Abstract

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 only where 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.

1. 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 [4–10]. 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 , , where is a bounded constant, which is along the lines of high accuracy techniques for singularly perturbed systems [5]. Note that, for , 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 into slow and fast and , and 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].

Assumption 1. is nonsingular.

In the following we consider five cases for the reduced-order observer-based controller design for singularly perturbed linear systems depending on combinations of measured states [1].

2. 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 is the control input. The reduced-order observer for (2) is given by [13]whereThe state estimation of the fast variables is obtained fromso thatThe matrix 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 is controllable, which is equivalent to the pair 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 equationsThe 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 equationwhere The feedback gain is chosen thus to place the fast eigenvalues at the desired locations; that is,To obtain the reduced-order feedback gains and , the following controllability assumption is needed [14].

Assumption 3. The pairs and are controllable.

Based on information from (3), (7), and (10), we present in Figure 1 the block diagram for the reduced-order observer-based 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 and can be used to control the system and provide corresponding design requirements.

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.

3. 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]whereThe slow state estimation is obtained fromso thatThe matrix is chosen to stabilize the reduced-order observer (16)-(17); that is,The reduced-order observer (16) can be designed under the following assumption [14].

Assumption 4. The pair is controllable, which is equivalent to the pair being observable. It was shown in Appendix that this is equivalent to being observable.

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 aswhere and are obtained from (11) and (14). The feedback gains and 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.

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.

4. Cases (III)–(V): Controller Design When Some Combinations of Slow and Fast Variables Are Measured

In Case (III), the measurable states are parts of the slow state vector in the singularly perturbed linear system defined in (1), aswhereUse the following partitioning:where dimensions of matrices , and are, respectively, , , and , with corresponding dimension of , , matrices. System (22) with (23)-(24) can be represented aswhereThe reduced-order observer for this case is derived in [1], and it is given bywhereFor 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 bywhere and are obtained from , with defined bywhere where and are matrices that satisfywith 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:whereThe original coordinates and and the coordinates and are related viawhere satisfies the algebraic Sylvester equation represented byThe steps used in the sequential and parallel observer design structures are summarized in Figure 3.

Figure 3 

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

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 and can be obtained from asThe control input in the - coordinates is given bywithThe 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.

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.

The remaining matrices defined in (29) are given by, , and can be obtained from the following formulas:In Case (IV), the measurable states are parts of the fast state vector in the singularly perturbed linear system defined in (1):whereusing the following partitioning:where dimensions of matrices , , and are, respectively, , , and , with the corresponding dimensions of , matrices. System (43) with (44)-(45) can be represented aswhereThe corresponding reduced-order observer structure is given by [1]whereThe observer gain and matrices and [7] are obtained fromFor the eigenvalue assignment in , we encounter singularly perturbed structure so that the two-stage design is applied to the slow and fast subsystems.