New Designs of Reduced-Order Observer-Based Controllers for Singularly Perturbed Linear Systems

The slow and fast reduced-order observers and reduced-order observer-based controllers are designed by using the two-stage feedback design technique for slow and fast subsystems. The new designs produce an arbitrary order of accuracy, while the previously known designs produce the accuracy of O(ε) only where ε is a small singular perturbation parameter. Several cases of reduced-order observer designs are considered depending on the measured state space variables: only all slow variables are measured, only all fast variables are measured, and some combinations of the slow and fast variables are measured. Since the two-stage methods have been used to overcome the numerical ill-conditioning problem for Cases (III)–(V), they have similar procedures. The numerical ill-conditioning problem is avoided so that independent feedback controllers can be applied to each subsystem. The design allows complete time-scale separation for both the reduced-order observer and controller through the complete and exact decomposition into slow and fast time scales. This method reduces both offline and online computations.


Introduction
The presentation in this paper is based on the doctoral dissertation [1].The fundamental technique used is the two-stage feedback controller design [2,3].The full-order observers for singularly perturbed linear systems were considered in [4][5][6][7][8][9][10].The reduced-order observer for singularly perturbed discrete-time systems has been studied only in two papers [11,12], both of them producing accuracy of () (an () is defined by () ≤ , where  is a bounded constant), where  is a small singular perturbation parameter.There are no corresponding results reported in the literature for the reduced-order continuous time observers.Since () accuracy might not be acceptable, our motivation is to design reduced-order observers for this class of systems with the accuracy of (  ) ≤   ,  = 1, 2, 3, . .., where  is a bounded constant, which is along the lines of high accuracy techniques for singularly perturbed systems [5].Note that, for  < 1,   → 0 rapidly.Consider a singularly perturbed linear system that contains slow and fast modes [6]: where  is a small positive singular perturbation parameter that indicates separation of state variables () ∈ R  into slow  1 () ∈ R  1 and fast  2 () ∈ R  2 and  1 +  2 = , and () ∈ R  are the system measurements.The problem matrices are constant and of appropriate dimensions.It was assumed that the matrix  has full rank equal to .The singularly perturbed system is studied under the following standard assumption [4]. 2 Mathematical Problems in Engineering Assumption 1.  22 is nonsingular.
In the following we consider five cases for the reducedorder observer-based controller design for singularly perturbed linear systems depending on combinations of measured states [1].

Case I: Controller Design When Only All Slow Variables Are Measured
Consider the linear time invariant singularly perturbed control system [1], in which only slow variables are directly measured: where () ∈ R  is the control input.The reduced-order observer for ( 2) is given by [13]  ż 2 () =   ẑ2 () +    () +    () , The state estimation of the fast variables is obtained from x2 () = ẑ2 () + 1   11  () (5) so that The matrix  11 is chosen to stabilize the reduced-order observer (3); that is, To design this reduced-order observer the following assumption is needed [1].
Assumption 2. The pair (  22 ,   12 ) is controllable, which is equivalent to the pair ( 22 ,  12 ) being observable, which is equivalent to the requirement that the original system is observable.
In the following, the Chang transformation matrix [4] will be needed: where matrices  and  satisfy the algebraic equations 0 =  ( The solutions for  and  can be obtained using either the fixed-point iterations or Newton method or eigenvector method [5].
Using the separation principle, the observer-based controller design via the two-stage design [2] produces [1] The feedback gain   is chosen thus to place slow eigenvalues at the desired locations; that is, The matrix  is obtained from the Sylvester algebraic equation where The feedback gain  2 is chosen thus to place the fast eigenvalues at the desired locations; that is, To obtain the reduced-order feedback gains   and  2 , the following controllability assumption is needed [14].
Based on information from (3), (7), and (10), we present in Figure 1 the block diagram for the reduced-order observerbased controller when only all slow state variables are perfectly measured.In (11) and (14) we have chosen the feedback gains for the eigenvalue assignment problem.However, any feedback gains  1 and  2 can be used to control the system and provide corresponding design requirements.

System
Reduced-order observer Figure 1: Case I: reduced-order observer-based controller design for a singularly perturbed linear system when only all slow state variables are directly measured.

Case II: Controller Design When Only All Fast Variables Are Measured
Consider the linear time invariant singularly perturbed control system [1], when only all fast state variable are directly measured: The reduced-order observer for system (15) is given by [7] ż where =  desired robs . (20) The reduced-order observer (16) can be designed under the following assumption [14].Additional matrices needed in this design can be obtained from ( 9) and ( 12)- (13).Using the separation principle, the observer-based controller can be designed via the two-stage design as where   and  2 are obtained from (11) and (14).The feedback gains   and  2 can be obtained under Assumption 3.
In Figure 2, the block diagram for the reduced-order observer-based controller when only all fast variables are perfectly measured is presented.

System
Reduced-order observer

Cases (III)-(V): Controller Design When Some Combinations of Slow and Fast Variables Are Measured
In Case (III), the measurable states  11 () are parts of the slow state vector  1 () in the singularly perturbed linear system defined in (1), as where Use the following partitioning: where The reduced-order observer for this case is derived in [1], and it is given by where For the eigenvalue assignment in    , we encounter the singularly perturbed structure, so that the two-stage method is applied for a two time-scale problem.
The reduced-order sequential observer configuration obtained using the two-stage method [1] is given by where  *  and  * 2 are obtained from  −1 4    , with  4 defined by where where    and    are matrices that satisfy with   matrices defined in (26).The reduced-order observer (29) has a sequential structure.It can be block diagonalized and used in a parallel structure as follows: where The original coordinates ẑ12 () and ẑ2 () and the coordinates ẑ () and ẑnew2 () are related via where  2 satisfies the algebraic Sylvester equation represented by The steps used in the sequential and parallel observer design structures are summarized in Figure 3.We use the parallel observers structure (33) and consider the reduced-order observer-based controller design for singularly perturbed linear systems.Observer ( 33) is now driven by the system measurements and control inputs; that is, where  2 and  2 can be obtained from  −1   as The control input in the ẑ -ẑ new2 coordinates is given by with The corresponding block diagram for the observer driven controller is presented in Figure 4.This block diagram clearly indicates full parallelism of the slow controller driven by the slow observer and the fast controller driven by the fast observer.
The remaining matrices defined in (29) are given by  In Case (IV), the measurable states  21 () are parts of the fast state vector  2 () in the singularly perturbed linear system defined in (1): where using the following partitioning: where dimensions of matrices The corresponding reduced-order observer structure is given by [1] ż where The observer gain  3 and matrices   ,2 and   ,2 [7] For the eigenvalue assignment in   ,2 , we encounter singularly perturbed structure so that the two-stage design is applied to the slow and fast subsystems.
The two-stage method provides the parallel observer [1].Using the parallel observer we can form the reduced-order observer-based controller given as where where requiring that the following assumption is satisfied [14].
The fast feedback gain  2,2 is obtained from the eigenvalue assignment problem; that is, The following observability assumption is needed [14].
The control input in ẑ,2 -ẑ new2,2 coordinates is given by  [ with The remaining matrices  ,2 and  2,2 satisfy the algebraic Sylvester equation represented by The corresponding block diagram for the observer driven controller is presented in Figure 5.
In Case (V), the measurable states  11 () and  21 () are parts of the slow state vector  1 () and the fast state  2 () in the singularly perturbed linear system defined as where using the following partitioning:  62)-( 63) can be represented as where () are the measurable states and   () are the unmeasurable states.  1 and   3 are elements in (62) relevant to the measurable states and   2 and   4 are elements in (62) relevant to the unmeasurable states.The reduced-order observer is given by The observer gains are obtained from For the eigenvalue assignment in   ,3 , we encounter again the singularly perturbed structure.
The slow feedback gain  ,3 can be obtained from the eigenvalue assignment problem; that is, assuming that the following assumption is satisfied [14].
The fast feedback gain  2,3 can be obtained from the eigenvalue assignment problem as The following observability assumption is needed [14].
The control input in the ẑ,3 -ẑ new2,3 coordinates is given by  () = − ,3 x () = − [  The corresponding block diagram for the observer driven controller is presented in Figure 6.

Numerical Example for Case I
Consider a 4th-order system with the system matrices , , and  defined in [1].The controllability matrix has full rank and therefore the pair (, ) is controllable.The results obtained using MATLAB are given below.We locate the feedback system slow eigenvalues at  desired (79)

Numerical Example for Case II
Consider a 4th-order system with the system matrices , , and  defined in Section 5. We locate the feedback system slow eigenvalues at

Numerical Example for Case III
Consider a 4th-order magnetic tape system from Section 5. We locate the feedback system slow eigenvalues at  desired  (83)

Numerical Example for Case IV
Consider a 4th-order system with the system matrices , , and  defined in Section 5. We locate the feedback system slow eigenvalues at  desired  (85)

Numerical Example for Case V
Consider a 4th-order system with the system matrices , , and  defined in Section 5. We locate the feedback system slow eigenvalues at  desired  (87)

Conclusion
We have designed with high accuracy the reduced-order observer-based controllers for singularly perturbed linear systems.The numerical ill-conditioning of the original problem is removed.We have demonstrated that the full-order singularly perturbed system can be successfully controlled with the state feedback coming from the reduced-order observer-based controllers fully designed on the subsystem levels.The two-stage method is successfully implemented for both the observer and controller designs.

Mathematical Problems in Engineering
In Case (II), the pair (

Figure 2 :
Figure 2: Case II: slow and fast observer-based controller design for a singularly perturbed linear system when only fast state variables are measured directly.

Figure 3 :
Figure 3: Case III: design outlines for the sequential and parallel reduced-order observer.

Figure 4 :
Figure 4: Case III: slow and fast observer-based controller designs for singularly perturbed linear systems with the system feedback gains obtained in (39) when parts of slow variables are measured.

Figure 5 :
Figure 5: Case IV: slow and fast observer-based controller design for a singularly perturbed linear system measurable variables being part of the fast variables only.

Figure 6 :
Figure6: Case V: slow and fast observer-based controller design for a singularly perturbed linear system when some parts of the slow variables and some parts of the fast variables are measured.
given in the previous numerical example.Following the design procedure, the observer matrices   ,   ,  12 ,  1 , and  2 are given as 11 , (1/) 21 ) is observable, which implies