98 PERFORMANCE SPECIFICATIONS AND LIMITATIONS
PERFORMANCE SPECIFICATIONS AND LIMITATIONS
Chapt.r B.l.nce M o el R e uction Simple linear models/controllers are normally preferred over complex ones in control system design for some obvious reasons:they are much easier to do analy sis and sy n- thesis with.Furthermore,simple cont rollers are also easier to implement and has higher reliability for there are fewer things to go wrong in the hardware or bugs to fix in the software.In the case when the system is infinite dimensional,the model/controller approximation becomes essential.In this chapter we consider the problem of reducing the order of a linear multivariable dynamical system.There are many ways to reduce the order of a dynamical sy stem.However,we shall study only one of them:balanced truncation method.The main advantage of this method is that it is simple and performs fairly well. A model order reduction problem can in general be stated as follows:Given a full order model G(s),find a lower order model,say,an r-th order model Gr,such that G and Gr are close in some sense.Of course,there are many ways to define the closeness of an approximation.For example,one may desire that the reduced model be such that G=Gr+△a and Aa is small in some norm.This model reduction is usually called an additive model reduction problem.We shall be only interested in Coo norm approximation in this book. Once the norm is chosen,the additive model reduction problem can be formulated as infG2 Grlloo· deg(G.)<r In general,a practical model reduction problem is inherently frequency weighted,i.e., the require ment on the approximation accuracy at one frequency range can be drastically different from the requirement at another frequency range.These problems can in general be formulated as frequency weighted model reduction problems 、inf Wo(G2Gr)Wil deg(G,)≤r 99
Chapter Balanced Model Reduction Simple linear modelscontrollers are normally preferred over complex ones in control system design for some obvious reasons they are much easier to do analysis and syn thesis with Furthermore simple controllers are also easier to implement and has higher reliability for there are fewer things to go wrong in the hardware or bugs to x in the software In the case when the system is innite dimensional the modelcontroller approximation becomes essential In this chapter we consider the problem of reducing the order of a linear multivariable dynamical system There are many ways to reduce the order of a dynamical system However we shall study only one of them balanced truncation method The main advantage of this method is that it is simple and performs fairly well A model order reduction problem can in general be stated as follows Given a full order model G s nd a lower order model say an rth order model Gr such that G and Gr are close in some sense Of course there are many ways to dene the closeness of an approximation For example one may desire that the reduced model be such that G Gr a and a is small in some norm This model reduction is usually called an additive model reduction problem We shall be only interested in L norm approximation in this book Once the norm is chosen the additive model reduction problem can be formulated as inf deg Gr r kG Grk In general a practical model reduction problem is inherently frequency weighted ie the requirement on the approximation accuracy at one frequency range can be drastically dierent from the requirement at another frequency range These problems can in general be formulated as frequency weighted model reduction problems inf deg Gr r kWo G Gr Wik
100 BALAN ED MODEL REDU CTION with appropriate choice of Wi and Wo.We shall see in this chapter how the balanced realization can give an effective approach to the above model reduction problems. ff6 Lyapunov Equations Testing stability,controllability,and observability of a system is very important in linear system analysis and synthesis.However,these tests often have to be done indirectly.In that respect,the Lyapunov theory is sometimes useful.Consider the following Lyapunov equation AQ+QA+H=0 (7.1) with given real matrices A and It is well known that this equation has a unique solution iff入：(A)+入(A)≠O,Vj.In this section,we shall study the relationships between the stability of A and the solution of Q.The following results are standard. Le mma7a Assume that A is stable,then the following statements hold: (iQ=t ()Q>0fH>0amdQ≥0fH≥02 ()ifH≥0，then(H,A)is observable iff Q>02 An immediate consequence of part (iii)is that,given a stable matrix A,a pair(C,A) is observable if and only if the solution to the following Lyapunov equation AQ+QA+CC=0 is positive definite where Qis the observabilty Gramian. Similarly,a pair (A, B controllable if and only if the solution to AP+PA*+ is positive definite where P is the controllability Gramian. In many applications,we are given the solution of the Lyapunov equation and need to conclude the stability of the matrix A. Le mmd72 Suppose Q is the solution of the Cyapunov equation (721),then ()Re入(A)≤0ifQ>0amdH≥02 (i)A is stable if Q>0 and H>02 (ii)A is stable if Q≥0，H≥0amd(H,A)is detectable2
BALANCED MODEL REDUCTION with appropriate choice of Wi and Wo We shall see in this chapter how the balanced realization can give an eective approach to the above model reduction problems Lyapunov Equations Testing stability controllability and observability of a system is very important in linear system analysis and synthesis However these tests often have to be done indirectly In that respect the Lyapunov theory is sometimes useful Consider the following Lyapunov equation AQ QA H with given real matrices A and H It is well known that this equation has a unique solution i i A j A i j In this section we shall study the relationships between the stability of A and the solution of Q The following results are standard Lemma Assume that A is stable then the fol lowing statements hold i Q R eA tH eAtdt ii Q if H and Q if H iii if H then H A is observable i Q An immediate consequence of part iii is that given a stable matrix A a pair C A is observable if and only if the solution to the following Lyapunov equation AQ QA CC is positive denite where Q is the observability Gramian Similarly a pair A B is controllable if and only if the solution to AP P A BB is positive denite where P is the control lability Gramian In many applications we are given the solution of the Lyapunov equation and need to conclude the stability of the matrix A Lemma Suppose Q is the solution of the Lyapunov equation then i Rei A if Q and H ii A is stable if Q and H iii A is stable if Q H and H A is detectable
7.2.Balanced Realizations 101 Proof.Let A be an eigenvalue of A and v0 be a corresponding eigenvector,then Av =Av.Pre-multiply equation (7.1)by v*and postmultiply (7.1)by v to get 2Re A(v*Qv)+v*Hv=0. Now if Q>0 then v*Qu>O,and it is clear that Re,入≤0ifH≥0 and Re.入<0if H>0.Hence (i)and (ii)hold.To see (iii),we assume ReA>0.Then we must have v*Hv =0,i.e.,Hv 0.This implies that A is an unstable and unobservable mode, which contradicts the assumption that (H,A)is detectable. 7.2 Balanced Realizations Although there are infinitely many different state space realizations for a given transfer matrix,some particular realizations have proven to be very useful in control engineering and signal processing.Here we will only introduce one class of realizations for stable transfer matrices that are most useful in control applications.To motivate the class of realizations,we first consider some simple facts. 9 Lemma 7.3 Let AB be a state space realization of a (not necessarily stable) CD transfer matrir G(s).Suppose that there erists a symmetric matrir 9 P 0 P=P* 0 with P.nonsin qular such that AP+PA*+BB*=0. Now partition the realization (A,B,C,D)compatibly with P as A:: A:2 B. A2: A22 B2 C2 D 9 Then 4:B C.D is also a realization of G.Moreover,(A::,B:)is controllable if A:: is stable. Proof.Use the partitioned P and (A,B,C)to get 0=AP+PA*+BB*= 「A:R+R4:+BB时RA+BB段 9 A2:P.B2B*
Balanced Realizations Proof Let be an eigenvalue of A and v be a corresponding eigenvector then Av v Premultiply equation by v and postmultiply by v to get Re v Qv v H v Now if Q then v Qv and it is clear that Re if H and Re if H Hence i and ii hold To see iii we assume Re Then we must have v H v ie H v This implies that is an unstable and unobservable mode which contradicts the assumption that H A is detectable Balanced Realizations Although there are innitely many dierent state space realizations for a given transfer matrix some particular realizations have proven to be very useful in control engineering and signal processing Here we will only introduce one class of realizations for stable transfer matrices that are most useful in control applications To motivate the class of realizations we rst consider some simple facts Lemma Let A B C D be a state space realization of a not necessarily stable transfer matrix G s Suppose that there exists a symmetric matrix P P P with P nonsingular such that AP P A BB Now partition the realization A B C D compatibly with P as A A B A A B C C D Then A B C D is also a realization of G Moreover A B is control lable if A is stable Proof Use the partitioned P and A B C to get AP P A BB AP PA BB PA BB AP BB BB
(02 BALANCED MODEL REDUCTION which gives B2=0 and A21=0 since P is nonsingular+Hences part of the realization is not controllabler A11 A12|B11 421A22B2 [8] C1 C2 D [8] Finallys it follows from Lemma 7 that (A11,B1)is controllable if A is stable+ We also have Lemma 7.4 Let AB be a state space realization of a (not necessarily stable) transfer matrir G(s).Suppose that there erists a symmetric matrir Q-Q- Q101 00 with Q nonsingular such that Q4+4Q+C*C=0. Now partition the realization (A,B,C,D)compatibly with Q as A11 A12 B11 421 A22 B2 Then is also a realization of G.Moreover,(C1,An)is observable if A is stable. The above two lemmas suggest that to obtain a minimal realization from a stable non-minimal realizationsone only needs to eliminate all states corresponding to the zero block diagonal term of the cont rollability Gramian P and the observability Gramian Q+ In the case where P is not block diagonals the following procedure can be used to eliminate non-controllable subsystemsi (+Let G(s)= 「AB] be a stable realization+ 2+Compute the controllability Gramian P 0 from AP+PA*+BB*=0. 3 DigPtoP=[][8]【ts广ithA>0amd [U1 U2 unitary+
BALANCED MODEL REDUCTION which gives B and A since P is nonsingular Hence part of the realization is not controllable A A B A A B C C D A A B A C C D A B C D Finally it follows from Lemma that A B is controllable if A is stable We also have Lemma Let A B C D be a state space realization of a not necessarily stable transfer matrix G s Suppose that there exists a symmetric matrix Q Q Q with Q nonsingular such that QA AQ CC Now partition the realization A B C D compatibly with Q as A A B A A B C C D Then A B C D is also a realization of G Moreover C A is observable if A is stable The above two lemmas suggest that to obtain a minimal realization from a stable nonminimal realization one only needs to eliminate all states corresponding to the zero block diagonal term of the controllability Gramian P and the observability Gramian Q In the case where P is not block diagonal the following procedure can be used to eliminate noncontrollable subsystems Let G s A B C D be a stable realization Compute the controllability Gramian P from AP P A BB Diagonalize P to get P U U U U with and U U unitary