DESIGN OF OBSERVERS FOR SYSTEMS WITH SLOW AND FAST MODES

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DESIGN OF OBSERVERS FOR SYSTEMS WITH SLOW AND FAST MODES by HEONJONG YOO A thesis submitted to the Graduate School-New Brunswick Rutgers, The State University of New Jersey In partial fulfillment of the requirements For the degree of Master of Science Graduate Program in Electrical and Computer Engineering Written under the direction of Professor Zoran Gajić And approved by New Brunswick, New Jersey May, 2014

ABSTRACT OF THE THESIS Design of Observers for systems with Slow and Fast Modes by HEONJONG YOO Thesis Director: Professor Zoran Gajić This thesis considers the design of observers for systems with slow and fast modes. The existing design method were able to achieve that goal with O() accuracy. Where is a small positive parameter that indicates seperation of system state space variables into slow and fast. We design independent slow and fast reduced-order observer with exact accuracy, and place their closed-loop eigenvalues exactly at the desired location. Furthermore, we apply the two stage method to design the full-order observer. The two stage method results in full-state feedback control for each subsystems which makes the subsystems asymptotic stable. Lastly, the crucial theme of the online savings is the fact that the design allows complete time-scale seperation for both the observer and controller through the complete and exact decomposition into slow and fast time scales. The above method reduces both off-line and on-line computations. In this thesis, we demonstrate the effectiveness of the two stage methods through theoretical and simulation results. ii

Acknowledgements Firstly, I express my deepest and sincerest gratitude to my supervisor, Prof. Zoran Gajić, who has supported me and gave a inspiration thoughout my thesis with his guidance and patience. His constant help gives me better direction whenever I encounter impediment throughout researching this subject. I wish express thanks to Prof. Quingze. Zou for his valuable teaching last semester. He gave me an important guidance to remind me ideas of controls and the fundamental concepts inspire me to come up with some of new concepts in this thesis. I would like to thank to my committee members, Prof. Quingze. Zou and Prof. Zingang. Yi, Prof. Zoran Gajić, I also would like to thank to my senior, Park.Gunyoung, for his help to get some backgrounds for my research. My deepest gratitude goes to my family for their constant love and support. My wife and son as well as my family in South Korea, have supported me for their love. This thesis is completed with their help and support. Lastly, I offer my regards and blessings to all of those who supported me in any respect during the completion of the project. iii

Table of Contents Abstract........................................ ii Acknowledgements................................. iii 1. Introduction................................... 1 1.1. Singularly Perturbed Systems....................... 1 1.1.1. Slow and Fast Seperation of Singularly Perturbed Systems... 2 1.2. Chang Transformation : Block Triangular Form.............. 3 1.2.1. Recursive Algorithm for Solving L-Equation........... 5 1.3. Chang Transformation : Block Diagonal Form............... 6 1.3.1. Recursive Algorithm for Solving the H-Equation......... 7 1.4. Chang Transformation : Composite Method............... 8 1.5. Introduction to Observers and Observer Based Controllers....... 9 1.5.1. Full-Order Observer Design..................... 9 1.5.2. Seperation Principle........................ 11 1.6. An Observer for Singularly Perturbed Systems............. 12 1.6.1. An Observer for the Slow Subsystem............... 12 1.6.2. An Observer for the Fast Subsystem............... 13 1.6.3. State Reconstruction for the Composite System......... 13 1.7. Observer-based Controllers for Singularly Perturbed Systems..... 14 1.7.1. A Controller for the Slow Subsystem............... 15 1.7.2. A Controller for the Fast Subsystem............... 15 1.7.3. A Composite Observer-based Controller............. 16 1.8. Controller Eigenvalues Assignment..................... 17 1.8.1. Design Procedure.......................... 17 iv

1.9. Observer Eigenvalues Assignment...................... 18 1.9.1. Design Procedure.......................... 18 1.10. Conclusion.................................. 19 2. Two-Time Scale Design for Singularly Perturbed Systems...... 20 2.1. Introduction................................. 20 2.2. State Feedback Control via the Chang Transformation.......... 20 2.2.1. Slow and Fast Time Scales : Uncorrected Method........ 22 Eigenvalue Assignment in Slow and Fast Time Scales : Uncorrected Method....................... 24 2.2.2. Slow and Fast Time Scales : Corrected Method......... 25 Eigenvalues Assignment in Two-time Scale via the Corrected Method.......................... 26 Numerical Example : Uncorrected Method............ 27 Numerical Example : Corrected Method............. 29 2.3. Two-Stage Design.............................. 30 2.3.1. Two-Stage Design......................... 32 2.3.2. Exact Eigenvalues Assignment.................. 34 2.3.3. Numerical Example : Two Stage Design............. 35 3. Two-Stage Observer Design for Singularly Perturbed Systems... 37 3.1. Introduction................................. 37 3.2. Mode Seperation of Singularly Perturbed Linear Systems........ 37 3.3. Two-Stage Design of the Full-Order Observer............... 39 3.3.1. An Observer for the Slow Subsystem................ 42 3.3.2. An Observer for the Fast Subsystem................ 43 3.4. State Reconstruction for the Composite System.............. 43 3.5. Design Algorithm for an Observer s Gain................. 45 3.6. A Numerical Example : = 0.1...................... 45 3.7. Observer Controller Duality........................ 47 v

3.8. Conclusion.................................. 49 4. Slow and Fast Observer-based Controller for Singularly Perturbed System......................................... 50 4.1. Introduction................................. 50 4.2. Slow and Fast Observer-Based Controller Derivation.......... 50 4.2.1. Design Algorithm.......................... 52 4.2.2. Numerical Example......................... 52 4.2.3. Simulation Results......................... 53 4.3. Conclusion.................................. 61 5. Conclusions and Future Work........................ 62 5.1. Conclusions.................................. 62 5.2. Future Work................................. 62 References....................................... 63 vi

1 Chapter 1 Introduction 1.1 Singularly Perturbed Systems Large time scale linear systems are encountered frequently in engineering problems. The crucial theme is how to reduce a large time scale system into a reduced form to enhance analysis, design and simulation. Consider a linear time invariant system ẋ(t) = Ax(t) + Bu(t) y(t) = Cx(t) + Du(t) (1.1) where x(t) R n, u(t) R r and y(t) R p are state, control input, and output variables respectively. As proposed by (Anderson, 1982), the above system (1.1) will be classified as a two-time-scale system if the eigenvalues of A matrix, denoted as λ(a) can be seperated into two disjoint sets q i r i for all q i in Q, and r i in R. (1.2) The small parameter is the eigenvalue ratio = max i q i min j r i (1.3) which presents the system s time scale seperation, and it defines (1.1) as a singularly perturbation system. Now one can regard (1.1) as two coupled subsystems represented by ẋ 1 (t) = A 11 x 1 (t) + A 12 x 2 (t) + B 1 u(t) ẋ 2 (t) = Ã21x 1 (t) + Ã22x 2 (t) + B 2 u(t) (1.4) y(t) = C 1 x 1 (t) + C 2 x 2 (t)

2 where the general system matrices are A = A 11 A 12 and B = B 1. Note that à 21 à 22 B 2 à 21, à 22 and B 2 have large elements. It can be shown that variations of x 2 (t) are fast, which is a characteristic of singularly perturbed systems. Later, we will introduce the Chang transformation to decouple the slow and fast subsystems. Using the spectral norm, can be represented by (Anderson, 1982) A s A f (1.5) where A s and A f are matrices obtained after the application of the Chang transfor- mation to the system matrix A = A 11 A 12. This fact will be discussed in later. à 21 à 22 Multiplying by the second equation in (1.4), the singularly perturbed system is given by ẋ 1 (t) = A 11 x 1 (t) + A 12 x 2 (t) + B 1 u(t) ẋ 2 (t) = A 21 x 1 (t) + A 22 x 2 (t) + B 2 u(t) (1.6) y(t) = C 1 x 1 (t) + C 2 x 2 (t) where A 21 = Ã21, A 22 = Ã22 and B 2 = B 2. 1.1.1 Slow and Fast Seperation of Singularly Perturbed Systems A singularly perturbed system is in the explicit state variable form in which the derivatives of some of the sates are multiplied by a small positive scalar. Consider the singular perturbed system (1.6), where is a small singular perturbation parameter, and x 1 (t) R n 1 and x 2 (t) R n 2 are state vectors. The system (1.6) may be approximately decomposed into a reduced system showing n 1 slow modes and a fast subsystem showing n 2 fast modes, (O Reilly, 1980). The reduced subsystem, in other word, slow subsystem is obtained by setting = 0 in (1.6), that is ẋ s (t) = A 11 x s (t) + A 12 x 2s (t) + B 1 u s (t) (1.7) 0 = A 21 x s (t) + A 22 x 2s (t) + B 2 u s (t) (1.8) y s (t) = C 1 x s (t) + C 2 x 2s (t) (1.9)

3 x 2s (t) is the slow part for x 2 (t). Since x 2 (t) = x 2s (t) + x 2f (t), the reduced states x 2s (t) can be obtained when is set zero. The following assumption should be satisfied for obtaining the slow subsystem. Assumption 1.1.1. A 1 22 exists, Under Assumption 1.1.1, x 2s (t) can be obtained from (1.8) as x 2s (t) = A 1 22 A 21x s (t) A 1 22 B 2u s (t) (1.10) Substitution of (1.10) into (1.7) results in with ẋ s (t) = A 0 x s (t) + B 0 u s (t) y s (t) = C 0 x s (t) + D 0 u s (t) A 0 = A 11 A 12 A 1 22 A 21, B 0 = B 1 A 12 A 1 22 B 2 C 0 = C 1 C 2 A 1 22 A 21, D 0 = C 2 A 1 22 B 2 The fast subsystem (Kokotovic et al., 1999) is defined by ẋ f (τ) = A 22 x f (τ) + B 2 u f (τ) y f (τ) = C 2 x f (τ) (1.11) (1.12) (1.13) where τ = (t t 0) (1.14) In this section, we present how the original singularly perturbed system (1.6) can be decomposed exactly into two sub-systems corresponding to slow and fast variables. 1.2 Chang Transformation : Block Triangular Form The purpose of the Chang transformation is to decouple the fast subsystem from the slow subsystem. The open loop system is considered regarding the Chang transformation in this section. The design of the Chang transformation is achieved in two steps. Firstly, the upper triangular form is achieved and secondly, the block diagonal from is

4 obtained to completely decouple the fast and slow sub-systems. Consider a general continuous-time linear system which is given by ẋ(t) = Ax(t) + Bu(t) (1.15) where x(t) R n, u(t) R m, and A, B are constant matrices of appropriate dimensions. There exists a similarity transformation defined by x(t) = T 1 1 x(t) (1.16) where T 1 = I n 0 (1.17) L I m which transforms (1.15) into ẋ(t) = T 1 T 1 1 AT 1 T 1 1 x(t) + Bu(t) (1.18) Multiplying from the left by T 1 1 both side of (1.18), (1.18) becomes x(t) = Ā x(t) + Bu(t) (1.19) where Ā = T 1 1 AT 1, B = T 1 1 B (1.20) The above method is a state transformation of a general linear system. Now we apply this method to the singularly perturbed system. Consider the singularly perturbed system. with ẋ = Ãx(t) + Bu(t), x(t 0 ) = x 0 (1.21) Ã = A 11 A 12, B = 1 A 21 1 A 22 B 1 1 B 2, x(t) = x 1 (1.22) Using the state tranformation (1.16), the above singularly perturbed system (1.21) has Ā = T 1 ÃT1 1 represented by Ā = I n 0 A 11 A 12 L I m 1 A 21 1 A 22 I n 0 L I m x 2 = A 11 A 12 L A 12 (1.23) f(l) LA 12 + 1 A 22

5 with f(l) = L(A 11 A 12 L) + 1 (A 21 A 22 L) (1.24) After the first step of the Chang tranformation, the block-triangular form (1.23) can be obtained if f(l) = 0. Note that A 21 and A 22 are divided by when applying the first step of the Chang tranformation. Furthermore, eigenvalues are preserved under Step 1 of the Chang tranformation. The following L equation is obtained by multiplying (1.24) by 0 = L(A 11 A 12 L) + (A 21 A 22 L) (1.25) The newly obtained system will be block-triangular provided the n 2 n 1 matrix L satisfies (1.25). 1.2.1 Recursive Algorithm for Solving L-Equation We use the fixed-point iteration to find the solution of (1.25), (Gajić and Shen, 1989) Setting = 0, (1.25) becomes The unique solution of (1.25) is The sought value of matrix L is approximated by A 21 A 22 L o = 0 (1.26) L o = A 1 22 A 21 (1.27) L = L 0 + O() (1.28) The recursive algorithm proposed by (Gajić and Shen, 1989) is L (i+1) = A 1 22 (A 21 + L (i) A 11 L (i) A 12 L (i) ) (1.29) It was shown in Gajic(1986) that algorithm (1.29) converges with the rate of convergence of O(), that is, after i-iterations, we have L (i) L 0 = 0 i (1.30)

6 1.3 Chang Transformation : Block Diagonal Form One additional trasnformation matrix T 2 leads to complete seperation of slow and fast states of the singularly perturbed system (1.21). Consider general linear system (1.19). From the upper triangular form, the state x(t) changes into the state ˆx(t). ˆx(t) = T 1 2 x(t) (1.31) with T 1 2 = I n H (1.32) 0 I m where x(t) are the states with the upper block-triangular form of Ā and ˆx(t) are the states corresponding to the block-diagonal form Â. The above state transformation changes (1.19) into x(t) = T 2 T2 1 ĀT 2 T2 1 x(t) + Bu(t) (1.33) The new coordinates of the state ˆx(t) can be obtained by multiplying (1.33) by T 2 from the left side. ˆx(t) = ˆx(t) + ˆBu(t) (1.34) with  = T2 1 ĀT 2, ˆB = T2 1 B (1.35) In the case of the singularly perturbed system, ˆx(t) contains fast and slow states given by ˆx(t) = x s(t) (1.36) x f (t) Consider the singularly perturbed system (1.21). After the first step of the Chang transformation, (1.21) becomes ẋ = Ā x(t) + Bu(t), x(t 0 ) = x 0 (1.37) with Ā = A 11 A 12 L A 12 0 LA 12 + 1 A 22 B = T 1 1 B (1.38)

7 In the same way, the block triangular form, obtained using (1.31), produces  = T2 1 ĀT 2, as  = I n H A 11 A 12 L A 12 I n H 0 I m 0 1 A 22 + LA 12 0 I m = A 11 A 12 L (A 11 A 12 L)H + A 12 H(A 22 / + LA 12 ) 0 1 A 22 + LA 12 (1.39) To make (1.39) block-diagonal, the element (A 11 A 12 L)H + A 12 H(A 22 / + LA 12 ) should be zero. The H equation is given by 0 = (A 11 A 12 L)H + A 12 H(A 22 + LA 12 ) (1.40) Equation (1.40) is the Sylvester algebraic equation. Its unique solution always exists since matrices A 22 + LA 12 and (A 11 A 12 L) have no eigenvalues in common (Chen, 1999) 1.3.1 Recursive Algorithm for Solving the H-Equation The iterative method is utilized to obtain the solution for H. Set = 0 in (1.40), to obtain H o as H o = A 12 A 1 22 (1.41) The solution value H has an approximation H o and the error of O(), that is H = H 0 + O() (1.42) In the same process, applying the fixed-point recursive algorithm for L, we have (A 12 A 22 L)H (i) H (i+1) A 22 H (i) LA 12 + A 12 = 0 (1.43) Equation (1.43) can be rearranged for H (i+1) as H (i+1) = (A 12 A 22 L)H (i) A 1 22 H(i) LA 12 A 1 22 + A 12A 1 22 (1.44) It is no difficult to show that the convergence rate of (1.44) is O().

8 1.4 Chang Transformation : Composite Method This section presents a combination of two previous sections and fully defined the Chang transformation. The composite state transformation matrices T c and T 1 c represented by and T 1 c T c = T 1 T 2 = = T 1 2 T 1 1 = I n 0 L I m I n H = I n 0 I m L I n H I n 0 = I n HL 0 I m L I m L H LH + I m are (1.45) H (1.46) The matrix T c is said to be the Chang transformation. Consider a general continuoustime linear system (1.15) where x(t) R n, u(t) R m, and A, B are constant matrices of appropriate dimensions. There exists a state transformation defined by where T 1 c (1.15) into I m x(t) = T 1 c x(t) (1.47) is given by (1.46) which transforms a general continuous-time linear system ẋ(t) = T c Tc 1 AT c Tc 1 x(t) + Bu(t) (1.48) By multiplying T 1 c on both side of (1.48), (1.48) becomes x(t) = Ā x(t) + Bu(t) (1.49) with x(t) = T 1 c x(t), Ā = T 1 c AT c, B = T 1 c B (1.50) The fast and slow states can be decomposed using x s(t) = I n HL H x 1(t) (1.51) x f (t) L x 2 (t) Where x s (t) and x f (t) are states in the slow and fast coordinates and x 1 (t) and x 2 (t) are states in the original coordinates. The original state can be recostructed by x 1(t) = I n H x s(t) (1.52) x 2 (t) L LH + I m x f (t) I m

9 Note that the complete slow and fast decomposition holds when we only consider unforced response. If we consider the input vector, we need an alternative way to exactly decompose the slow and fast states. This fact will be considered later. 1.5 Introduction to Observers and Observer Based Controllers Sometimes all state space variables are not available for measurements, or it is not practical to measure all of them, or it is too expensive to measure all state space variables. In order to be able to apply the state feedback control to a system, all of its state space variables must be available at all times. Thus, we face the problem of estimating system state space variables. 1.5.1 Full-Order Observer Design Consider a linear time invariant system given as ẋ(t) = Ax(t) + Bu c (t), x t0 = x 0 = unknown y(t) = Cx(t) (1.53) where x(t) R n, u(t) R r, y(t) R p with constant matrices A, B, C having appropriate dimensions. We may construct a full-order observer having the same matrices A, B, C such that ˆx(t) = Aˆx(t) + Bu c (t), ˆx t0 = ˆx 0 ŷ(t) = C ˆx(t) (1.54) Then we compare the output y(t) of the system (1.53) and the output ŷ(t) of the full-order observer (1.54). These two outputs will be different since in the first case the system initial condition is unknown, and in the second case it has been chosen arbitrarily. The difference between these two outputs will generate an error signal y(t) ŷ(t) = Cx(t) C ˆx(t) = Ce(t) (1.55) which can be used as the feedback signal to the full-order observer such that the estimation error e(t) is reduced to zero. Considering the feedback signal (1.55), the observer

10 structure is given by ˆx(t) = Aˆx(t) + Bu c (t) + K(y(t) ŷ(t)) (1.56) Note that the observer has the same structure as the system plus the driving feedback term that contain information about the observation error. The observer is implemented on line as a dynamic system driven by the same input as the original system and the measurements coming from the original systems, that is ˆx(t) = (A KC)ˆx(t) + Bu c (t) + Ky(t) (1.57) with y(t) = Cx(t), u c (t) = F ˆx(t) (1.58) This can be realized by proposing the system-observer structure as given in Figure. 1.1. Figure 1.1: Full-order observer-based controller It is easy to derive an expression for dynamics of the observation error as ė(t) = ẋ(t) ˆx(t) = (A KC)e(t) (1.59) If the observer gain K is chosen such that the matrix A KC is asymptotically stable, then the error e(t) can be reduced to zero at steady state. At this point, we need the following assumption. Assumption 1.5.1. The pair (A, C) is observable

11 In practice, the observer eigenvalues should be chosen to be about 5 6 times faster than the system eigenvalues so that the minimal real part of observer eigenvalues to be 5 6 times bigger than the maximal real part of system eigenvalues, that is R(λ min ) observer > (5 or 6) R(λ max ) system (1.60) 1.5.2 Seperation Principle This section presents the fact that the observer-based controller preserves the closedloop system eigenvalues. The system under state feedback control, that is u(t) = F x(t) has the closed-loop form as ẋ(t) = (A BF )x(t) (1.61) so that the eigenvalues of the matrix A BF are the closed-loop system eigenvalues under state feedback. In the case of the observer-based controller, as given in Figure 1.1, the control input signal applied to the observer-based controller is given as u c (t) = F ˆx(t) = F x(t) + F e(t) (1.62) Substituting equation (1.62) in the full-order observer (1.57) and the system (1.61), we obtain the following augumented closed-loop matrix form ẋ(t) = A BF x(t) (1.63) ˆx(t) KC A KC BF ˆx(t) At this point, we introduce the state transformation matrix given by x(t) = I 0 x(t) = T aug x(t) (1.64) e(t) I I ˆx(t) ˆx(t) Since matrix T aug is nonsingular, we can apply the similarity transformation to the closed-loop matrix form (1.63), which leads to ẋ(t) = A BF BF x(t) (1.65) ė(t) 0 A KC e(t) It is well known that the similarity transformation preserves the same eigenvalues as in the original system. The state matrix of the system (1.65) is upper block triangular and its eigenvalues are equal to the eigenvalues λ(a BF ) λ(a KC), which indicates that the independent placement of observer and controller eigenvalues is possible.

12 1.6 An Observer for Singularly Perturbed Systems The singularly perturbed system (1.6) may be rewritten as with ẋ(t) = Ãx(t) + Bu(t), x(t 0 ) = x 0 y(t) = Cx(t) Ã = A 11 A 12, B = B 1 A 21 x(t) = x 1 x 2 A 22 B 2 (1.66) (1.67) The corresponding full-order observer for the singularly perturbed system (1.66) is given as ˆx(t) = (Ã KC)ˆx(t) + Bu(t) + Ky(t) ŷ(t) = C ˆx(t) (1.68) where ˆx(t) is an estimate of the state x(t) in (1.66) and the state error is defined as e(t) = ˆx(t) x(t) (1.69) The role of the observer (1.68) reconstruct the state x(t) of (1.66) in a uniformly asymptotic manner in the sense that lim e(t) = 0 (1.70) t The observability Assumption 1.5.1 is needed for (1.70) to hold 1.6.1 An Observer for the Slow Subsystem An observer for the reduced system (1.11) is given as (O Reilly, 1979a) ˆx s (t) = (A 0 K 0 C 0 )ˆx s (t) + K 0 y s (t) + B 0 u s (t) (1.71) where the state reconstruction error is given as e s (t) = ˆx s (t) x s (t) (1.72)

13 The slow error dynamics can be represented by ė s (t) = (A 0 K 0 C 0 )e s (t), e s (t 0 ) = e 1 (t 0 ) (1.73) The observer (1.71) will uniformly aymptotically reconstruct the state x s (t), that is If the following assumption holds lim e s(t) = 0 (1.74) t Assumption 1.6.1. The pair (A 0, C 0 ) is observable. 1.6.2 An Observer for the Fast Subsystem A full-order observer for the fast system (1.13) is given as (O Reilly, 1979a) ˆx f (τ) = ( 1 A 22 1 K 2C 2 )ˆx f (τ) + 1 K 2y f (τ) + 1 B 2u f (τ) (1.75) where the state reconstruction error is defined as e f (τ) = ˆx f (τ) x f (τ) (1.76) Similarly, the fast error dynamics can be represented by ė f (τ) = ( 1 A 22 1 K 2C 2 )e f (τ), e f (t 0 ) = ˆx f (0) x f (0) (1.77) The observer (1.75) will uniformly aymptotically reconstruct the state x f (τ) if lim e f (τ) = 0 (1.78) τ The fast subsystem observability assumption is needed for (1.78) to hold Assumption 1.6.2. The pair (A 22, C 2 ) is observable. 1.6.3 State Reconstruction for the Composite System This section presents a composite observer design based on the two slow and fast observers (1.71) and (1.75) (O Reilly, 1979a).

14 Lemma 1.6.1. If the observer (1.68) is coupled to the system (1.66) with where K = K 1 (1.79) 1 K 2 K 1 = 1 2 A 12A 1 22 K 2 + K 0 [I 1 2 C 2A 1 22 K 2] (1.80) and if A 0 + K 0 C 0 and A 22 + K 2 C 2 are uniformly asymptotically stable, then the eigenvalues related to the error dynamics in the original coordinates satisfy λ i = λ i (A 0 + K 0 C 0 ) + O(), i = 1,..., n 1 λ j = λ j ( 1 A 22 + 1 K 2C 2 ) + O() (1.81), i = n 1 + j, j = 1,..., n 2 1.7 Observer-based Controllers for Singularly Perturbed Systems A dynamical feedback controllers for the singularly perturbed system (1.66) is given by u c (t) = F ˆx(t) (1.82) where ˆx(t) is an estimate of the state x(t) and is generated by the full-order observer (1.68). The overall closed-loop system for the original system (1.66) is given by ẋ(t) = Ã + BF BF x(t) (1.83) ė(t) 0 Ã KC e(t) It is required that the controller (1.83) be uniformly asymptotically stable in the sense that lim x(t) = 0 (1.84) e(t) t Obviously, this may be achieved if and only if (1.66) is stabilizable by feedback (1.82) and the observer reconstruction error system, that is ė(t) = (Ã KC)e(t), e(t 0) = ˆx(t 0 ) x(t 0 ) (1.85) The observability Assumption 1.5.1 and the following Assumption is needed for (1.84) to hold Assumption 1.7.1. The pair (Ã, B) is controllable.

15 1.7.1 A Controller for the Slow Subsystem An observer-based controller for the slow subsystem (1.11) is given by (O Reilly, 1980) u s (t) = F 0ˆx s (t) (1.86) where ˆx s (t) is an estimate of the original state x s (t) generated by the slow reducedorder observer (1.71). The slow state reconstruction error e s (t) defined by (1.72) satisfies (1.73). Hence, the closed-loop slow subsystem is given by ẋs(t) = A 0 + B 0 F 0 B 0 F 0 x s(t) (1.87) ė s (t) 0 A 0 K 0 C 0 e s (t) The slow subsystem (1.11) is uniformly completely stabilizable by the controller (1.86) and the slow reduced-order observer (1.71) if lim x s(t) = 0 (1.88) e s (t) t The observability Assumption 1.6.1 and the following Assumption is needed for (1.88) to hold Assumption 1.7.2. The pair (A 0, B 0 ) is controllable. 1.7.2 A Controller for the Fast Subsystem An observer-based controller for the fast subsystem (1.13) is given in the fast time-scale τ by u f (τ) = F 2ˆx f (τ) (1.89) where ˆx f (τ) is an estimate of the original state x f (τ) generated by the fast reduced-order observer (1.75). The slow state reconstruction error e f (τ) defined by (1.76) satisfies (1.77). Hence, the closed-loop fast subsystem is given by (τ) ẋf = A 22 + B 2 F 2 B 2 F 2 x f (τ) (1.90) ė f (τ) 0 A 22 K 2 C 2 e f (τ)

16 Furthermore, the fast subsystem (1.13) is uniformly completely stabilizable by the controller (1.89) and the fast reduced-order observer (1.75) if lim τ x f (τ) = 0 (1.91) e f (τ) The observability Assumption 1.6.2 and the following Assumption is needed for (1.91) to hold Assumption 1.7.3. The pair (A 22, B 2 ) is controllable. 1.7.3 A Composite Observer-based Controller At this point, we need to introduce the observer driven controller proposed by (O Reilly, 1980). Lemma 1.7.1. If the observer and controller are coupled to the system (1.66) with ] F = [F 1 F 2 (1.92) F 1 = [I + K 2 A 1 22 B 2]F 0 + F 2 A 1 22 A 21 (1.93) K = K 1 (1.94) 1 K 2 K 1 = 1 2 A 12A 1 22 K 2 + K 0 [I 1 2 C 2A 1 22 K 2] (1.95) and if the slow subsystem and fast subsystem are each uniformly stabilizable by two observers (1.71) and (1.75) and controllers (1.86) and (1.89), then there exists a positive sufficiently small such that the original system (1.66) is uniformly completely stabilizable for any (0, ]. This lemma indicates that the state and error dynamic can be reconstructed within O() approximation. There are several papers for observers and observer driven controllers for singularly perturbed systems and all of them did design with O() accuracy (O Reilly, 1980), (O Reilly, 1979a),(O Reilly, 1979b), (Gardner JrandCruz Jr, 1980), and (P orter, 1974).

17 1.8 Controller Eigenvalues Assignment The goal is to find a similarity transformation P and the feedback gain F such that P 1 (A BF )P = Λ desired (1.96) with λ(a BF ) = λ(λ desired ) = λ desired (1.97) If (A, B) is controllable, λ(a BF ) can be arbitrarily located (Chen, 1999). Premultiply (1.96) by P produces AP BF P = P Λ desired (1.98) Rearranging (1.98), it becomes AP P Λ desired = B F (1.99) with F = F P (1.100) The following lemma was presented in (Chen, 1999, pp. 240). Lemma 1.8.1. If A and Λ desired have no eigenvalues in common, then the unique solution P of AP P Λ desired = B F exists if and only if (A, B) is controllable and (Λ desired, F ) is observable. 1.8.1 Design Procedure The procedure for computing the feedback gain through the Lyapunov method is presented in (Chen, 1999) Assume a controllable pair (A, B), where A is R n n and B is R n m. Find a R m n real matrix F such that (A BF ) has a set of desired eigenvalues that contains no eigenvalues of A. Step 1. Select an R n n matrix Λ desired that has the desired set of eigenvalues. The form of Λ desired can be chosen arbitrarily, often it is a diagonal matrix Step 2. Select an arbitrary R m n vector F such that (Λ desired, F ) is observable Step 3. Solve the Sylvester(Lyapunov) equation AP P Λ desired = B F for the unique

18 P. Step 4. Compute the feedback gain F = F P 1 if the matrix P is invertible. If P is not invertible, go back to Step 2 and choose another F. 1.9 Observer Eigenvalues Assignment The corresponding Lyapunov method for obtaining the observer gain is to find the observer gain in the original coordinates. To find the observer gain, we need to transpose matrix (A KC). Consider the similarity transformation P 1 (A T C T K T )P = Λ obs desired (1.101) where λ(a T C T K T ) = λ(λ obs desired ) = λ desired (1.102) If (A, C) is observable, λ(a KC) can be arbitrarily located according to (Chen, 1999). It is well known that the closed-loop eigenvalues of the observer should be located 5 6 times farther to the left from the closed-loop system eigenvalues. Multiplying both side of (1.101) by P, (1.101) becomes the following Lyapunov equation A T P P Λ obs desired = CT KT (1.103) with K T = K T P (1.104) 1.9.1 Design Procedure For this section we introduce the procedure to compute the observer gain through the Lyapunov method. The following design procedure is presented in (Chen, 1999).Consider the observable pair (A, C), where A is R n n and C is R p n. Find a R n p real K such that (A KC) has any set of desired eigenvalues that contains no eigenvalues of A. Step 1. Select an arbitrary matrix Λ obs desired that has no common eigenvalues with those

19 of A. Step 2. Select an arbitrary R p n vector K T such that (Λ obs desired, K T ) is observable. Step 3. Solve for the unique P from the Sylvester equation A T P P Λ obs desired = CT KT. Step 4. Obtain the transposed observer gain from K T = K T P 1. If P is not invertible, go back to Step 2 and choose another K T. Step 5. Obtain the observer gain from K = (K T ) T. 1.10 Conclusion In this thesis, we will show that it is possible to design reduced-order slow and fast controllers exactly and independently in slow and fast time scales.

20 Chapter 2 Two-Time Scale Design for Singularly Perturbed Systems 2.1 Introduction In this section, we will consider how one can place eigenvalues at the desired locations for each subsystems. Since a singular perturbed system has two sets of seperated eigenvalues, some of which are large, it might not be possible to place some of eigenvalues at the desired location in the case of the inaccessible states. 2.2 State Feedback Control via the Chang Transformation This section presents state feedback control via the Chang transformation and shows how to obtain the feedback gain. Firstly, we show that the Chang transformation can t seperate slow and fast states when we consider the input signals. Consider the singular perturbed system We define matrices A and B as ẋ 1 (t) = A 11 x 1 (t) + A 12 x 2 (t) + B 1 u(t), x 1 (t 0 ) = x 10 ẋ 2 (t) = A 21 x 1 (t) + A 22 x 2 (t) + B 2 u(t), x 2 (t 0 ) = x 20 (2.1) A = A 11 A 12, B = B 1 (2.2) A 21 A 22 B 2 Dividing by,the following matrix form is obtained ẋ1(t) = A 11 A 12 x 1(t) + B 1 u c (t) (2.3) ẋ 2 (t) x 2 (t) A 21 A 22 Similarly, we define matrices A new and B new as A new = A 11 A 12, B new = B 1 (2.4) A 21 A 22 B 2 B 2

21 The above general singularly perturbed system (2.3) can be transformed into the following form via the Chang transformation. ẋs(t) = A s 0 x s(t) + B s u c (t) (2.5) ẋ f (t) 0 x f (t) System (2.5) can be divided into two subsystems represented by A f ẋ s (t) = A s x s (t) + B s u c (t) ẋ f (t) = A f x f (t) + B (2.6) f u c(t) It looks like that the fast states are seperated from the slow states. However, they are coupled through the control signal as substitute (2.7) into (2.6) B f u c (t) = F s x s (t) F f x f (t) (2.7) ẋ s (t) = (A s B s F s )x s (t) + B s F f x f (t) ẋ f (t) = 1 (A f B f F f )x f (t) + 1 B f F s x s (t) (2.8) From above (2.8), the pure seperation of fast and slow states are impossible, since the first part of equation in (2.8) contains fast states and the second part of equation in (2.8) contains slow states. Therefore, the different method to seperate fast and slow states are proposed (Kokotovic et al., 1999). Consider the composite state feedback form of a singularly perturbed system, with ẋ(t) = (A new + B new F )x(t), x(t 0 ) = x 1(t 0 ) (2.9) x 2 (t 0 ) A new = A 11 A 12, B new = B 1(t), A 21 A 22 B 2 (t) F = [F 1 F 2 ], x = x 1(t) x 2 (t) (2.10) The first step of the Chang transformation produces ẋs(t) = A 11f A 12f L A 12f x s(t) (2.11) ẋ f (t) 0 LA 12f + 1 A 22f x f (t)

22 with A 11f = A 11 +B 1 F 1, A 12f = A 12 +B 1 F 2, A 21f = A 21 +B 2 F 1, A 22f = A 22 +B 2 F 2 (2.12) and LA 11f LA 12f L + A 21f A 22f L = 0 (2.13) Note that feedback matrices from (2.12) are present in equation (2.13). Applying the second step of the Chang transformation we obtain with ẋs(t) = (A 11f ) (A 12f )L 0 x s(t) (2.14) ẋ f (t) 0 L(A 12f ) + A 22f x f (t) (A 11f LA 12f )H + H(LA 12f A 22f ) + A 12f = 0 (2.15) Where A 11f, A 12f and A 22f are feedback matrices. If L equation (2.13) and H equation (2.15) are satisfied, it is possible to decouple the slow and fast subsystems represented by equation (2.14) 2.2.1 Slow and Fast Time Scales : Uncorrected Method The seperation of slow and fast sub-systems is discussed in the previous section. In this section, the method to obtain feedback gains for slow and fast subsystems will be considered. The composite input u c (t) is given by u c (t) = u s (t) + u f (t) = F 0 x s (t) + F 2 x f (t) (2.16) However, a realizable composite control requires that the system states x s (t) and x f (t) be expressed in terms of the actual system states x 1 (t) and x 2 (t) is proposed by Kokotovic, (Kokotovic et al., 1999) u c (t) = F 0 x 1 (t) + F 2 [x 2 (t) + A 1 22 (A 21x s (t) + B 2 F 0 x 1 (t))] = F 1 x 1 (t) + F 2 x 2 (t) (2.17) with F 1 = (I r + F 2 A 1 22 B 2)F 0 + F 2 A 1 22 A 21 (2.18)

23 In the above design procedure the gain matrices F 0 and F 2 are seperately designed in the slow and fast time scales. The following theorem is presented by Kokotovic (Kokotovic et al., 1999) Lemma 2.2.1. Let F 2 be designed such that Reλ(A 22 + B 2 F 2 ) < 0, then an > 0 such that the composite control u c (t) = F 0 x s (t) + F 2 [x 2 (t) + A 1 22 (A 21x s (t) + B 2 F 0 x s (t))] (2.19) applied to the system (2.3), produces the closed-loop system, ( starting from any bounded initial condition x 10 and x 20 ), with the following property x 1 (t) = x s (t) + O() (2.20) x 2 (t) = A 1 22 (A 21 + B 2 F 0 )x s (t) + x f (t) + O() (2.21) u c (t) = u s (t) + u f (t) + O() (2.22) for all finite time t t 0 and all (0, ]. If in addition F 0 is designed such that Reλ(A 0 + B 0 F 0 ) < 0, there exists an > 0 such that the resulting closed-loop system is asymptotically stable and (2.20) - (2.22) hold for all (0, and t [t 0, ). Proof. (Kokotovic et al., 1999) Consider a seperation of fast and slow states through the Chang transformation applied to equation (2.14), which produces ẋs = (A 11f ) (A 12f )L 0 x s (2.23) ẋ f 0 A 22f + L(A 12f ) x f with the initial conditions x 0 s = x 0 1 Hx 0 2 x 0 f = x0 2 + Lx 0 1 (2.24) Where A 11f, A 12f, A 21f and A 22f are feedback matrices of (2.12). Instead of using F 1 and F 2 as design parameters, we use F s and F 2 as design parameters given by F 1 = F s + F 2 L (2.25) Substituting equations (2.12) and (2.25) into L equation (2.13) results in the following equation L = A 1 22 (A 21 + B 2 F s ) + A 1 22 L[(A 11 + B 1 F s ) A 12 L] (2.26)

24 and substituting (2.26) into (2.23), using (2.25) and (2.12), leads to the closed-loop system ẋs = A s + B s F s 0 x s (2.27) ẋ f 0 A f + B f F f x f with A s = A 0 A 12 A 1 22 L(A 11 A 12 L) B s = B 0 A 12 A 1 22 LB 1 A f = A 22 + A 12, B f = B 2 + LB 1 (2.28) Since matrices A 0, B 0, A 22 and B 2 are approximation of matrices A s, B s, A f and B f, we can say that the eigenvalues of matrices λ(a s + B s F s ) and λ(a f + B f F f ) are approximated by eigenvalues of λ(a 0 + B 0 F 0 ) and λ(a 22 + B 2 F 2 ). Eigenvalue Assignment in Slow and Fast Time Scales : Uncorrected Method Through the application of the Chang transformation, an n-dimensional eigenvalue placement can be reduced to seperate eigenvalue placement problems of dimension n 1 and n 2. The following theorem is proposed by Anderson (Anderson, 1982) Lemma 2.2.2. If a (A, B) is controllable, there exists at least one real m n dimensional feedback matrix F such that the closed-loop eigenvalues given by λ(a BF ) can be placed arbitrary by designer s decision. If the original system is controllable, it can be shown by linear algebra that the slow and fast subsystems in equation (2.27) are also controllable. The above lemma is considered in the original coordinates. However, one can ask a question about whether a designer can place slow and fast eigenvalues seperately. The singularly perturbed system has different sets of eigenvalues Q and R. It can be shown that it is possible to relocate the n 1 slow open-loop eigenvalues Q to n 1 new eigenvalue location Q. The following theorem is presented by kokotovic (Kokotovic et al., 1999) Lemma 2.2.3. If A 1 22 exits and if the slow subsystem pair (A 0, B 0 ) and the fast system pair (A 22, B 2 ) are both controllable, and feedback gains F 0, F 2 are designed to assign distinct eigenvalues λ i, i = 1,..., n 1 and λ j, j = 1,..., n 2, to the matrices A 0 + B 0 F 0 and

25 A 22 + B 2 F 2 respectively, then an > 0 such that for all (0, ] the application of the composite feedback control (uncorrected method) u c (t) given by u c (t) = [(I r + F 2 A 1 22 B 2)F 0 + F 2 A 1 22 A 21]x 1 (t) + F 2 x 2 (t) (2.29) results in a closed-loop system containing n 1 slow eigenvalues (λ 1, λ 2,..., λ n1 ) and n 2 fast eigenvalues (λ n1 +1, λ n1 +2,..., λ n1 +n 2 ), which are approximated by λ i = λ i (A 0 + B 0 F 0 ) + O(), i = 1,..., n 1 λ j = [λ (2.30) j(a 22 + B 2 F 2 ) + O()], i = n 1 + j, j = 1,..., n 2 2.2.2 Slow and Fast Time Scales : Corrected Method This method is different from the aforementioned uncorrected method since the state trajectories of the singularly perturbed system (2.1) are approximated by O( 2 ). The following Lemma is proposed by Kokotovic (Kokotovic et al., 1999, pp. 99-101). Lemma 2.2.4. Using A s, B s in (2.28), the corrected slow model is ẋ sc (t) = A 0c x sc (t) + B 0c u sc (t) A 0c = A 0 A 12 A 1 22 L 0(A 11 A 12 L 0 ) (2.31) with the initial condition B 0c = B 0 A 12 A 1 22 L 0B 1 x sc (t 0 ) = x 0 1 H 0 (x 0 2 + L 0 x 0 1) (2.32) Similarly, the corrected fast model in the τ = t t 0 scale is given by ẋ fc (τ) = A 22c x fc (t) + B 2c u fc (t) A 22c = A 22 + L 0 A 12 (2.33) B 2c = B 2 + L 0 B 1 with the initial condition x 2fc (t 0 ) = x 0 2 + L 1 x 0 1 (2.34) If the composite control u c (t) = F 1 x 1 (t) + F 2 x 2 (t) (2.35)

26 with F 1 = F 0c + F 2 L 1 (2.36) is applied to the system (2.3), and if A 22 + B 2 F 2 is a Hurwitz matrix, then there exists an such that the state and control of the closed-loop system, starting from any bounded initial conditions x 0 1 and x0 2, are approximated based by x 1 (t) = x sc (t) + H 0 x fc (τ) + O( 2 ) (2.37) x 2 (t) = L 1 x sc (t) + (I m L 0 H 0 )x fc (τ) + O( 2 ) (2.38) u c (t) = u sc (t) + u fc (τ) + O( 2 ) (2.39) with L 0 = A 1 22 (A 21 + B 2 F 0 ), L 1 = A 1 22 (A 21 + B 2 F 0c ) + A 1 22 L 0(A 0 + B 0 F 0 ) (2.40) H 0 = (A 12 + B 1 F 2 )(A 22 + B 2 F 2 ) 1 If A 0 + B 0 F 0 is also Hurwitz then there exists an such that the closed-loop system is asymptotically stable. The above Lemma shows that state trajectories is approximated by O( 2 ). The aforementioned uncorrected method has states trajectories (2.20),(2.21) and (2.22) with O() which indicates corrected method is a better method to approximate states and input. Eigenvalues Assignment in Two-time Scale via the Corrected Method Now that we consider the eigenvalue assignment via the corrected method, firstly, we present the following lemma (Kokotovic et al., 1999) Lemma 2.2.5. If A 1 22 exists and if the slow subsystem pair (A 0, B 0 ) and the fast subsystem pair (A 22, B 2 ) are each controllable then there exists an > 0 and gain matrices F 0c and F 2 which arbitrarily assign eigenvalues λ i, i = 1,..., n 1 and λ j, j = 1,..., n 2 to the closed-loop matrices A 0c + B 0c F 0c and A 22c + B 2c F 2, respectively, such that for all (0, ] the application of the composite feedback control u c (t) = [F 0c + F 2 L 1 ]x 1 (t) + F 2 x 2 (t) (2.41)

27 with L 1 = A 1 22 (A 21 + B 2 F 0c ) + A 1 22 (A 21 + B 2 F 0 )(A 0 + B 0 F 0 ) (2.42) to the singularly perturbed system (2.3) results in the closed-loop system containing n 1 slow eigenvalues (λ c 1, λc 2,..., λ n c 1 ) and n 2 fast eigenvalues (λ c n 1 +1, λc n 1 +2,..., λc n 1 +n 2 ) that are approximated by λ c i = λ i (A 0c + B 0c F 0c ) + O( 2 ), i = 1,..., n 1 λ c j = [λ j(a 22c + B 2c F 2c ) + O( 2 (2.43) )], i = n 1 + j, j = 1,..., n 2 The fast eigenvalues can be approximated by O(), since O( 2 ) approximation of the fast eigenvalues are devided by. Numerical Example : Uncorrected Method This section illustrates the uncorrected method, with the accuracies O() and corrected method O( 2 ) specifically. The matrices of the singularly perturbed system are given by A = A 11 A 12, B = B ] 1, C = [C 1 C 2 A 21 A 22 B 2 (2.44) For the singularly perturbed system (Kokotovic et al., 1999, pp. 124-125), the matrices A,B and C are given by 0 0.4 0 0 0 0 0 0.345 0 0 A =, B = 0 1 0.465 1 0.262 0 0 0 0 1 1 C = 1 0 0 0 0 0 1 0 The partitions matrices of A are represented by A 11 = 0 0.4, A 12 = 0 0, 0 0 0.345 0 A 21 = 0 0.524 0.465 0.262, A 22 = 0 0 0 1 (2.45) (2.46)

28 The open-loop system has four eigenvalues at (0, 0.4282, 4.2218, 10). A 0 and B 0 are defined in (1.12) and given by A 0 = 0 0.4 0 0.3888 B 0 = 0 0.1944 Hence, the uncorrected slow gain F 0 is given by F 0 = [ ] 12.8571 5.2741 (2.47) (2.48) The choice of F 0 places the slow eigenvalues of A 0 +B 0 F 0 at 0.707 ± j0.707. However, the eigenvalues of the actual full system under this feedback control are approximated by O( 1 10 ) in the case of = 0.1. We also place the eigenvalues of the fast modes at ( 7, 8). The fast gain matrix F 2 is given by F 2 = [ ] 0.3005 0.0350 (2.49) The ill-conditioning of computation can be eliminated by this two-time scale method. Once we find the slow gain and the fast gain in the new coordinates, then the composite gain matrix F in the original coordinates is given by ] F = [F 1 F 2 (2.50) with F 1 = [ ] [ ] 12.8571 5.2741, F 2 = 0.3005 0.0350 (2.51) The poles of (A + BF ) are given by 0.6412 + 0.8832i 0.6412 0.8832i λ(a + BF ) = 4.0273 9.6904 (2.52) of which the slow eigenvalues are observed to be within O() of the prescribed eigenvalues ( 0.707 ± j0.707) and the fast eigenvalues are also within O(1) of the prescribed eigenvalues ( 7, 8).

29 Numerical Example : Corrected Method Now that we consider the corrected method from the uncorrected method. As previously mentioned, this method assigns eigenvalues with O( 2 ) approximation. Hence, the corrected method improves the uncorrected method regarding the eigenvalues assignment. From (2.31), A 0c and B 0c are given by A 0c = 0 0.4 1.0162 0.6488 [ ] B 0c = 0 0.1944 From (2.33), A 22c and B 2c are given by 0.3236 A 22c = 0.2620 0.1820 1.0000 [ ] B 2c = 0 1 The slow gain F 0c and the fast gain F 2c are given by F 0c = [ ] 7.6293 3.9367 (2.53) (2.54) (2.55) and F 2c = [ ] 0.8664 0.1764 (2.56) Once we find the slow gain and the fast gain in the slow and fast coordinates through the corrected method, then the gain matrices (2.35)-(2.36) in the original coordinates are given by with F 1c = ] F = [F 1c F 2c [ ] [ ] 15.4838 8.2228, F 2c = 0.8664 0.1764 (2.57) (2.58) The eigenvalues of (A + BF ) are 0.7100 + 0.6922i 0.7100 0.6922i λ(a + BF ) = 7.4970 + 0.8539i 7.4970 0.8539i (2.59)

30 The slow eigenvalues are observed to be within O( 2 ) of the prescribed eigenvalues ( 0.707 ± j0.707) and the fast eigenvalues are within O() of the prescribed fast eigenvalues ( 7, 8). 2.3 Two-Stage Design The above uncorrected method and the corrected method shows O() and O(1) approximation for the fast eigenvalues, and O( 2 ) and O() approximation for the slow eigenvalues. This section provides the exact assignment of both the slow and fast eigenvalues via state feedback. If the original system is given by ẋ(t) = Ax(t) + Bu c (t) (2.60) with A = A 11 A 12, B = B 1, A 21 x(t) = x 1 x 2 A 22 B 2 (2.61) Then (2.60) becomes the block-diagonal form using the Chang transformation (1.45), that is x s x f = T 1 c x 1 (2.62) x 2 Applying the Chang transformation (2.62) to the original singularly perturbed system (2.60), we have x(t) = à x(t) + Bu c (t) (2.63) with à = T 1 c AT c = x(t) = x s x f A s 0 0 A f, B = T 1 c B = B s B f (2.64)

31 System (2.63) can be expressed as ẋ s = A s x s + B s u(t) ẋ f = A f x f + B f u(t) (2.65) It looks like that the fast states are seperated from the slow state by only considering matrix à in (2.63). Consider the composite input is given by Substituting (2.66) to (2.65), so that (2.65) becomes u c (t) = F s x s (t) F f x f (t) (2.66) ẋ s (t) = (A s B s F s )x s (t) + B s F f x f (t) ẋ f (t) = 1 (A f B f F f )x f (t) + 1 B f F s x s (t) (2.67) It follows from (2.67) that the seperation of fast and slow states is not obtained, since the slow and fast states are mixed again. Hence, the different method is needed to seperate exactly the fast and slow states via feedback control. Suppose, we desire to relocate only n 1 slow open-loop eigenvalues to n 1 new closed eigenvalues locations. Firstly, we only consider the driving term for the slow states represented by u c (t) = v(t) F s x s (t) (2.68) Substituting the driving terms from slow states (2.68) in (2.65), the following equations are obtained ẋ s (t) = (A s B s F s )x s (t) ẋ f (t) = A f x f (t) + B (2.69) f F sx s (t) This shows that the slow subsystem has no terms from the fast states. However, the fast subsystem has the terms with the slow state denoted by B f F sx s. Therefore, the second transformation T 2 needed to remove the remaining slow terms in ẋ f (t) is presented in the following lemma. Lemma 2.3.1. (Anderson, 1982) x fnew (t) = P x s (t) + x f (t) (2.70) and P (A s B s F s ) B f F s A f P = 0 (2.71)

32 If the above algebraic Lyapunov equation (2.71) is satisfied, it is possible to completely decouple the slow and fast states. Proof. From the second transformation (2.70), we take the derivative of equation (2.70) represented by ẋ fnew (t) = P ẋ s (t) + ẋ f (t) = P (A s B s F s )x s (t) + A f (x fnew(t) P x s (t)) B f F sx s (t) = [P (A s B s F s ) B f F s A f P ]x s(t) + A f x fnew(t) Therefore, if the Lyapunov equation (2.71) is satisfied, (2.72) becomes (2.72) ẋ fnew (t) = A f x fnew(t) (2.73) This result is quite amazing, since we realize that there is a method to assign the slow eigenvalues without affecting the fast eigenvalues. To summarize the proposed exact relocation of eigenvalues, the state transformation is represented by x 1 = T c x s = T c T 2 x 2 x f x s x fnew = T tot x s x fnew (2.74) with T 2 = I n1 0 P I n2 T c = I n1 H L I n2 LH (2.75) This method is derived so as to obtain the gain F in the original coordinate from the two stage transformed coordinate. 2.3.1 Two-Stage Design This section summurizes previous method to assign exact eigenvalues assignment by proposing the following design algorithm. Stage 1.

33 Step 1. Apply the Chang transformation to the singularly perturbed system. Then, apply slow states feedback control denoted by u(t) = v(t) F s x s (t). Substituting u(t) = v(t) F s x s (t) into (2.63) produces ẋs(t) = A s B s F s 0 x s(t) + B s v(t) (2.76) ẋ f (t) x f (t) B f F s From the above equation, we can place slow eigenvalues in the desired location represented by At this point, we need the following assumption. A f λ(a s B s F s ) = λ desired s (2.77) B f Assumption 2.3.1. The pair (A s, B s ) is controllable. Under this assumption, according to (Chen, 1999), the eigenvalues of A s B s F s can be arbitrarily located into the desired locations. Step 2. Introduce another transformation T 2 x s(t) = T2 1 x s(t) = I 0 x s(t) (2.78) x fnew (t) x f (t) P I x f (t) Apply the second transformation to (2.76) to obtain a block diagonal form as ẋs(t) = A s B s F s 0 x s(t) + ẋ fnew (t) 0 x fnew (t) A f B s P B s + B f Which requires that the algebraic Lyapunov equation be satisfied v(t) (2.79) Stage2. P (A s B s F s ) B f F s A f P = 0 (2.80) Firstly, we take v(t) = F f2 x fnew (t), then obtain the gain F f using the eigenvalues placement such that λ( A f Similarly, we need the following assumption. (P B s + B f )F f2) = λ desired f (2.81)

34 Assumption 2.3.2. The pair (A f, P B s B f ) is controllable. Taking v(t) = F f2 x fnew (t), the subsystem (2.79) can be represented as x(t) = Āfcl x(t) (2.82) with Ā fcl = A s B s F s 0 A f B s F f (P B s + B f )F f2 (2.83) x(t) = x s x fnew From the block triangular form of matrix Āfcl in (2.82), the independent slow and fast eigenvalues assignment can be represented by λ(āfcl) = λ(a s B s F s ) λ[a f (P B s + B f )F f2] (2.84) This two stage method shows the independent slow and fast eigenvalues assignment. 2.3.2 Exact Eigenvalues Assignment After obtaining gain F s and F f2 via the two stage design, there is a way to go back to the original coordinates without changing eigenvalues location. We define u s (t) = F s x s (t) and v(t) = F f x fnew (t). Therefore the input in (2.68) can be represented by u c (t) = v(t) + F s x s (t) = F f x fnew (t) + F s x s (t) (2.85) Formula (2.85) can be expressed in the following matrix form. ] u c (t) = [F s F x s(t) ] f2 = [F s F f2 x fnew (t) ] = [F s F I 0 f2 T c x 1(t) P I x 2 (t) ] = [F s + F f2 P F f2 T c x 1(t) x 2 (t) T 1 2 x s(t) x f (t) (2.86)

35 where T c is the Chang transformation (1.45) and T 2 is the second transformation (2.70). Hence the gains F 1 and F 2 in the original coordinates are given by u(t) = F x 1(t) ] = [F 1 F x 1(t) 2 x 2 x 2 (t) ] = [F s + F f2 P F f2 T c x 1(t) x 2 (t) (2.87) and Q.E.D. F 1 = F s + F f2 P F 2 = F f2 (2.88) The formula (2.88) is the final step to obtain the gain F in the original coordinates after we assign the eigenvalues for each subsystems via the two stage method. 2.3.3 Numerical Example : Two Stage Design The singularly perturbed system is given by (2.45). For the design algorithm Step 1 in Stage 1, we apply the Chang transformation to (2.45). The matrices A s, B s and A f, B f are given by A s = 0 0.4000, B s = 0.0325 0 0.4282 0.2489 A f = 4.2218 2.6200, B f = 0 0 10.0000 10 (2.89) Considering slow state feedback, the matrix form (2.76) can be obtained so that we can assign the slow eigenvalues at the desired location λ[a s + B s G s ] = ( 0.707 ± 0.707 j). The slow gain F s is given by F s = [ ] 11.6731 5.4857 (2.90) For Step 2 in Stage 1, after taking the second transformation (2.78), we solve the algebraic Lyapunov equation (2.80) and matrix P is given by 9.7886 P = 3.9027 (2.91) 12.8635 5.7780

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