Hybrid Predictive Control Based on High-Order Differential State Observers and Lyapunov Functions for Switched Nonlinear Systems

In this paper, a hybrid predictive controller is proposed for a class of uncertain switched nonlinear systems based on high-order differential state observers and Lyapunov functions. The main idea is to design an output feedback bounded controller and a predictive controller for each subsystem using high-order differential state observers and Lyapunov functions, to derive a suitable switched law to stabilize the closed-loop subsystem, and to provide an explicitly characterized set of initial conditions. For the whole switched system, based on the high-order differentiator, a suitable switched law is designed to ensure the whole closed-loop’s stability. The simulation results for a chemical process show the validity of the controller proposed in this paper.


Introduction
Switched system is a typical hybrid dynamic system made up of some subsystems and a switched law.In recent years, the stabilization of constrained switched systems became an attractive research subject [1].
Model predictive control (MPC) is a receding horizon control (RHC) method to handle constraints within an optimal control setting [2].There have been many results to show the performance of constrained MPC [3].In MPC design, the initial feasibility of the optimization problem is always assumed.Due to uncertainties and constraints of the practical process, this assumption may not be satisfied.Furthermore, the set of initial conditions, starting from where a given MPC formulation is guaranteed to be feasible, has not been explicitly characterized.
In recent years, controller design methods based on Lyapunov functions have been developed, which can give an explicitly characterized set of initial conditions from which the closed-loop system is stable [4].By embedding the Lyapunov-based design methods into the MPC design, we can obtain the set of initial conditions from where the closed-loop system is stable.In refs.[5,6], two Lyapunov-based predictive controllers were derived for constrained nonlinear systems.In refs.[7,8], two Lyapunov-based predictive controllers were proposed for constrained switched systems and constrained switched systems with uncertainties, respectively.In these papers, the states of the system are observable.
However, in real processes the system's states are often not measurable, and hence, state-feedback controllers and switched laws cannot be realized.One of the methods to overcome this difficulty is to construct a state observer to estimate the states for constructing the controller and switched law.In ref. [9], an output feedback bounded controller was given for a class of nonlinear systems which was not switched system.In ref. [10], for a kind of nonlinear switched systems without uncertainties and disturbance, a bounded nonlinear controller was given.But it was not guaranteed to be optimal with respect to an arbitrary performance criterion which incurporates requested performance in the design.In ref. [11], a hybrid output feedback predictive controller was proposed for a class of switched nonlinear systems without uncertainties.In papers [9][10][11], the processes' states were estimated using a high-gain observer, but many adjustable parameters of the observer need to be chosen expe-rientially.Sometimes the wrong selection of parameters can cause stability problems and an undesired transient performance of the observer.In refs.[12][13][14], a high-order differential state observer was designed to estimate the states of a nonlinear system.Theoretically the parameters are chosen according to the performance and stability of the observer and theoretically few parameters with explicit meanings have to be selected based on the performance and stability of the observer.
In this paper, an output feedback hybrid predictive controller is proposed for a class of uncertain switched nonlinear systems based on high-order differential state observers and Lyapunov functions.The main idea is to design a hybrid predictive controller based on Lyapunov functions and high-order differential state observers, which switches between a bounded feedback controller and a predictive controller for each subsystem, and to provide an explicitly characterized set of initial conditions to stabilize the closed-loop subsystem.Here, we use high-order differentiators as state observers.This high-order differential state observer has simple structure with few parameters.A suitable switched law based on the high-order differentiator is designed to guarantee the whole closed-loop system's stability.Finally, the simulation results for a chemical process show the validity of the procedure proposed in this paper.

Problem Description
Consider the constrained switched nonlinear system where denotes the vector of continuous-time state variables, denotes the vector of manipulated inputs taking values in a nonempty compact subset , where  is the Euclidian norm, and is the magnitude of the constraints.
denotes the bounded uncertain parameter vector taking values in a nonempty compact subset

 
: is the switching signal assuming to be a piece-wise continuous (from the right) function of time, i.e., for all , implying that only a , ,  denote the set of switching times at which the kth subsystem is switched in and out, respectively.It is assumed that all entries of the vector functions are sufficiently smooth and that x , and The objective of this paper is to design a nonlinear output feedback predictive controller based on Lyapunov functions and a high order differential state observer for the case where state measurements are not available for each mode of the uncertain switched nonlinear system given by Equation (1).Then, for the whole switched system, based on state estimations, a suitable switched law is designed to ensure the whole closed-loop system's stability.

High-Order Differential State Observers
In order to construct an output feedback controller to stabilize the controlled system (1), we use high-order differential state observers [12][13][14] to estimate the unmeasurable states of the system (1).
Firstly, we give some assumptions.Assumption 1: Consider system (1), for every k K  , there exist an integer and a set of invertible coordinates are nonlinear scalar functions of x, such that the system (1) takes the form where ,

 
, , , , is input-to-state stable (ISS) [9], where T 1 , , The following assumptions are given to reduce the influence of uncertainties.
This formula is different from formula (4) since it does not depend on the uncertain parameter k  .We also assume this subsystem is ISS stable.
In order to construct a controller to stabilize the controlled system (1), we use high-order differential state observers [12][13][14] to estimate the un-measurable states of system (1).The high-order differential state observer for each mode can be described as , 1, ,  (7).Note that the HOD is independent of the model of the original system (1).
Proposition 1.The HOD does not rely on the model of the estimated system, parameters are chosen using (8), and has following characteristics: 1) The HOD is an asymptotically stable system.

State Feedback Bounded Controller Based on Lyapunov Functions
We recall the design of a state feedback bounded controller to obtain the set of initial conditions from which the system is stable [9].Define the tracking error variables T and the tracking error vector is the reference input vector, where is a reference input and is its ith time derivative.Then the where function, and The Lyapunov function is chosen as V  e P e , where the positive-definite matrix is chosen to sat- is non-increasing, and x and define the set The continuous bounded control law is constructed as follows where where k g is the it column of and is the column of ; Remark 1.For convenience, this bounded controller ( 13)-( 14) is redefined as .

 
k B x Remark 2. Here, the Lyapunov functions used in verifying the switching conditions at any given time, , are based on .Note that the Lyapunov functions V are in general different from k used in bounded controllers.For the systems with relative degree Based on this bounded controller ( 13)-( 14), an estimation of the stability region is computed as where 0   is the largest number for which , and The robustness property of the bounded controller in ( 13)-( 14) is formalized by the following proposition: Proposition 2. Consider the system (1) for a fixed value   t k   .Under the Assumptions 1-4, compute the bounded control law of ( 13)-( 14) using the Lyapunov functions k and V 0 k   , and then give the stability Then, given any positive real number k , there exists positive real numbers , and     and the output of the closedloop system satisfies: is similar to the proof of Theorem 1 in ref. [9]).

Output Feedback Bounded Controller Based on State Estimations and Lyapunov Functions
In this section, we consider the case when some states of system (1) are not measurable.The bounded controller based on state estimations and Lyapunov functions should be designed and the stable region of initial conditions should be described.
Based on the high-order differential state observer ( 6)-( 8), the following presents the output feedback controller used for each mode and characterizes its stability properties: Proposition 3. Considering the nonlinear system (1), for a fixed mode     , design the output feedback controller with a high-order differential state observer ( 6)-( 8) where 1 exp Rem as a two ark 4. The ith closed-loop subsystem can be cast time-scale system given by ( 19) where e is a vector of the auxiliary error variables , and Proposition 4 establishes the existence of a set, an a rolled r such that once the state estimation error is smaller th certain value (note that the decay rate can be cont by adjusting ), the presence of the state is output feedb stability region, Propositio Given any po e real numbe : : , where where Owing to the existence of parameter uncertainties and constraints, the initial fe of the MPC in (32) is not guaranteed.If it is infeasible, the control action is switched to the bounded controller (17).To describe the whole control action arg M asibility , we cast the kth subsystem (1) as a switched system of the form where       : 0, 1,2 i t   is the switching signal which is assumed to be a piecewise continuous (from the right) function of time.When   i t 1  , the control input takes i.e., the MPC is used; and when , it takes : 2) Design the MPC controller given by ( 21)-( 31 , if then the whole closed-loop system is stable (See the proof in Appendix B).Remark 7. The controller presented in Theore be implemented using the following steps: 1) Given the system model (1) with constraints on the inputs, and a control Lyapunov function to design the bounded controller (17) with suita compute the stability regions (15) and ( 16).Here the staller design only th able region est n the mth subsystem is switched in, the con- ; if the state is in the neighborhood of origin, then and the closed-loop x ing to Proposition rk 8. [10].The time interval b es sho ld be long enough to ecreased to a suff value such that the closed-loop system is stable.Furased on , bu state t w at me, t system is stable accord 4.

Rema
etween two consecutive switch u ensure that the estimation error d iciently small thermore, the decision to switch is not b

Simulation
Consider a continuously stirred tank reactor where three parallel, irreversible, first-order exothermic reactions of the form where A is the reactant species, s the desired product species, U, R denote the by-product species.Under standard modeling assumptions, the mathematical and D i model for the process takes the form [8]   where A C and The boundary of parameters is   .For this system, perform the following sformation Two quadratic, posi e-definite fun ons of the form, tiv cti Note that these positiv e-define function is given for system (39).To estimate the stability regions, the Lyapunov functions where 3

Conclusion
In this paper, a hybrid predictive control method is pro-   posed for a class of uncertain switched nonlinear systems with input constraints and unavailable state measurements.The main objectives were to design a hybrid controller which switches between a bounded controller and a predictive controller based on Lyapunov functions and a high-order differential state observer with a suitable switched law to stabilize the closed-loop subsystem, and to provide an explicitly characterized set of initial conditions.For the whole switched system, a suitable switched law based on the state estimation was derived to ensure of the controller proposed in this paper.

Appendix A Proof of Proposition 4
The proof uses the result of Proposition   By Proposition 4, we have , and . This completes the proof of Proposition 5.

Appendix B
Proof of Theorem 1. (Similar to the proof of Theorem in ref. [7]) Based on Propositions 3-5, we need only to prove that, with the switched law (35)-(37), the whole closed-loop system is still stable.
Let t satisfy in

Assumption 3 :kVAssumption 5 : 1 ,
There exists a known constant bk  Before designing the output feedback controller, we have to revise Assumption 1.There exists an invertible coordinate transformation 

4 )
When the MPC is infeasible   k e estimation x of the closed-loop s  , i.e., when

5 )
bility regions are only signs, for the states cannot be measured, and in the contro imation   k  x is used.And choose Lyapunov function c k V for the system (19); 2) Determine suitable parameters to design the MPC in (21)-(31).Give the size of the ball to whic the state is required to converge, max d At the time of switch e mth subsystem onstraints and time of switching into the kth subsystem), consider whether the state estimation belongs to the stable regi ; , consider the c in Theorem 1, and choose  y m M satisfying (36) and (37), respectively;

2 e
, are then used to synthesize tw ne for each mode of the form o bounded nonlinear controllers (o )

Figure 2 .
Figure 2. Closed-loop state (the reactor concentration C A ) profile.

Figure 4 .
Figure 4.The input Q profile.
This work was supported by the National Natural Science Foundation of Peoples Republic of China under Grants 61374004, 61004013, 61104007 and 60804033, the National Specialized Research Fund for the Doctoral Program of Higher Education under Grant 20113705120003, Higher Educational Science and Technology Program Foundation of Shandong Province under Grant J10LG28, nd h Fund from the whole closed-loop system's stability.The simulation results for a continuously stirred tank reactor showed the validity J11LG08 a th the Doctoral Starting Researc e Qufu Normal University.
long as d k is small enough, we can have , i.e., k, constraint (35) ensures the initial conditions switched on mode k, using the result of Proposition 5, we can have the mode k is stable.So we need only to prove the stability at the switched time.If is switched out and then switched back in.So we can have the feasibility of constraints (28)-(29), then the value of   k V x j continuously decreases.If this mode is not switched in, there exists at lease some such that mode is active and Lyapunov function 1, , j  p j V continues to decrease until j j V    .Similar to discussion before, the constraint (35) ensures that j V continues to be less than j to denote the time at which, for the rth time, the kth subsystem is switched in and out, re-