Notes on Linear Robust Control - Robust Control Systems I | MEM 633, Study notes of Mechanical Engineering

Download Notes on Linear Robust Control - Robust Control Systems I | MEM 633 and more Study notes Mechanical Engineering in PDF only on Docsity! Notes on Linear Robust Control MEM 633 October 2, 2002 Professor Harry G. Kwatny Office: 3-151A hkwatny@coe.drexel.edu http://www.pages.drexel.edu/faculty/hgk22 MEM 633-634 Notes Professor Kwatny Contents 1 Introduction to the Robust Control Problem......................................................... 1 2 State Space Models ................................................................................................... 2 2.1 Solutions of Linear Systems ............................................................................... 2 2.2 The Matrix Exponential ...................................................................................... 2 2.3 Controllability ..................................................................................................... 2 More on Invariance ..................................................................................................... 7 2.4 Observability....................................................................................................... 7 More on Invariance ..................................................................................................... 9 2.5 Kalman Decomposition ...................................................................................... 9 2.6 Thorp-Morse Form.............................................................................................. 9 2.7 Zeros ................................................................................................................... 9 3 Nominal Controller Design: State Space Perspective.......................................... 12 3.1 State Feedback .................................................................................................. 12 Pole Placement.......................................................................................................... 12 The Linear Regulator Problem.................................................................................. 12 3.2 Observers & the Separation Principle............................................................... 12 3.3 Disturbance Rejection....................................................................................... 12 4 Transfer Function Models...................................................................................... 13 4.1 State Space to Transfer Function ...................................................................... 13 4.2 Frequency Response ......................................................................................... 13 4.3 Poles & Zeros of Transfer Functions ................................................................ 13 4.4 Realizations....................................................................................................... 13 5 Closed Loop Transfer Functions ........................................................................... 15 5.1 Well Posedness ................................................................................................. 15 5.2 Closed Loop Transfer Functions....................................................................... 15 Output ....................................................................................................................... 15 Error .......................................................................................................................... 15 Control ...................................................................................................................... 15 ii MEM 633-634 Notes Professor Kwatny 1 Introduction to the Robust Control Problem 1 MEM 633-634 Notes Professor Kwatny 2 State Space Models 2.1 Solutions of Linear Systems ( ) ( )x A t x B t u= + 0 0 0 0 ( , ) ( , ) ( ) t x(t; x ,u) t t x t s B s u(s)ds= Φ + Φ∫ ( , ) ( ) ( , ), ( , )d t A t t t t dt τ τΦ = Φ Φ = I 2.2 The Matrix Exponential Cayley-Hamilton Theorem: Every square matrix satisfies its own characteristic equation, i.e. if φ λ λ λ λ( ) = − = + + +− −I A a an n n 1 1 0 Then φ( )A A a A a In n n= + + + =− − 1 1 0 0 From this we obtain: e t I t A tAt n n= + + + − −Aα α α0 1 1 1( ) ( ) ( ) 2.3 Controllability We briefly review some basic concepts and results for linear autonomous systems BuAxx += Cxy = where . Recall that given an initial state x(0) = x0 and a control u(t), t > 0, the corresponding trajectory is define by the variations of parameters formula pmn RyRuRx ∈∈∈ ,, ∫+= t A(t-s)At Bu(s)dse x e,u) x(t;x 0 00 2 MEM 633-634 Notes Professor Kwatny Definition: A state is reachable from x0 if there exists a finite time t > 0 and a piecewise continuous control u such that . nRx ∈1 10 x,u)x(t;x = 0xR denotes the set of states reachable from x0. Let us make a few preliminary observations. If 1x is reachable from 0x in some time , it is reachable in every time t. To see this simply rescale s: 1 0t > ( ) ( ) 1 11 1 1 1 1 1 1 1 0 0 0 st t st t t t A(t - )A(t -s) st st t t t A(t- ) A(t -t) st st t t e Bu(s)ds e Bu( )d e Be u( )d = = ∫ ∫ ∫ Thus, we have the replacement 1 1 ( ) A(t -t) st tu s e u( )→ . Notice that 1x is reachable from 0x if and only if 1 At 0x e x− is reachable from the origin for , viz 0 t< < ∞ 1 0 1 0 0 0 t t At A(t-s) At A(t-s)x e x e Bu(s)ds x e x e Bu(s)ds= + ⇔ − =∫ ∫ As a result, we focus on characterizing the set of states reachable from the origin. Let denote the linear vector space of control functions , and U 1( ), [0, ]u tτ τ ∈ nR≅X the space of states 1( )x t . The inner product on is the usual U 1 0 ˆ ˆ, ( ) ( t Tu u u u dτ τ= ∫ ) τ The mapping defined by :A →U X 1 1( )1 0( ) ( ) t A tx t e Buτ τ τ−= ∫ d (1.1) defines the state reached from the origin at time when the control u is applied on the time interval [0 . 1t 1, ]t A state 1x is reachable from the origin over the time interval [0 if and only if the relation 1, ]t 1( )A u x= has a solution u t . Such a solution exists if and only if ( ) 1 Imx A∈ . It is more convenient to apply the equivalent condition *1 Imx AA∈ because . So let’s us prove this result. * :AA →X X 3 MEM 633-634 Notes Professor Kwatny Since is symmetric (⇒ ), this is equivalent to ( )CG t Im kerCG= ⊕X CG ker ( ), 0CA G t ⊥ = >B t First show, ker ( )Cx G t x A ⊥ ∈ ⇒ ∈ B . If ker ( )Cx G t∈ , then so that 0 T Cx G x = 2( ) 0 0, 0 t T A tx e B dτ τ τ− = <∫ t< Therefore 0, 0T Asx e B s t= < < Expanding Ase and comparing coefficients leads to 0, 0, , 1T ix A B i n= = … − ] 0 Consequently, which implies that x is orthogonal to 1[T nx B AB A B− =… A B , i.e. x A ⊥ ∈ B . Now show ker ( )Cx G t x A ⊥ ∈ ⇐ ∈ B . But x A ⊥ ∈ B 0 implies so reversing the above steps leads to . This is true only if 1[ ]T nx B AB A B− =… ker ( )C 0 T Cx G x = x G t∈ . Qed Theorem: The system or the matrix pair (A,B) is (completely) controllable if and only if 0 nR R= , or equivalently nBAABB n =− ]rank[ 1… Proof: Qed We wish to emphasize the geometric aspects of controllability and observability. To do so fully requires the concept of an invariant subspace. Definition: A subspace V is invariant with respect to A if nR⊆ VAV ⊆ Clearly, every eigenvector defines a one-dimensional invariant subspace. Furthermore, the set of all vectors satisfying h Ah hλ= is called the eigenspace of A associated with the eigenvalue λ . Every eigenspace of A is an invariant subspace as is every subspace that can be constructed as the sum of 6 MEM 633-634 Notes Professor Kwatny eigenspaces. Perhaps less obvious is the fact that every invariant subspace is the direct sum of eigenspaces. ) is A-invariant, i.e., . In fact is the smallest A-invariant subspace of 0(R )0()0( RAR ⊆ )0(R nR containing B. Moreover, if there exists a system of coordinates in which the state equations take the form: 1) n=0(dim R x x A A A x x B u1 2 11 12 22 1 2 1 0 0 L NM O QP = L NM O QP L NM O QP + L NM O QP , x1∈R n1 , x2∈Rn-n1 such that the pair (A11,B1) is completely controllable, i.e., nRA =111 B , and in fact x1 are coordinates in R( )0 . Hence the restriction of the system to R( )0 (x2=0) results in a controllable system. Thus, we refer to R( )0 as the controllable subspace. More on Invariance Recall that application of the linear control u K , results in the closed loop system x v= + ( )x A BK x Bv= + + This motivates the following definition: Definition: A subspace V⊆Rn is (A,B)–invariant if there exists a state feedback matrix K such that ( )A BK V V+ ⊆ Now , the following theorem can be established. Theorem: V is (A,B)–invariant if and only if Rn⊆ AV V⊆ + B 2.4 Observability We briefly review some basic concepts and results for linear autonomous systems BuAxx += Cxy = where . Recall that given an initial state x(0) = x0 and a control u(t), t > 0, the corresponding trajectory is define by the variations of parameters formula pmn RyRuRx ∈∈∈ ,, 7 MEM 633-634 Notes Professor Kwatny ∫+= t A(t-s)At Bu(s)dse x e,u) x(t;x 0 00 Definition: A state x1∈Rn is indistinguishable from x0 if for every finite time t and piecewise continuous control u(t), y t x u y t x u( ; , ) ( ; , )1 2= . I x( 0 ) denotes the set of states indistinguishable from x0. Define ( )1 1 ker n i i CA − = =N ∩ Theorem: (0)I =N proof: Definition: The system or the matrix pair (C,A) is said to be (completely) observable if knowledge of u(t) and y(t) on a finite time interval determines the state trajectory on that interval. Theorem: The system or the matrix pair (C,A) is (completely) observable if and only if I( )0 = ∅. proof: (Wonham) I( )0 is A-invariant. In fact I( )0 n is the largest A-invariant subspace of Rn contained in . If dim there exists a set of coordinates in which the system equations are in the form keraCf ( )I 0 1a f =      ẋ1 ẋ2 =      A11 0 A21 A22      x1 x2 +      B1 B2 u, x1∈R n1 , x2∈R n-n1 y = [C1 0 ]      x1 x2 8 MEM 633-634 Notes Professor Kwatny 11 MEM 633-634 Notes Professor Kwatny 3 Nominal Controller Design: State Space Perspective 3.1 State Feedback Pole Placement The Linear Regulator Problem 3.2 Observers & the Separation Principle 3.3 Disturbance Rejection 12 MEM 633-634 Notes Professor Kwatny 4 Transfer Function Models 4.1 State Space to Transfer Function 4.2 Frequency Response 4.3 Poles & Zeros of Transfer Functions 4.4 Realizations 13 MEM 633-634 Notes Professor Kwatny 6 Performance in the Frequency Domain 6.1 Sensitivity Functions E s I L R s I L GD s I L LD s( ) ( ) ( ) ( )= + − + + +− − −1 1 1 1 2 , where L GK:= Sensitivity function: S I L:= + −1 Complementary sensitivity function: T I L:= + −1 L L Consider a scalar system in which is the open loop transfer function and is the closed loop transfer function. Then compute the (relative) variation of the closed loop with respect to (relative) variation of the open loop transfer function: L GK= T L= + −[ ]1 1 dT T dL L dT dL L T L L L L L L L L L S = = − + + + + = − + + = + = − − − − [ ] [ ] [ ] [ ] [ ] 1 1 1 1 1 1 2 1 1 1 m r This is Bode’s original reason for the terminology ‘sensitivity function’ for S. A fundamental tradeoff Note that I L I L L I+ + + =− −1 1 S T I+ = T I L L I L L I L= + = + = +− − − −1 1 1 1 G I KG I GK G+ = +− −1 1 GK I GK G I KG K I GK GK+ = + = +− − −1 1 1 E s S s R s S s GD s T s D s( ) ( ) ( ) ( ) ( ) ( ) ( )= − +1 2 U s K s I G s K s R s K s I G s K s G s D s K s I G s K s I G s K s D s ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) = + − + + + + − − − 1 1 1 1 22l q 16 MEM 633-634 Notes Professor Kwatny 6.2 Sensitivity Peaks )(max ω ω jSM S = , )(max ωω jTT =M Sensitivity peaks are related to gain and phase margin. Sensitivity peaks are related to overshoot and damping ratio. -1 a L j( )ω L plane− S a= 1 S L− = +1 1 S > 1 S < 1 Re Im 6.3 Bandwidth Bandwidth (sensitivity) { }),0[21)(:max vjSv vBS ∈∀<= ωωω Bandwidth (complementary sensitivity) { }),(21)(:min ∞∈∀<= vjSv vBT ωωω Crossover frequency { }),0[1)(:max vjLv vc ∈∀≥= ωωω Bandwidth is related to rise time and settling time. 6.4 Limits on Performance 17 MEM 633-634 Notes Professor Kwatny 7 Nominal Controller Design: Frequency Domain Perspective 7.1 Full State Feedback Controllers The Quadratic Regulator Problem ( , ) ( ) ( ) ( ) ( ) ( ) ( ) TT T T f t J x t x T Q x T x Qx u Ru dτ τ τ τ τ = + + ∫ Principle of Optimality & HJB Equation Stability of Sol’ns to LQR Consider the linear system x Ax= . Suppose V x with . Compute the time rate of change of V along trajectories ( ) Tx Px= 0P > T T TVV x x PAx x A Px x ∂= = + ∂ (1.1) Suppose V x , . Then, clearly, the system is asymptotically stable, as . Actually, if , we require only . Hence, if there exists a that satisfies the Liapunov equation TQ= − x 0Q > det A ( ) 0x t → 0P >t → ∞ 0≠ 0Q ≥ (1.2) TPA A P Q+ = − for any , the system is asymptotically stable provided det . As a matter of fact this requirement is necessary, as well as sufficient, for asymptotic stability. 0Q ≥ 0A ≠ Now, consider the linear control system x Ax Bu= + (1.3) Suppose that so that the closed loop system is u Kx= ( )x A BK x= + (1.4) Moreover, choose where 1 TK R B P−= − x T (1.5) 1T TPA A P PBR B P Q−+ − = − Now, (1.5) can be rewritten 1( ) ( )TP A BK A BK P Q PBR B P−+ + + = − − 18 MEM 633-634 Notes Professor Kwatny Given the detectability/stabilizabilty assumptions it is possible to prove there exists a minγ so that there are no eigenvalues of H on th imaginary axis provided minγ γ> . In fact, when γ → ∞ we approach the standard LQR solution. For minγ γ= n , the controller is the full state feedback controller. All other values of H∞ miγ γ≤ < ∞ produce valid min- max controllers. The condition that Re ( ) 0Hλ ≠ is equivalent to stability of the matrix 2 1 1T TA EE B γ ρ + − B Notice that the closed loop system matrix is 1 TA BB ρ − Since 2/TEE γ is destabilizing, the feedback system has some margin of stability. Solving the Riccati Equation Constructing the Solution Computing Tools CARE Solve continuous-time algebraic Riccati equations. [X,L,G,RR] = CARE(A,B,Q,R,S,E) computes the unique symmetric stabilizing solution X of the continuous-time algebraic Riccati equation -1 A'XE + E'XA - (E'XB + S)R (B'XE + S') + Q = 0 or, equivalently, -1 -1 -1 F'XE + E'XF - E'XBR B'XE + Q - SR S' = 0 with F:=A-BR S'. When omitted, R,S and E are set to the default values R=I, S=0, and E=I. Additional optional outputs include the gain matrix -1 G = R (B'XE + S') , the vector L of closed-loop eigenvalues (i.e., EIG(A-B*G,E)), 21 MEM 633-634 Notes Professor Kwatny and the Frobenius norm RR of the relative residual matrix. [X,L,G,REPORT] = CARE(A,B,Q,...,'report') turns off error messages and returns a success/failure diagnosis REPORT instead. The value of REPORT is * -1 if Hamiltonian matrix has eigenvalues too close to jw axis * -2 if X=X2/X1 with X1 singular * the relative residual RR when CARE succeeds. [X1,X2,L,REPORT] = CARE(A,B,Q,...,'implicit') also turns off error messages, but now returns matrices X1,X2 such that X=X2/X1. REPORT=0 indicates success. 7.2 Output Feedback Controllers The Classical H2 Problem – LQG The classical output feedback optimal control problem for SISO systems was solved during the 2nd World War using a frequency domain formulation. It is referred to as the Wiener-Hopf-Kolmogorov problem. Attempts to extend this result to the MIMO case using frequency domain techniques were not fruitful. The MIMO problem was formulated and solved in the state space by Kalman and coworkers around 1960. We summarize the result here. The Linear Quadratic Guassian (LQG) Problem - Setup The standard problem formulation is as follows. The plant is described by x Ax Bu w y Cx v = + + = + The disturbances are independent, zero-mean white noise processes have covariances, ,w v { } { } ( ) ( ) ( ) ( ) ( ) ( ) T T E w t w W t E v t v V t τ δ τ δ τ = − = − τ and { }( ) ( ) 0TE w t v τ = 22 MEM 633-634 Notes Professor Kwatny We seek that minimizes the performance index: ( )u t 0 1lim T T T T J E x Qx u Ru dt T→∞   = +   ∫   0, 0T TR R= ≥ = >, Q Q Solution Summary 1ˆ( ), Tu Kx t K R B P−= = − , 1T TPA A P PBR B P Q−+ − = − 0P ≥ ( ) 1ˆ ˆ ˆ , Tx Ax Bu L y Cx L SC V −= + + − = , 1T TSA AS SC V CS W−+ − = − 0S ≥ The Modern Paradigm We view the control design problem in terms of the diagram shown in Figure 1. P K zw u y Figure 1. The so-called ‘modern paradigm’ views the plant in terms of two input sets: disturbance and control inputs, and two out put sets: performance and measured variables. In the frequency domain the plant is characterized in terms of a transfer matrix: z v P P P P w u L NM O QP = L NM O QP L NM O QP 11 12 21 22 Closing the loop with u Kv= Produces the closed loop transfer function z Fw= , F P P I P K P= + − −11 12 22 1 21b g 23 MEM 633-634 Notes Professor Kwatny 1( ) ( ) ( ) ( ) 2 1tr ( ) ( ) 2 T T T z t z t dt Z j Z j d Z j Z j d ω ω ω π ω ω π ∞ ∞ −∞ −∞ ∞ −∞ = − ω = −    ∫ ∫ ∫ { } { }22 2 2 1tr ( ) ( ) 2 1tr ( ) ( ) 2 T T E z E Z j Z j d F j F j d F ω ω π ω ω ω π ∞ −∞ ∞ −∞ ω = −     = −    = ∫ ∫ The H∞ Problem We consider the same control design problem as above, except with the class of disturbances defined by 2 ( ) 1w t = We seek to choose K such that the maximum of 2 ( )z t over all disturbance inputs is a minimum. Thus, we seek to ( )w t 2 21 min max ( ) K w z t = Now, compute 1( ) ( ) ( ) ( ) 2 1 ( ) ( ) ( ) ( ) 2 T T T T z t z t dt Z j Z j d W j F j F j W j d ω ω ω π ω ω ω ω π ∞ ∞ −∞ −∞ ∞ −∞ = − = − − ∫ ∫ ∫ ω Now, we seek the maximum performance energy over all disturbances with unit norm. It occurs when W j is aligned with the maximum eigenvalue of , ( )ω *F F ( ) ( ) 2 2 1 22 1max ( ) ( ) ( ) ( ) ( ) 2 max ( ) T T w z t z t dt W j F j W j d F j F ω ω σ ω ω ω π σ ω ∞ ∞ −∞ −∞= ∞ = = = ∫ ∫ Solution Summary 26 MEM 633-634 Notes Professor Kwatny Proposition (H2 Output Feedback): Suppose is a unit intensity white noise signal, ( )w t { }( ) ( ) ( )TE w t w I tτ δ= −τ . Then the unique, stabilizing, optimal controller that minimizes 2 ( )zwT s is 2 22 2 0 A L K F  − =      were ( ) ( ) 2 2 2 2 2 2 22 1 2 2 2 1 2 2 2 T T uu xu T xy yy A A B F L C L D F F R R B X L Y C V V − − = + + + = − + = − + 2 22 ,X Y satisfy the two Riccati equations 1 1 2 2 2 2 2 2 1 1 2 2 2 2 2 2 T T r r uu xx xu uu xu T T e e yy xx xy yy xy T T X A A X X B R B X R R R R A Y Y A Y C V C Y V V V V − − − − + − = − + + − = − + ( ) ( ) 1 2 1 2 T r uu e xy yy A A B R R A A V V C − − = − = − xu ( ) 1sI A −− 2F2L 2B 2C 22D y u x̂- 27 MEM 633-634 Notes Professor Kwatny Proposition (H∞ Output Feedback): Suppose is a bounded signal, . A stabilizing controller that satisfies ( )w t 2L ( ) ( )Tw t w t dt ∞ −∞ < ∞∫ ( )zwT s γ∞ < is 0 A Z L K F ∞ ∞ ∞ ∞ ∞ −  =    where ( ) ( ) ( ) 1 21 2 2 22 1 2 1 2 12 2 1 1, T T uu uu T xy yy T A A B L D W B F Z L C Z L D F F R R B X L Y C V V W B X Z I Y X γ γ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ − ∞ ∞ − ∞ ∞ ∞ ∞ ∞ ∞ ∞ = + + + + + = − + = − +   = = −    0X ∞ ≥ and Y satisfy the Riccati equations 0∞ ≥ 1 1 2 2 1 12 1 1 2 2 1 12 1 1 T T T r r uu xx xu uu xu T T T e e yy xx xy yy xy T T X A A X X B R B B B X R R R R A Y Y A Y C V C C C Y V V V V γ γ − − ∞ ∞ ∞ ∞ − − ∞ ∞ ∞ ∞   + − − = − +      + − − = − +    and the following conditions are satisfied: 1. The Hamiltonian matrix ( ) 1 1 2 2 2 2 1 1 2 1T T uu xu uu TT T xx xu uu xu uu xu A B R R B R B B B R R R R A B R R γ − − − −  − − +     − + − −  1 1 T has no eigenvalues on the imaginary axis, or equivalently, 1 2A BW B F∞ ∞+ + is stable. 2. The Hamiltonian matrix ( )1 12 2 2 12 1 1 2 1T T T xy yy yy T xx xy yy xy xy yy A V V C C V C C C V V V V A V V C γ − − − −  − − +    − + − +   1 has no eigenvalues on the imaginary axis, or equivalently, 28 MEM 633-634 Notes Professor Kwatny Guaranteed Gain and Phase Margin Proposition: Suppose is the maximum sensitivity peak. Then SM α± ≥ 1 1GM ,       ±≥ 2 sin2 αPM , SM1=α Proof: The proof is easily obtained from the geometry shown in the figure GH-plane -1 1 1R S a −= = 1 12sin 2a θ −  =     1 1 a− 1 1 a+ 8.2 MIMO Nyquist Analysis Recall that the closed loop poles are roots of the polynomial [ ]( ) ( )det ( )Ld s d s I L s= + If the open loop system is stable, then to assess stability of the closed loop system we need only be concerned with the zeros of { }( ) det ( )F s I L= + s Nyquist analysis still applies with { }det ( )I L s+ replacing 1 . Specifically we have: ( )L s+ Z P N= − where: Z=number of closed loop poles in the RHP P=number of open loop poles in the RHP N=number of counterclockwise encirclements of the F-plane origin by . ( )F C 31 MEM 633-634 Notes Professor Kwatny Suppose that the closed loop system is asymptotically stable. Then the Nyquist { }det ( ) 0, , 0I L s s jσ ω σ+ ≠ ∀ = + ≥ 8.3 M ∆ - Structure Models of Uncertainty G G E G G I E G I E G G G I EG G G I E G I E G p p p p p p = + = + = + = − = − = − − − − ( ) ( ) ( ) ( ) ( ) 1 1 1 , , E W W= 1∆ 2 ∆ ∞ < 1 M W M W= 1 0 2 , M K I GK KS M K I GK G T M GK I GK T M I GK G SG M I GK S M I GK S I I 0 1 0 1 0 1 0 1 0 1 0 1 = + = = + = = + = = + = = + = = + = − − − − − − ( ) ( ) ( ) ( ) ( ) ( ) 8.4 Small Gain Theorem Consider a feedback loop with open loop transfer matrix . Then we define the spectral radius: L s( ) ρ ω λ ωL j L j i i ( ) max ( )b g b g= Theorem (Spectral radius stability theorem). Consider a system with stable open loop transfer function . Then the closed loop is stable if L s( ) ρ ω ωL j( ) ,b g < ∀1 Theorem (Small gain theorem). Consider a system with stable open loop transfer function . Then the closed loop is stable if L s( ) L j( ) ,ω ω< ∀1 where L denotes any norm satisfying AB A B≤ ⋅ . 32 MEM 633-634 Notes Professor Kwatny 8.5 Robust Stability of the M ∆ - Structure Theorem (Robust stability for unstructured perturbations). Assume that the nominal system M s( ) is stable and that the perturbations are stable. Then the -system is stable for all perturbations satisfying ∆( )s M ∆ ∆ ∞ ≤ 1 if and only if σ ω ωM j( ) ,b g < ∀1 ⇔ M ∞ < 1 Notice that: RS ⇔ M ∞ < 1 ⇔ W M W1 0 2 1∞ < Special case G G I wp = + <∞( ),∆ ∆ 1 ⇔ wT ∞ < 1 33 MEM 633-634 Notes Professor Kwatny Poisson Integrals Various integral formulas can be derived from the Cauchy integral formula by choosing specific contours. Consider the contour shown, with . The integral will exist only if the function has restricted behavior at infinity. In particular, we require ∞→R )(sf 0)(lim = ∞→ R Rm R , where )(sup:)( θ θ jeRfRm = , ]2/,2/[ ππθ −∈ R If has no singularities in RHP two equivalent formulas an be derived: )(sf ∫ ∞ ∞− −+ = dv v jvfsf )( )(1)( 2 ωσ σ π , ωσ j+=s ∫ ∞ ∞− −+ −= dv v jvfv j sf 22 )( )()(1)( ωσ ω π The last equation can be broken down into real and imaginary parts to yield: ∫ ∞ ∞− −+ −= dv v jvfvsf 22 )( )(Im)(1)(Re ωσ ω π , ∫ ∞ ∞− −+ −−= dv v jvfvsf 22 )( )(Re)(1)( ωσ Im ω π Thus )(Im)(Re sfjf ⇒ω . However, we can not evaluate on the imaginary axis by simply setting )(sf 0=σ , because the integrals do not exist since the integrands have a pole at ω=v . However, a careful limiting process leads to 36 MEM 633-634 Notes Professor Kwatny αα α ω π ω α d d ejfdjf ∫ ∞ ∞− = 2 cothlog)(Re1)(Im Suppose is a transfer function. Then consider )(sH )(log)( sHsf = for which )()(Re sHsf = and . Then )()(Im sHsf ∠= αα α ω π ω α d d ejHd jH ∫ ∞ ∞− =∠ 2 cothlog )(log1)( Since must be analytic in RHP, can have neither poles nor zeros in RHP. )(sf )(sH Bode Waterbed Formula Application of the Cauchy Integral Formula to systems with relative degree 2 or greater: (Waterbed effect) ln ( )S j d pi ORHP poles ω ω π 0 ∞z ∑= ln ( )T j d qiORHP zeros ω ω ω π 0 2 1∞z ∑= Example: Stable plant L s s ( ) ( ) = + 1 1 2 È 2 4 6 8 w -0.6 -0.4 -0.2 S Bode Integral Formula 37 MEM 633-634 Notes Professor Kwatny 10.2 Parseval’s Theorem A simple, but important formula is given by Parseval’s theorem. Suppose the functions f t f t1 2( ), ( ) have Laplace transforms , respectively. Then we can write F s F s1 2( ), ( ) f t f t dt L F s f t dt j F s e ds f t dt j f t e dt F s ds j F s F s ds st j j st j j j j 1 2 1 1 2 1 2 2 1 2 1 1 2 1 2 1 2 ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) −∞ ∞ − −∞ ∞ − ∞ ∞ −∞ ∞ −∞ ∞ − ∞ ∞ − ∞ ∞ z z zz zz z = = = = − π π π f t dt j F s F s ds F j F j d F j d j j2 2 1 2 1 2 1 2 ( ) ( ) ( ) ( ) ( ) ( ) −∞ ∞ − ∞ ∞ −∞ ∞ −∞ ∞ z z z z = − = − = π π ω ω ω π ω ω 38 MEM 633-634 Notes Professor Kwatny 5. The function spaces consisting of (complex-valued) integrable functions with bounded norm: L a b pp[ , ], 1≤ ≤ ∞ a b, ]f t t( ), [∈ f f t dp p a b p = tLNM O QPz ( ) /1 , 1≤ < ∞p and f f t a b∈ = sup ( [ , ] t)∞ . 6. The set of complex valued m matrices, denoted n× Cm n× , with norm 1/ 2 2 * , 1 trace( ) ( ) n ij i i j i A a A A σ =   = = =    ∑ ∑ A , or the norm A A= σ b g This is sometimes called the Frobenius norm. 7. Time-Domain (Signal) spaces. Consider complex vector valued time functions defined on the interval (or, ). The appropriate p- norm is f t Rn( ) ∈ t ∈ −∞ ∞[ , ] t ∈ ∞[ , ]0 f f tp i p i n p = dtFHG I KJ=−∞ ∞ ∑z ( ) / 1 1 , 1≤ < ∞p and f f t i i∞ = sup max ( )e jt . For p = 2 we can use Parseval’s theorem to obtain f f t dt F j d F F di i n i i n 2 2 1 1 2 2 1 1 2 1 2 = FHG I KJ = F HG I KJ = FH IK = −∞ ∞ = −∞ ∞ −∞ ∞ ∑z ∑z z( ) ( ) / / * / ω ω ω 8. Frequency-Domain Spaces. Consider the functions , F j Cn( )ω ∈ −∞ < < ∞ω F F jp p i n p = dFHG I KJ=−∞ ∞ ∑z12 1 1 π ω ω( ) / , 1≤ < ∞p and F F j F∞ = sup ( ) ( ) * ω ω ωe jj . The space of all frequency functions with bounded p-norm is often designated . Lp 9. The space of functions that are analytic in the open right half plane, R with F s F C Cn( ), : → e( )s > 0 F F jp p i n p = + dFHG I KJ> =−∞ ∞ ∑zsup ( ) / ξ π ξ ω ω 0 1 1 1 2 , 1≤ < ∞p and F F s F s s ∞ > = sup ( ) ( )* 0 e j The space of functions with bound p-norm is the Hardy space . Hp 41 MEM 633-634 Notes Professor Kwatny For almost all ω, lim ( ) ( ) ξ ξ ω ω → + = ∈ 0 F j F j pL x . Moreover, the supremum occurs on the imaginary axis. 11.2 System Norms/ Induced Norms Consider a mapping A from one normed linear space to another . That is, y A= One way to view the ‘magnitude’ of the action A is consider its ‘gain,’ that is the ratio: y x/ . The ratio will depend on the specific choice of input x. So, we choose the one that produces the largest ratio, i.e.: A y x Ax xx x = = ≠ ≠ max max 0 0 Because A is linear, it is clear that the gain is independent of the size of x so we could rescale and equivalently define A A x = = max 1 x Now, the definition depends on the specific choices of input and output spaces and . Suppose that the p-norm is appropriate for both spaces, then we can define A Ax xxαβ β α = ≠ max 0 These induced norms have the product property: A B A B≤ Transfer matrix norms Consider systems described by l m× stable, proper transfer matrices: Y s G s U s( ) ( ) ( )= We will consider two transfer function norms: H2 Norm G s G j G j d G j di i G ( ) tr ( ) ( ) ( )* / rank[ ] / 2 1 2 1 1 2 1 2 1 2 = FHG I KJ = F HG I KJ−∞ ∞ = −∞ ∞z ∑zπ ω ω ω π σ ω ω H∞ Norm 42 MEM 633-634 Notes Professor Kwatny G s G j( ) sup ( )∞ = ω σ ω Suppose that the input space is equipped with the norm and the output space with the . Then we have L2 L∞ Y U G∞ = sup * * ω { }GU so that G U U2 12 ∞ = = max sup * * ω { }G GU Suppose that the input and output spaces are equipped with the norm. Then we have L2 Y U G G2 2 = * * U so that G U j G j G j U j d U j 22 1 = = −∞ ∞zmax ( ) ( ) ( ) ( )* *ω ω ω ω ω We need to choose U to maximize the result. Clearly, at each ω we maximize the kernel by aligning U with the largest eigenvector of G G* so that we have U G GU j U j* * ( ) ( ) ( )ω σ ω ω= 2 where σ ωG j( )b g is the maximum singular value of G. Then we need to concentrate all of the energy in U at the frequency at which σ is a maximum G G∞ = sup ( ) ω jσ ωb g Computing the H2 Norm AL L A BBc c T T+ + = 0 , A L L A C CT o o T+ + = 0 G CL C B L Bc T T o2 1 2 1 2 = =tr( ) tr( ) / / Computing the H∞ Norm Find smallest γ > 0 such that H has no eigenvalues on the imaginary axis. H A BB C C A T T T = − − L N MM O Q PP 1 2γ 43