long expression at the point where the function
Usually AD is implemented in either of the two ways: 1) operator overloading, 2) source transformation. In operator overloading the existing classes or types are redefined, initialised with appropriate values for the function and its derivative, invoke the function, and find the values of the derivatives. Source transformation is a method that requires suitable sophisticated compilerlike software to read the computer program containing the function and determine which statements for the function
There are fundamentally two approaches for computing the derivatives using AD.
1) Forward mode: The function is evaluated for its derivative in the forward direction starting from the input (or independent) variable. In this forward mode of calculation, the sensitivities for finding the derivatives of the intermediate stages are propagated forward until the final result.
2) Reverse mode: This works in two stages. In the first stage the original code is run and is augmented with the statements to store data. Then the derivative is evaluated starting from the output (dependent variable) towards the input using all the sensitivities (also called adjoints) in the reverse direction in its second run.
There are many AD packages that work either in forward and/or reverse mode by implementing usually operator and function overloading techniques. ADOL C, ADIC, Sacado etc. are developed for working in C/C++, ADF, OpenAD, ADIFOR runs with Fortran. Python based AD packages are pyadolc, algopy, etc. MATLAB compatible version of AD is developed by TOMLAB Inc. called MAD.
In the present work, TOMLABMAD has been used for the purpose of linearisation, so as to reduce the error, incurred due to the evaluation of derivative values computationally, by methods such as finite difference. The use of AD for finding the gradient or Jacobian using MAD software package is efficient and fast enough to implement it online with a regular PC.
5 Numerical exampleTo demonstrate the performance of mixed
$ \begin{align*} A =&\left[ \begin{array}{cc} 0.80&0.09 \\ 1.00&0.00 \end{array} \right] \end{align*} $ 
$ \begin{align*} B_{0} =&\left[ \begin{array}{c} 0.1 \\ 1.0 \end{array} \right] , B_{1}=\left[ \begin{array}{c} 0.1 \\ 0.0 \end{array} \right], B_{2}=\left[ \begin{array}{c} 0.0 \\ 1.0 \end{array} \right]\\ C_{0} =&\left[ \begin{array}{ll} 1&1 \end{array} \right] , C_{1}=\left[ \begin{array}{ll} 1&0 \end{array} \right] , C_{2}=\left[ \begin{array}{ll} 1.1&0.07 \end{array} \right] \\ D_{00} =&1.0, D_{01}=0.1, D_{02}=0.001, \\ D_{10} =&0.01, D_{11}=1.0, D_{12}=0.1, \\ D_{20} =&0.1, D_{21}=0.0. \end{align*} $ 
The system is perturbed with the deterministic disturbance,
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Fig. 1. Input and output profiles of the Nash game based mixed 
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Fig. 2. State estimation error profiles of the Nash game based mixed 
It can be observed that the effect of the measurement noise component
Control of bioreactor is of much interest due to its highly nonlinear behaviour. A mechanistic model^{[14]} of the bioreactor is given in the Appendix B. The dynamics of bioreactor, which exhibits inputmultiplicity nonlinearity, is shown elsewhere^{[14, 15]}. At the low concentration of input feed substrate, the growth rate of the cellbiomass dominates the reaction rate. Thus, increase in feed concentration results in an increase in productivity. However, beyond a certain value of the input feed concentration, the product and substrate inhibition prevails and thereupon productivity shows a negative gradient with respect to the input feed concentration. Thus, there exists an optimum value of the bioreactor productivity. The necessary condition for optimality implies that the steadystate gain (product concentration vs. manipulated feed substrate concentration) changes its sign across the optimum point and the steady state gain is zero at the optimum. Thus, the system exhibits input multiplicity and there exists a singularity at the optimum operating point.
The robust control of the bioreactor is one of the objectives of the paper. For achieving this, the dynamics of bioreactor system is efficiently captured through the nonlinear system identification method using LWN model from the process inputoutput data^{[3]}. The mechanistic model (Appendix B) of the bioreactor^{[16]} is assumed to be the actual process for the simulation case studies.
6.1 Control of bioreactor using mixedIn the present study on the bioreactor process, the closed loop robustness due to mixed
For some processes the plantmodel mismatch may result in deterioration of the performance or may even lead to catastrophic effects, depending on the nature of the process. The statefeedback controller design^{[3, 10]} cannot be deployed for such processes, where the system states are not directly measurable for control, or, for model based control cases, where the states of the model do not reflect the actual process states. For any blackbox model, usually, the model states fall under the latter category. In such cases, it is wise to use successively linearized deviation (SLD) model, than simply employing successive linearisation technique. Statedeviation (SD) model is one in which the model, whose states represent deviation of the states from a reference value (usually the previous value of the model state) as
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Fig. 3. SLDLWN model response for sinusoidal input signal 
It could be observed that there is not much distortion in the performance of the LWN model due to successively linearised deviation approach. Hence the approximation error of the nominal model due to the linearisation approach is minimal and thereby the controller
The closedloop performance of the output feedback NGMMPC controller for step change in
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Fig. 4. Parametric uncertainty: Output and input profiles of the bioreactor using Nash game based mixed 
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Fig. 5. Parametric uncertainty: Output and input profiles of the bioreactor using Nash game based mixed 
6.2 Comparison of output feedback MPC Schemes
The efficacy of the proposed mixed
In this paper, an observer based infinitehorizon output feedback mixed
For a discrete time linear system given by
$ \begin{align} x_{k+1} =&\bar{A}x_{k}+\bar{B}u_{k}+\bar{E}w_{k}^0 \end{align} $  (A1) 
$ \begin{align} y_{k} =&\bar{C}_{1}x_{k}+\bar{D}_{1}w_{k}^0 \end{align} $  (A2) 
$ \begin{align} z_{k} =&\bar{C}_{2}x_{k}+\bar{D}_{2}u_{k} \end{align} $  (A3) 
let the energy (Lyapunov) function associated with it is defined as
As the horizon tends towards infinity in an infinitehorizon control problem, it is known that
$ \begin{align} \Delta V_k(x_k)=V_k(x_{k+1})V_k(x_k)=0. \end{align} $  (A4) 
Moreover, the controlled output
$ \begin{align} \label{zTz} z_k^{\rm T}z_k=0. \end{align} $  (A5) 
Using the definition of the Lyapunov function and the dynamic state equation,
$ \begin{align} \Delta V_{k}(x_{k}) =&(\bar{A}x_{k}+\bar{B}u_{k})^{\rm T}P_{k}(\bar{A}x_{k}+ \bar{B}u_{k})x_{k}^{\rm T}P_{k}x_{k} = \notag \\ &x_{k}^{\rm T}\bar{A}^{\rm T}P\bar{A}x_{k}+x_{k}^{\rm T}\bar{A}^{\rm T}P_{k}\bar{B} u_{k}+u_{k}^{\rm T}\bar{B}^{\rm T}P_k\bar{A}x_k+\notag \\ &u_{k}^{\rm T}\bar{B}^{\rm T}P_{k}\bar{B}u_{k}x_{k}^{\rm T}P_{k}x_{k}.\label{dVex} \end{align} $  (A6) 
On the other hand, (A5) gives
$ \begin{align} z_{k}^{\rm T}z_{k}=[x_{k}^{\rm T}\;\;u_{k}^{\rm T}]\left[ \begin{array}{c} \bar{C}_{2}^{\rm T} \\ \bar{D}_{2}^{\rm T} \end{array} \right] [\bar{C}_{2}\;\;\bar{D}_{2}]\left[ \begin{array}{c} x_{k} \\ u_{k} \end{array} \right] =0. \label{zTzex} \end{align} $  (A7) 
Therefore, the summation of (A6) and (A7) also equates to zero i.e.,
$ \begin{align} S &=[x_{k}^{\rm T}\;\;u_{k}^{\rm T}]\left( \begin{array}{c} \left[ \begin{array}{cc} \bar{A}^{\rm T}P\bar{A}C&A^{\rm T}PB \\ \bar{B}^{\rm T}P\bar{A}&\bar{B}^{\rm T}P\bar{B} \end{array} \right] + \\[4mm] \left[ \begin{array}{cc} \bar{C}_{2}^{\rm T}\bar{C}_{2}&\bar{C}_{2}^{\rm T}\bar{D}_{2} \\ \bar{D}_{2}^{\rm T}\bar{C}_{2}&\bar{D}_{2}^{\rm T}\bar{D}_{2} \end{array} \right] \end{array} \right) \left[ \begin{array}{c} x_{k} \\ u_{k} \end{array} \right] = \notag \\ &\quad [x_{k}^{\rm T}\;\;u_{k}^{\rm T}]\times \notag \\ &\left[ \begin{array}{cc} \bar{A}^{\rm T}P\bar{A}P+\bar{C}_{2}^{\rm T}\bar{C}_{2}&\bar{A}^{\rm T}P\bar{B}+\bar{C }_{2}^{\rm T}\bar{D}_{2} \\ \bar{B}^{\rm T}P\bar{A}+\bar{D}_{2}^{\rm T}\bar{C}_{2}&\bar{B}^{\rm T}P\bar{B}+\bar{D} _{2}^{\rm T}\bar{D}_{2} \end{array} \right]\times \notag \\ & \left[ \begin{array}{c} x_{k} \\ u_{k} \end{array} \right]=0. \end{align} $  (A8) 
From the above, we get the following algebraic Riccati equation for the controller part, as given below:
$ \begin{align} P =&\bar{A}^{\rm T}P\bar{A}+\bar{C}_{2}^{\rm T}\bar{C}_{2}(\bar{C}_{2}^{\rm T}\bar{D} _{2}+\bar{A}^{\rm T}P\bar{B})\times \notag \\ &(\bar{D}_{2}^{\rm T}\bar{D}_{2}+\bar{B}^{\rm T}P\bar{B})^{1}(\bar{D}_{2}^{\rm T}\bar{C }_{2}+\bar{B}^{\rm T}P\bar{A}). \label{AlgRic1} \end{align} $  (A9) 
Defining
$ \begin{align} {M}=\left[ \begin{array}{cc} \bar{A}^{\rm T}P\bar{A}P+\bar{C}_2^{\rm T}\bar{C}_2&\bar{A}^{\rm T}P\bar{B}+\bar{C}_2^{\rm T}\bar{ D}_2 \\ \bar{B}^{\rm T}P\bar{A}+\bar{D}_2^{\rm T}\bar{C}_2&\bar{B}^{\rm T}P\bar{B}+\bar{D}_2^{\rm T}\bar{ D}_2 \end{array} \right] \end{align} $  (A10) 
the stabilising controller
$ \begin{align} {M}\left[ \begin{array}{c} 0 \\ I \end{array} \right]=0 \end{align} $  (A11) 
for a positive definite matrix
$ \begin{align} \bar{K}=(\bar{B}^{\rm T}P\bar{B}+\bar{D}_2^{\rm T}\bar{D}_2)^{1}(\bar{B}^{\rm T}P\bar{A}+ \bar{D}_2^{\rm T}\bar{C}_2). \end{align} $  (A12) 
The property of duality is utilised for the estimation part of the
Accordingly, the algebraic Riccati equation for the estimation part is obtained from (A9) and the above given duality, as follows
$ \begin{align} Q =&\bar{\bar{A}}Q\bar{\bar{A}}^{\rm T}+\bar{E}\bar{E}^{\rm T}(\bar{\bar{A}}Q\bar{C% }_{1}^{\rm T}+\bar{E}\bar{D}_{1}^{\rm T})\times \notag \\ &(\bar{D}_{1}\bar{D}_{1}^{\rm T}+\bar{C}_{1}Q\bar{C}_{1}^{\rm T})^{1}(\bar{D}_{1} \bar{E}^{\rm T}+\bar{C}_{1}Q\bar{\bar{A}}^{\rm T}) \end{align} $  (A13) 
where
$ \begin{align} \bar{L}=(\bar{\bar{A}}Q\bar{C}_{1}^{\rm T}+\bar{E}\bar{D}_{1}^{\rm T})(\bar{D}_{1} \bar{D}_{1}^{\rm T}+\bar{C}_{1}Q\bar{C}_{1}^{\rm T})^{1}. \end{align} $  (A14) 
The mechanistic model of a bioreactor is given by the following differential equations ^{[14, 15]}
$ \begin{align} \frac{{\rm d}X}{{\rm d}t} =&DX+\mu X \end{align} $  (B1) 
$ \begin{align} \frac{{\rm d}S}{{\rm d}t} =&D(S_{f}S)\frac{1}{Y_{\frac{X}{S}}}\mu X \end{align} $  (B2) 
$ \begin{align} \frac{{\rm d}P}{{\rm d}t} =&DP+(\alpha \mu +\beta )X \end{align} $  (B3) 
where
$ \begin{align} \mu =\frac{\mu _{m}(1\dfrac{P}{P_{m}})}{K_{m}+S+\dfrac{S^{2}}{K_{i}}} \end{align} $  (B4) 
where
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