A Method for Determining Stabilizeability of a Class of Switched Systems

Transcription

1 Proceedings of the 7th WSEAS International Conference on Systems Theory and Scientific Computation Athens Greece August A Method for Determining Stabilizeability of a Class of Switched Systems SALLEHUDDIN MOHAMED HARIS Universiti Kebangsaan Malaysia Dept of Mech and Materials Eng Faculty of Engineering 4600 UKM Bangi MALAYSIA salleh@engukmmy MOHAMAD HANIF MD SAAD Universiti Kebangsaan Malaysia Dept of Mech and Materials Eng Faculty of Engineering 4600 UKM Bangi MALAYSIA hanif@engukmmy ERIC ROGERS University of Southampton School of Electronics and Comp Sc SO7 BJ UNITED KINGDOM etar@ecssotonacuk Abstract: In this paper the problem of finding a stabilizing feedback controller for single input single output switched systems of order two is addressed To ensure stability under arbitrary switching the existence of a common Lyapunov function (CLF) needs to be established A method for establishing the existence of CLFs given suitable feedback controllers is presented The method presented here is computationally less demanding compared to those that depend on linear matrix inequalities solvers Key Words: Hybrid systems Common Lyapunov functions Stability under arbitrary switching Brunovsky canonical form State transformation Introduction Switched systems are a class of hybrid systems that is made up of a collection of linear subsystems with rules that govern switching between these subsystems Stability of switched systems is an area of important study as switching sequences and dwell time affect system behaviours and performances Several methods for determining stability of switched systems have been recognised Among these are the existence of common or multiple Lyapunov functions modifying theorems Poincaré mappings and Lagrange-based methods In this paper our attention is focussed on finding the existence of common Lyapunov functions as this ensures stability for arbitrary switching sequences Some progress in this area have been made for example in Here we present a method for determining stabilizeability of single input single output (SISO) switching systems where each subsystem is a second order linear time invariant (LTI) system Problem Definition In this paper we consider the class of switching system denoted by ẋ = A i x + B i u () where x IR u IR A IR B IR and i =N The problem is to ensure quadratic stabilization of () Formally it is defined as follows Definition System () is said to be quadratically stabilizeable if there exists a set of state feedback control laws u i = A i + B i K i such that A i + B i K i i= N share a common quadratic Lyapunov function x T Mx The LMI Approach One method of obtaining solutions to the problem posed by Definition is by solving the following linear matrix inequalities (LMI) 6: Then MA i + A T i M + B iz i + Zi T BT i < 0 M > 0 () K i = Z i M () ensures stability of System () While this is a powerful method which has been shown to be capable of providing stable pole placement feedback control of switched systems 6 the computational burden could prove to be a limiting factor As an example consider the case of four switching models given by A = b =

2 Proceedings of the 7th WSEAS International Conference on Systems Theory and Scientific Computation Athens Greece August A = A = A 4 = b = b = b 4 = (4) It was found that feasibility of () could not be determined using Matlab s LMI Lab 4 The Proposed Method We now present our methodology which is founded on the well known fact that a single input system can be uniquely transformed to its Brunovsky controllable canonical form 7 Lemma Let (A b) be a single input linear system Then there exists a unique state transformation matrix C which converts the system into the Brunovsky canonical form CAC = where with and a a a a n a a a a n 0 0 Cb= = b Ab A b A n b A n b D n = b D i = AD i a i b; i = n n D = D D D D n C = D Lemma Given m m > 0 there exists a feedback control law u = Kx = k k x such that à = A + bk = α β (ie the resulting closed loop system) has M defining its quadratic Lyapunov function if and only if m > 0 Proof: No loss of generality arises from assuming m = Then for M>0to have the required property it is required that Mà + ÃT M<0 or equivalently Q := αm +αm + βm +αm + βm (m + βm ) < 0 Given that the system is in the Brunovsky form β is the trace of à and hence β<0is a necessary condition for closed loop stability Also m =0is not allowed and αm < 0 is necessary Suppose now that m < 0 and α > 0 Then for Q<0to hold det(q) is required to be positive It is also easy to see that the maximum value of this determinant occurs when βm = αm and that it is negative Hence m > 0 is a necessary condition for Q<0 If m > 0 and α<0 anyα<0can be chosen and then let β = αm m Then det(q) = 4α(m m ) > 0 and sufficiency is established Definition 4 A matrix m m is said to be an equivalence to the controllable canonical form if m > 0 Lemma 5 Given a nonsingular matrix c c C = c c 4 there exists an equivalence matrix M > 0 such that C T MC also has this property if and only if the quadratic equation c c 4 x +(c c + c c 4 )x + c c > 0 (5) has positive solution x>0

3 Proceedings of the 7th WSEAS International Conference on Systems Theory and Scientific Computation Athens Greece August Proof: To prove sufficiency proceed as follows Given M > 0 with m > 0 and calculating C T MC gives H(m ):= m = c c 4 m +(c c + c c 4 )m + c c m (6) and from (5) H(xx ) > 0 Hence by continuity there exists a small enough ɛ > 0 such that H(xx + ɛ) > 0 Setting m = m = x and m = x + ɛ now gives an M with the required property To prove necessity assume without loss of generality that there exists m > 0 such that both M and C T MC have the desired property ie m > 0 and H(xx ) > 0 Now if c c 4 > 0 clearly (5) has a positive solution If c c 4 < 0 then since m >m H(m m m ) H(m ) > 0 (7) and m is a positive solution of (5) For a given state transformation matrix C the solution of (5) consists of one or two open intervals or it could be empty Use I to denote the set of solutions Still with the single input assumption let Λ = { N } be a finite set Then for each switched model denote the state transformation matrix which converts it to the Brunovsky canonical form by C i i = N By Lemma each transformation matrix is uniquely defined Let z i = C i x and set T j = C Cj+ j = N In this notation T j is the state transformation matrix from z j+ to z ie z = C x = C C j+ z j+ = T j z j+ Next classify T j according to t j tj 4 into the following three categories: S p = {j Λ t j tj 4 > 0} S n = {j Λ t j tj 4 < 0} S z = {j Λ t j tj 4 =0} Then Λ=S p S n S z Also for j S z define r j = t j tj + tj tj 4 s j = t j tj and for the other cases define j S z (8) p j = tj t j q j = tj t j j S p S n (9) 4 For each j S z define a linear form as L j = r j x + s j j S z and for each j S p or j S n define a quadratic form as Q j = x +(p j + q j )x + p j q j j S p S n (0) Then by Lemma 5 obtain (or solve) x from Q j (x) < 0 j S n Q j (x) > 0 j S p L j (x) > 0 j S z Note also that the polynomial Q j (x) of (0) has roots { p j q j } and hence the solution set can be defined as follows: If j S p an open set is defined as I j =( min( p j q j ) (max( p j q j ) ) If j S n an open set is defined as I j = (min( p j q j ) (max( p j q j )) If j S z and since T j is nonsingular it follows that r j 0 Hence an open set can be defined as { s ( j r I j = j ) r j > 0 ( s j r j ) r j < 0 The following result now follows immediately from Lemma 5 Theorem 6 A sufficient condition for the switched system () to be quadratically stabilizeable is I = N j= I j () If all j S p j= N then the switched system () is always stabilizeable If all i S n i= N () is also necessary Proof: To prove choose m I and m = m + ɛ Then it follows immediately that for small enough ɛ the corresponding matrix (in z co-ordinates) m m + ɛ m becomes (for suitably chosen controls) the common quadratic Lyapunov function

4 Proceedings of the 7th WSEAS International Conference on Systems Theory and Scientific Computation Athens Greece August In the case of choose m large enough and then m I In the case of the inequality (7) from the proof of Lemma 5 shows that m I and therefore I In the case of N = the above result is from Lemma 5 necessary and sufficient In general however this property does not always hold The following result gives conditions for the general case Theorem 7 Let Λ={ N } Then the system considered here is quadratically stabilizeable if and only if there exists a positive x such that min Q j (x) j S n < 0 min Q j (x) > min Q j (x) j S n L j (x) > 0 I S z () Proof: Assume that there is a quadratic Lyapunov function in z coordinates which is expressed as m M = and by Lemma m > 0 It is easy to see from the proof of Lemma 5 that M is a common quadratic Lyapunov function for the other models if and only if H j ( ) > 0 j = N or m +(p j + q j )m + p j q j > 0 j S p m +(p j + q j )m + p j q j < 0 j S n r j m + s j > 0 j S z () Also since m >m the first two equations in () can be rewritten as e + m +(p j + q j )m + p j q j > 0 j S p e + m +(p j + q j )m + p j q j < 0 j S n(4) where e>0 and the necessity of () is obvious To prove sufficiency assume that there is a solution x such that min Q j (x) > 0 Then m and m can be chosen such that m = x and m = x + ɛ with ɛ>0small enough to ensure that () holds Otherwise set w = min Q j (x) 0 Then choose m = x and m = x + ( min Q j (x) max ) Q j (x) w and it is easy to see that () holds Hence the matrix m in this case meets the requirement Return now to Theorem 6 Then in fact it has been proven that the system is stabilizeable if all j S p or if all j S n j =N then () is also necessary When N however the following is true: Corollary 8 If N then () is also necessary Proof: For the case when N = it was proved in Lemma 5 To prove this result for the case when N = first construct T and T If both j =and j =are in S p (or S n ) the result has been proven in Theorem 6 Without loss of generality assume that j = S p and j = S n and to establish necessity assume I I = Then p p q q Again without loss of generality it can be assumed that p q and p q Now it is obvious that Q (x) Q (x) x (q p ) (5) and the result follows immediately Now note that the first inequality in () is equivalent to min {p k q k } <x<max {p k q k } k S n and the second inequality in this set is equivalent to (p j + q j p k q k )x + p j q j p k q k > 0 j S p k S n (or each Q j j S p is greater than each Q k k S n ) Also since a positive solution is required the following result ensues Corollary 9 The switched system considered here is quadratically stabilizeable if and only if the following set of linear inequalities have a solution min {p k q k } <x<max {p k q k } k S n (p j + q j p k q k ) x + p j q j a k q k > 0 j S p k S n r j x + s j > 0 j S z x>0 (6) Recalling the proof of Lemma a stabilizing control law is easily constructed Theorem 0 Let (A b) be a canonical system of the form

5 Proceedings of the 7th WSEAS International Conference on Systems Theory and Scientific Computation Athens Greece August and ẋ = a a x + 0 m m u (7) with m > 0 Then x T Mx is a quadratic Lyapunov function for the closed loop system under the action of the control law ( ( u = kx = (α a ) αm m m a ))x (8) where α<0is an arbitrary real number Note that (8) is not unique We now perform this method step by step on the example system (4) shown previously Step Use Lemma to obtain the state transition matrices C i such that in the coordinate frame z i the i-th switching model is in the Brunovsky canonical form C = C = 5 7 C = C 4 = 5 4 Step Define another set of state transformation matrices T j = C Cj+ j=n such that z = T j z j+ 4 T = T = T = Step Calculate p j and q j by (9) if j S p S n and r j and s j by (8) if j S z For T For T t t 4 = j = S p p = q = 4 t t 4 = j = S n p = q = 5 For T t t 4 = j = S p p = q = 6 Step 4 Construct the system of inequalities (6) and find a solution x = x 0 Note: If there is no solution the problem considered is not mathematically stabilizeable j = S p and k = S n : (p + q p q )x + p q p q =x 6 > 0 j = S p and k = S n : (p + q p q )x + p q p q = x +8> 0 The complete set of inequalities (6) is then < x < 5 x 6 > 0 x +8 > 0 x > 0 with solution <x<4 Choose for example x 0 =5 Step 5 Using the inequalities (4) set m = x 0 to find a positive solution e>0 Set m = m + e Construct a positive definite matrix M = m > 0 which is a common quadratic Lyapunov function for all switching models with certain feedback control laws Note: If Step 4 has a solution then mathematically there exist solutions for the inequalities (4) With x 0 =5 e +5+( 4) 5+4 > 0 e +5+( 6) 5+8 > 0 e +5+( 5) 5+0 > 0 e > 0 giving < e < Choose for example e = 5 Then 5 M = 5 4

6 Proceedings of the 7th WSEAS International Conference on Systems Theory and Scientific Computation Athens Greece August Step 6 Convert M to each canonical coordinate system using M j+ = T T j M T j j =N Then convert model (A i b i ) to its canonical representation à i = C i A i Ci b T i = i =N and use (8) to construct the feedback gainsk i M = à = à = M 4 = M = à = Ã4 = Set for example α = < 0 and hence k = 7 7 k = k = 0 k 4 = 0 Step 7 Revert to the original coordinate x where the controls are K i = k i C i i =N The common quadratic Lyapunov function for all closed loop switching models is then Hence C T M C K = 9 89 K = 8 K = 8 K 4 = 8 and Clearly M 0 = M 0 (A i + b i K i )+(A i + b i K i ) T M 0 < 0 i = 4 Hence this system is quadratically stabilizeable 5 Conclusion The problem of finding stabilizing feedback controllers for single input single output switching systems comprising of subsystems that are second order linear time-invariant has been addressed The problem relies on finding a common Lyapunov function (CLF) for all subsystems A method has been presented by which the existence of a CLF that is shared by all subsystems given appropriate control parameters could be determined Using appropriate transformations to and from the Brunovsky form the required control parameters can then be identified References: Z Sun and SS Ge Analysis and synthesis of switched linear control systems Automatica pp 8 95 G Davrazos and NT Koussoulas A review of stability results for switched and hybrid systems Proc 9th Mediterranean Conf Control and Automation Dubrovnik Croatia 00 KS Narendra and J Balakrishnan A common Lyapunov function for stable LTI systems with commuting A-matrices IEEE Trans Automatic Control pp T Ooba and Y Funahashi Two conditions concerning common QLFs for linear systems IEEE Trans Automatic Control pp JL Mancilla-Aguilar A condition for the stability of switched nonlinear systems IEEE Trans Automatic Control pp VF Montagner VJS Leite RCLF Oliveira and PLD Peres State feedback control of switched linear systems: An LMI approach J Computational and Applied Mathematics pp P Brunovsky A classification of linear controllable systems Kybernetika 970 pp 7 87