OBSERVABILITY AND CONTROLLABILITY

Transcription

1 ECE45/55: Multivariable Control Systems I. 5 OBSERVABILITY AND CONTROLLABILITY 5.: Continuous-time observability: Where am I? We describe dual ideas called observability and controllability. Both have precise (binary) mathematical descriptions, but we need to be careful in interpreting the result. We develop some other techniques to help quantify the concepts. Continuous-time observability If a system is observable, we can determine the initial condition of the state vector x./ via processing the input to the system u.t/ and the output of the system y.t/. Since we can simulate the system if we know x./ and u.t/ this also implies that we can determine x.t/ for t. e.g.,forlti, x.t/ D e At x./ C Z t e A.t / Bu./d: Consider the LTI SISO system LCCODE «y.t/ C a Ry.t/ C a Py.t/ C a 3 y.t/ D b «u.t/ C b Ru.t/ C b Pu.t/ C b 3 u.t/: If we have a realization of this system in state-space form Px.t/ D Ax.t/ C Bu.t/ y.t/ D Cx.t/C Du.t/; Lecture notes prepared by Dr. Gregory L. Plett. Copyright c 5,, 9,, 5, 3,,, Gregory L. Plett

2 ECE45/55, OBSERVABILITY AND CONTROLLABILITY 5 and we have initial conditions y./, Py./, Ry./,howdowefindx./? In general, or, y./ D Cx./C Du./ Py./ D C.Ax./ C Bu./ / C D Pu./ ƒ Px./ D CAx./ C CBu./ C D Pu./ Ry./ D CA x./ C CABu./ C CB Pu./ C D Ru./: y.k/./ D CA k x./ C CA k Bu./ CCCBu.k /./ C Du.k/./; 4 y./ Py./ Ry./ 3 5 D 4 C CA CA ƒ O.C;A/ where T is a (block) Toeplitz matrix D u./ 5 x./ C 4 CB D 5 4 Pu./ CAB CB D Ru./ ƒ T Thus, if O.C; A/ is invertible, then ˆ< y./ u./ >= x./ D O 4 Py./ 5 T 4 Pu./ 5 ˆ: >; : Ry./ Ru./ We say that fc;ag is an observable pair if O is nonsingular. CONCLUSION: For a SISO system, if O is nonsingular, then we can determine/estimate the initial state of the system x./ using only u.t/ and y.t/ (and therefore, we can estimate x.t/ for all t ). EXTENSION: For a MIMO system, if O is full rank, then we can determine/estimate the initial state of the system x./ using only u.t/ and y.t/ (and therefore, we can estimate x.t/ for all t ). Lecture notes prepared by Dr. Gregory L. Plett. Copyright c 5,, 9,, 5, 3,,, Gregory L. Plett 5 ;

3 ECE45/55, OBSERVABILITY AND CONTROLLABILITY 5 3 EXAMPLE: Observability canonical form: 3 Px.t/ D 4 5 x.t/ C 4 ˇ ˇ 3 5 u.t/ y.t/ D h a 3 a a i x.t/: ˇ3 Then O D 4 C CA CA 3 5 D 4 : :: 3 5 D I n : This is why it is called observability form! EXAMPLE: Two unobservable networks u x y u H x x F y H x (Redrawn) x F y In the first, if u.t/ D then y.t/ D 8 t. Cannotdeterminex./. In the second, if u.t/ D, x./ and x./ D, theny.t/ D and we cannot determine x./. [circuitredrawnforu.t/ D ]. Observers An observer is a device that has as inputs u.t/ and y.t/ the input and output of a linear system. The output of the observer is the (estimated) state of the linear system. Lecture notes prepared by Dr. Gregory L. Plett. Copyright c 5,, 9,, 5, 3,,, Gregory L. Plett

4 ECE45/55, OBSERVABILITY AND CONTROLLABILITY 5 4 The observer observes the internal state x (estimated as Ox) from external signals u and y. u.t/ Note that our equations yield an observer: A; B; C; D x Observer y.t/ Ox u.t/ A; B; C; D x y.t/ s s u Pu Ru T y Py Ry s s O Ox Later, we ll design more practical observers that don t use differentiators. Lecture notes prepared by Dr. Gregory L. Plett. Copyright c 5,, 9,, 5, 3,,, Gregory L. Plett

5 ECE45/55, OBSERVABILITY AND CONTROLLABILITY : Continuous-time controllability: Can I get there from here? Can we generate an input u.t/ to set an initial condition quickly? If u.t/ D ı.t/ and x. / D, then Px.t/ D Ax.t/ C Bu.t/ y.t/ D Cx.t/C Du.t/: X.s/ D.sI A/ BU.s/ D.sI A/ B: So, via the Laplace initial-value theorem, x. C / D lim s! sx.s/ D lim s.si A/ B s! D lim I A B s! s D B: Thus, an impulse input brings the state to B from. What if u.t/ D ı.k/.t/? Then X.s/ D.sI A/ Bs k D I A Bs k s s D As A I C C s s C ::: Bs k ƒ holds for large s D Bs k C ABs k C A Bs k 3 CC Ak B s C AkC B s The first terms are impulsive: they have zero value for t>. C Lecture notes prepared by Dr. Gregory L. Plett. Copyright c 5,, 9,, 5, 3,,, Gregory L. Plett

6 ECE45/55, OBSERVABILITY AND CONTROLLABILITY 5 Thus, A x. C k B / D lim s s! s D A k B: So, if u.t/ D ı.k/.t/ then x. C / D A k B. Now, consider the input C AkC B s C u.t/ D g ı.t/ C g P ı.t/ Cgn ı.n/.t/: Since x. / D, x. C / D g B C g AB CCg n A n B,or 3 x. C / D B AB A n B 4 ƒ C where C is called the controllability matrix. CONCLUSION: For a SISO system, if C is nonsingular, then there is an impulsive input u such that x. C / is any desired vector if x. / D. EXTENSION: For a MIMO system, if C is full rank, then there is an impulsive input u such that x. C / is any desired vector if x. / D. In fact, we may use where 4 g g g 3 u.t/ D Xn id where x d is the desired x. C / vector. 3 g i ı.i/.t/ g g g 3 5 D B AB A n B xd 5 Lecture notes prepared by Dr. Gregory L. Plett. Copyright c 5,, 9,, 5, 3,,, Gregory L. Plett

7 ECE45/55, OBSERVABILITY AND CONTROLLABILITY 5 If C is nonsingular, we say fa; Bg is a controllable pair and the system is controllable. EXAMPLE: Controllability canonical form: 3 a 3 Px.t/ D 4 a a h i y.t/ D x.t/: Then ˇ ˇ ˇ3 5 x.t/ C 4 C D ŒB AB A n B 3 D 4 : :: 5 D I n : This is why it is called controllability form! 3 5 u.t/ If a system is controllable, we can instantaneously move the state from any known state to any other state, using impulse-like inputs. Later, we ll see that smooth inputs can effect the state transfer (not instantaneously, though!). DUALITY: fa; B; C; Dg controllable fa T ;C T ;B T ;D T g observable. EXAMPLE: Two uncontrollable networks. u x y u F x F x Lecture notes prepared by Dr. Gregory L. Plett. Copyright c 5,, 9,, 5, 3,,, Gregory L. Plett

8 ECE45/55, OBSERVABILITY AND CONTROLLABILITY 5 8 In the first one, if x./ D then x.t/ D 8 t. Cannotinfluencestate! In the second one, if x./ D x./ then x.t/ D x.t/ independently alter state. 8 t. Cannot Diagonal systems, controllability and observability Recall the diagonal form 3 3 Px.t/ D : 4 :: 5 x.t/ C 4 : 5 u.t/ n n h i h i y.t/ D ı ı ı n x.t/ C u.t/: R x.t/ ı u.t/ : y.t/ n R x n.t/ ı n n When controllable? When observable? 3 C ı ı ı n O D CA 4 : 5 D ı ı n ı n 4 ::: CA n n ı n ı n n ı n 3 5 Lecture notes prepared by Dr. Gregory L. Plett. Copyright c 5,, 9,, 5, 3,,, Gregory L. Plett

9 ECE45/55, OBSERVABILITY AND CONTROLLABILITY ı D n ı 4 ::: : 5 4 :: 5 : n n n n ı n ƒ Vandermonde matrix V Singular? detfog D.ı ın/ detfvg D.ı ın/ Y. j i /: CONCLUSION: Observable i j, i j and ı i id ; ;n. i<j s C x.t/ s C x.t/ u.t/ y.t/ u.t/ y.t/ s C x.t/ s C x.t/ If D then not observable. Can only observe the sum x C x. If ık D then cannot observe mode k. What about controllability? Use duality and switch ıs ands. CONCLUSION: Controllable i j, i j and i id ; ;n. s C x.t/ s C x.t/ u.t/ y.t/ u.t/ y.t/ s C x.t/ s C x.t/ If D then not controllable. Can only control the sum x C x. If k D then cannot control mode k. Lecture notes prepared by Dr. Gregory L. Plett. Copyright c 5,, 9,, 5, 3,,, Gregory L. Plett

10 ECE45/55, OBSERVABILITY AND CONTROLLABILITY 5 5.3: Discrete-time controllability and observability Discrete-time controllability Similar concept for discrete-time. Consider the problem of driving a system to some arbitrary state xœn xœk C D AxŒk C BuŒk xœ D AxŒ C BuŒ xœ D AŒAxŒ C BuŒ C BuŒ xœ3 D A A xœ C ABuŒ C BuŒ C BuŒ : 3 h i uœn xœn D A n xœ C B AB A B A n B 4 : 5 : ƒ C uœ Which leads to 4 uœn : uœ 3 5 D C ŒxŒn A n xœ : If C has no inverse (det.c/ D ; C is not full-rank) then these control signals don t exist. In that case, the input is only partially effective in influencing the state. If C is full-rank, then the input can move the system to any arbitrary state for any xœ. NOTE I: State transition is not instantaneous. Takes n time steps. Lecture notes prepared by Dr. Gregory L. Plett. Copyright c 5,, 9,, 5, 3,,, Gregory L. Plett

11 ECE45/55, OBSERVABILITY AND CONTROLLABILITY 5 NOTE II: In continuous-time, we used input u.t/ D g ı.t/ C g ı.t/ P C,a signal we could only approximate in practice. Here, the input isa perfectly good input signal. Discrete-time reachability In the literature, there are three different controllability definitions:. Transfer any state to any other state.. Transfer any state to zero, called controllability to the origin. 3. Transfer the zero state to any state, called controllability from the origin, orreachability. In continuous time, because e At is nonsingular, the three definitions are equivalent. In discrete time, if A is nonsingular, the three definitions are also equivalent. However, if A is singular, () and (3) are equivalent but not () and (3). EXAMPLE: xœk C D xœk C uœk : Its controllability matrix has rank and the equation is not controllable in () or (3). However, A k D for k 3 so xœ3 D A 3 xœ D for any initial state xœ and any input uœk. Thus, the system is controllable to the origin but not controllable from the origin or reachable. Lecture notes prepared by Dr. Gregory L. Plett. Copyright c 5,, 9,, 5, 3,,, Gregory L. Plett

12 ECE45/55, OBSERVABILITY AND CONTROLLABILITY 5 Definition () encompasses the other two definitions, so is used as our definition of controllable. Discrete-time observability Can we reconstruct the state xœ from the output yœk and input uœk? yœk D CxŒk C DuŒk yœ D CxŒ C DuŒ yœ D CŒAxŒ CBuŒ C DuŒ yœ D C A xœ C ABuŒ C BuŒ C DuŒ : yœn D C A n xœ C A n BuŒ CCBuŒn C DuŒn : In vector form, we can write yœ C D uœ yœ 4 : 5 D CA 4 : xœ C CB D uœ 5 4 CAB CB 5 4 : yœn CA n : : ::: D uœn ƒ ƒ O T So, 3 33 yœ uœ xœ D O 44 : 5 T 4 : 55 : yœn uœn If O is full-rank or nonsingular, xœ may be reconstructed with any yœk ; uœk. WesaythatfC;Ag form an observable pair. 3 5 : Lecture notes prepared by Dr. Gregory L. Plett. Copyright c 5,, 9,, 5, 3,,, Gregory L. Plett

13 ECE45/55, OBSERVABILITY AND CONTROLLABILITY 5 3 Do more measurements of yœn ; yœn C ; : : : help in reconstructing xœ? No!(Caley Hamiltontheorem:nextsection). So, if the original state is not observable with n measurements, then it will not be observable with more than n measurements either. Since we know uœk and the dynamics of the system, if the system is observable we can determine the entire state sequence xœk, k once we determine xœ xœn D A n xœ C Xn id A n i BuŒk yœ uœ uœn D A n O 44 : 5 T 4 : 55 C C 4 : 5 : yœn uœn uœ Aperfectlygoodobserver(nodifferentiators...) Lecture notes prepared by Dr. Gregory L. Plett. Copyright c 5,, 9,, 5, 3,,, Gregory L. Plett

14 ECE45/55, OBSERVABILITY AND CONTROLLABILITY : Cayley Hamilton Theorem AsquarematrixA satisfies its own characteristic equation. That is, if./ D det.i A/ D then.a/ D : We can easily show this if A is diagonalizable. Let A D V ƒv: Then A D V ƒv V ƒv D V ƒ V A k D V ƒ k V: The characteristic polynomial is: so if we replace with A we get./ D n C a n n CCa.A/ D A n C a n A n CCa I D V ƒ n C a n ƒ n CCa I V: To prove the Cayley Hamilton theorem, we just need to show that the quantity inside the brackets is zero. It is adiagonalmatrix,andeachdiagonalelementhastheform because i is an eigenvalue of A. n i C a n n i CCa D Lecture notes prepared by Dr. Gregory L. Plett. Copyright c 5,, 9,, 5, 3,,, Gregory L. Plett

15 ECE45/55, OBSERVABILITY AND CONTROLLABILITY 5 5 So, each diagonal element is zero, and we have shown the proof. If A is not diagonalizable, the same proof may be repeated using the Jordan form and Jordan blocks: A D T JT: Consider a sketch of the proof for a Jordan block of size and. i / D 3 i C a i C a i C a D : Then J i D.J i / D " " i i 3 i 3 i 3 i " C a i i i C a " i i C a " : We can easily see that the diagonal and lower-diagonal components are zero so where.j i / D " D 3 i C a i C a but D d./ D which completes the sketch. d SIGNIFICANCE: The Cayley Hamilton theorem shows us that A n is a function of matrix powers A n down to A.Therefore,tocomputeany polynomial of A it suffices to compute only powers of A up to A n and appropriately weight their sum. A lot of proofs use the Cayley Hamilton theorem. Lecture notes prepared by Dr. Gregory L. Plett. Copyright c 5,, 9,, 5, 3,,, Gregory L. Plett

16 ECE45/55, OBSERVABILITY AND CONTROLLABILITY 5 As we just saw with the section on discrete-time observability, the Cayley Hamilton theorem implies that if we cannot observe the state with n measurements, we cannot observe it with more measurements either. " EXAMPLE: With A D we have./ D det.i A/, so 3 4 "!./ D det A./ D 5.A/ D A 5A I " " D 5 5 " D : 3 4 " Disclaimer: Does observability/controllability matter, practically? The singularity of C has only one bit of information: Is the realization mathematically controllable or not? This may not tell the whole story. " " " A D ; B D ; C D C C fa; Bg are a controllable pair, but barely. EXAMPLE: Controlling an airplane. (Ideas only, no details). System state h x D 4 P i T P ; D Pitch; D Roll: Control with elevator? Lecture notes prepared by Dr. Gregory L. Plett. Copyright c 5,, 9,, 5, 3,,, Gregory L. Plett

17 ECE45/55, OBSERVABILITY AND CONTROLLABILITY 5 " Px D 3 F x C F 4 5 ı e where ı e is the elevator angle. C is singular can t influence roll with elevators. Control with ailerons? Px D " F F x C ı a where ı a is the aileron angle. C is nonsingular! So, we can control both pitch AND roll with ailerons. THIS IS NONSENSE! Physicallythinkofthesystem!Doyouwantto roll plane over every time you need to pitch down? Physical intuition can be better than finding C. Other tools can help... Lecture notes prepared by Dr. Gregory L. Plett. Copyright c 5,, 9,, 5, 3,,, Gregory L. Plett

18 ECE45/55, OBSERVABILITY AND CONTROLLABILITY : Continuous-time Gramians Continuous-time controllability Gramian If a continuous-time system is controllable, then is nonsingular for t>. W c.t/ D Z t e A BB T e AT d SIGNIFICANCE: Consider x.t / D e At x./ C Z t e A.t / Bu./d: We claim that for any x./ D x and any x.t / D x the input u.t/ D B T e AT.t t/ Wc.t / e At x x will transfer x to x at time t. PROOF: Substitute the expression for u.t/ into the convolution expression: x.t / D e At x./ D e At x./ Z t Z t e A.t / BB T e AT.t / dwc.t / e At x x e AˇBB T e AT ˇ dˇ Wc.t / e At x x D e At x./ W c.t /Wc.t / e At x x D e At x./ e At x C x D x : Therefore, we can compute the input u.t/ required to transfer the state of the system from one state to another over an arbitrary interval of time. The solution is also the minimum-energy solution. Lecture notes prepared by Dr. Gregory L. Plett. Copyright c 5,, 9,, 5, 3,,, Gregory L. Plett

19 ECE45/55, OBSERVABILITY AND CONTROLLABILITY 5 9 EXAMPLE: Consider the system in diagonal form " " :5 :5 Px.t/ D x.t/ C u.t/: The controllability matrix is: C D " :5 :5 which has rank, sothesystemiscontrollable. Consider the input required to move the system state from x./ D ΠT to zero in two seconds. Z " " e :5 :5 h W c./ D e :5 " : :3 D ; :3 :498 and h u.t/ D :5 i " e :5. t/ e. t/ D 58:8e :5t C :9e t : W c./ " i " e :5 e e e If a continuous-time system is controllable, and if it is also stable,then W c D Z e A BB T e AT d can be found by solving for the unique (positive-definite) solution to the (Lyapunov) equation AW c C W c A T D BB T : W c is called the controllability Gramian. Lecture notes prepared by Dr. Gregory L. Plett. Copyright c 5,, 9,, 5, 3,,, Gregory L. Plett "! d

20 ECE45/55, OBSERVABILITY AND CONTROLLABILITY 5 PROOF: We proved this identity when considering Lyapunov stability. Wc measures the minimum energy required to reach a desired point x starting at x./ D (with no limit on t) ( Z ) t min ku./k d x./ D ; x.t/ D x ˇ D x T W c x : In fact, for any specific t, the minimum energy is x T W c.t/x. If A is stable, W c >which implies we can t get anywhere for free. If A is unstable, then W c can have a nonzero nullspace Wc D for some which means that we can get to using u s with energy as small as you like! (u just gives a little kick to the state; the instability carries it out to efficiently). Wc may be a better indicator of controllability than C. Continuous-time observability Gramian If a system is observable, Wo.t/ is nonsingular for t>where W o.t/ D SIGNIFICANCE: We can prove that where Ny.t/ D y.t/ C Z t x./ D W o.t / Z t e AT C T Ce A d: Z t e AT t C T Ny.t/dt e A.t / Bu./d Du.t/ D Ce At x./: Therefore, we can determine the initial state x./ given a finite observation period (and not use differentiators!). Lecture notes prepared by Dr. Gregory L. Plett. Copyright c 5,, 9,, 5, 3,,, Gregory L. Plett

21 ECE45/55, OBSERVABILITY AND CONTROLLABILITY 5 PROOF: We prove that the above equations are correct by substitution: W o.t / Z t e AT t C T Ny.t/dt D W o.t / Z t D Wo.t /W o.t /x./ D x./: e AT t C T Ce At dtx./ If a continuous-time system is observable, and if it is also stable, then W o D Z e AT C T Ce A d is the unique (positive-definite) solution to the (Lyapunov) equation A T W o C W o A D C T C: W o is called the observability Gramian. This relationship can be proven in the same way we proved the similar relationship for the controllability gramian. If measurement (sensor) noise is IID N.; I/ then W o is a measure of error covariance in measuring x./ from u and y over longer and longer periods lim E k Ox./ t! x./k D x./ T Wo x./: If A is stable, then W o >and we can t estimate the initial state perfectly even with an infinite number of measurements u.t/ and y.t/ for t (since memory of x./ fades). If A is not stable then W o can have a nonzero nullspace Wo x./ D which means that the covariance goes to zero as t!. Wo may be a better indicator of observability than O. Lecture notes prepared by Dr. Gregory L. Plett. Copyright c 5,, 9,, 5, 3,,, Gregory L. Plett

22 ECE45/55, OBSERVABILITY AND CONTROLLABILITY 5 5.: Discrete-time Gramians Discrete-time controllability Gramian In discrete-time, if a system is controllable, then W dc Œn D is nonsingular. In particular, W dc D Xn md A m BB T.A T / m X A m BB T.A T / m md is called the discrete-time controllability Gramian and is the unique positive-definite solution to the Lyapunov equation W dc AW dc A T D BB T : As with continuous-time, Wdc measures the minimum energy required to reach a desired point x starting at xœ D (with no limit on m) ( m ) X min kuœk k ˇˇˇˇˇ xœ D ; xœm D x D x T W dc x : kd ASIDE: When considering discrete-time stability, we showed that this form of equation is indeed a Lyapunov equation. Discrete-time observability Gramian In discrete-time, if a system is observable, then W do Œn D is nonsingular. In particular, Xn.A T / m CC T A m md Lecture notes prepared by Dr. Gregory L. Plett. Copyright c 5,, 9,, 5, 3,,, Gregory L. Plett

23 ECE45/55, OBSERVABILITY AND CONTROLLABILITY 5 3 W do D X.A T / m CC T A m md is called the discrete-time observability Gramian and is the unique positive-definite solution to the Lyapunov equation W do A T W do A D C T C: As with continuous-time, if measurement (sensor) noise is IID N.; I / then W do is a measure of error covariance in measuring xœ from u and y over longer and longer periods lim E k OxŒ t! xœ k D xœ T Wdo xœ : Lecture notes prepared by Dr. Gregory L. Plett. Copyright c 5,, 9,, 5, 3,,, Gregory L. Plett

24 ECE45/55, OBSERVABILITY AND CONTROLLABILITY : Computing transformation matrices Transformation to controllability form Given a system Px.t/ D Ax.t/ C Bu.t/ y.t/ D Cx.t/C Du.t/; can we find a transformation (similarity) matrix T to transform this system into controllability form? Recall, this looks like: 3 3 a n Px co.t/ D : a n 4 : ::: : 5 x co.t/ C 4 : 5 u.t/ a h i y.t/ D x co.t/ C Du.t/: ˇ ˇ ˇn Note that Cco D I so it is controllable. Thus, our original system must be controllable. The transformation is accomplished via Thus x D Tx co : T AT D A co ; T B D B co CT D C co ; D D D co : Let s find T explicitly; let T D Œt ;:::;t n : Lecture notes prepared by Dr. Gregory L. Plett. Copyright c 5,, 9,, 5, 3,,, Gregory L. Plett

25 ECE45/55, OBSERVABILITY AND CONTROLLABILITY 5 5 Note that B D TBco or 3 B D T 4 : 5 D t : So, t D B. FromAT D TA co we have AŒt ;:::;t n D Œt ;:::;t n 4 a n : a n : ::: : a 3 5 : So " nx ŒAt;:::Atn D t ;:::;t n ; a k t n kc : kd By induction, At D t D AB At D t 3 D A B; and so forth ::: so T D ŒB AB A n B D C: CONCLUSION: AsystemfA; B; C; Dg can be transformed to controllability canonical form if and only if it is controllable, in which case the change of coordinates is x D Cx co : EXTENSION I: If x old D Tx new then T D C old C new.thatis,toconvert between any two realizations, T is a combination of the controllability Lecture notes prepared by Dr. Gregory L. Plett. Copyright c 5,, 9,, 5, 3,,, Gregory L. Plett

26 ECE45/55, OBSERVABILITY AND CONTROLLABILITY 5 matrices of the two different realizations. C new D B new A new B new Anew n B new or D T B old D T C old ; T A old TT B old T A n old TT B old T D C old C new : EXTENSION II: If x old D Tx new then T D O old O new. Thiscanbeshownina similar way. Lecture notes prepared by Dr. Gregory L. Plett. Copyright c 5,, 9,, 5, 3,,, Gregory L. Plett

27 ECE45/55, OBSERVABILITY AND CONTROLLABILITY 5 5.8: Canonical (Kalman) decompositions What happens if fa; Bg not controllable or if fa; C g not observable? Is part of the system controllable? Is part of the system observable? Given a system with fa; Bg not controllable, Px.t/ D Ax.t/ C Bu.t/, y.t/ D Cx.t/,let strytotransformtocontrollabilitycanonicalform. Let t D B; t D AB; : : : t r D A r B,andsupposet ;t ;:::;t r are independent but for some constants ;:::; r. t rc D A r B D rt t r Then rank.c/ D r since the vectors A r B; A rc B;:::;A n B can all be expressed as a linear combination of t ;t ;:::;t r. Let src ;:::;s n be your favorite vectors for which h i CN old D t t r s rc s n is invertible. If we were able to change coordinates to controllability form via x old D Tx new,wewoulduset D C old Cnew. But, the system is not controllable, so we use T D CN old Cnew D CN old. We get (in the new coordinate system) " " " " Pxc.t/ Ac A xc.t/ Bc D C u.t/ Px Nc.t/ A Nc x Nc.t/ i y.t/ D hc " x c.t/ c C Nc C Du.t/: x Nc.t/ Lecture notes prepared by Dr. Gregory L. Plett. Copyright c 5,, 9,, 5, 3,,, Gregory L. Plett

28 ECE45/55, OBSERVABILITY AND CONTROLLABILITY 5 8 Ac is a right-companion matrix and B c is of the controllability-canonical form. We see that the uncontrollable modes x Nc are completely decoupled from u.t/. EXAMPLE: Consider s C D s.s C /.s / D s s : In observer-canonical form, " " Px.t/ D x.t/ C u.t/ h i y.t/ D x.t/: So, t D " and t D At D " D t : So, rank.c/ D. Lets D ΠT.Theconvertedstate-spaceformis " " PNx.t/ D Nx.t/ C u.t/ h i y.t/ D Nx.t/: Now suppose that we desire to transform a system that is not observable. The dual form separates out the unobservable states. Let t D C; t D CA; : : : t r D CA r,andsupposet ;t ;:::;t r are independent but t rc D CA r D rt t r for some constants ;:::; r. Lecture notes prepared by Dr. Gregory L. Plett. Copyright c 5,, 9,, 5, 3,,, Gregory L. Plett

29 ECE45/55, OBSERVABILITY AND CONTROLLABILITY 5 9 Then, let src ;:::;s n be your favorite row vectors for which 3 t : t r NO old D s rc 4 : 5 s n is invertible. If we were able to change coordinates to observability form via x old D Tx new,wewoulduset D O old O new. But, the system is not observable, so we use T D NO old O new D NO old. Then, we get the Kalman decomposition " " " " Pxo.t/ Ao xo.t/ Bo D C u.t/ Px No.t/ A A No x No.t/ B No h i " x o.t/ y.t/ D C o C Du.t/: x No.t/ Note: No path from x No to y! Full Kalman decomposition We can do both transformations in sequence to get full Kalman decomposition. See text chapter. Lecture notes prepared by Dr. Gregory L. Plett. Copyright c 5,, 9,, 5, 3,,, Gregory L. Plett

30 ECE45/55, OBSERVABILITY AND CONTROLLABILITY : Popov Belevitch Hautus controllability/observability tests PBH EIGENVECTOR TEST: fc;ag is an unobservable pair iff a non-zero eigenvector v of A satisfies Cv D. (i.e., C and v are perpendicular). PROOF: ) Suppose Av D v and Cv D for v. Then CAv D C v D and so forth up to CA n v D n Cv D. So, Ov D and since v this means that O is not full rank and fc;ag is not observable. ( Now suppose that O is not full rank (fc;ag unobservable). Let s extract the unobservable part. That is, find T such that " T A o AT D CT D h A C o where A o is size r where r D rank.o/ and therefore A No is size n r. A No i ; Let v be an eigenvector of A No.Then " " " T A o AT D v A A No v D " v so we have found an eigenvector of A A D where D T " v : Now, we just need to show that C D (note: ). " h i " C D ƒ CT D C o D v NC v and we are done. Lecture notes prepared by Dr. Gregory L. Plett. Copyright c 5,, 9,, 5, 3,,, Gregory L. Plett

31 ECE45/55, OBSERVABILITY AND CONTROLLABILITY 5 3 DUAL: fa; Bg is uncontrollable iff there is a left eigenvector w T of A such that w T B D. INTERPRETATION: In modal coordinates, homogeneous response is nx x.t/ D e it v i.wi T x.// and y.t/ D Cx.t/: Or, id y.t/ D nx e it Cv i.wi T x.//: id If fc;ag is unobservable, then it has an unobservable mode, where Av i D i v i and Cv i D. If fa; Bg is uncontrollable, then it has an uncontrollable mode, namely the coefficients of the state along that mode is independent of the input u.t/. The coefficients of x in the mode associated with are w T x. d dt.wt x/ D w T.Ax C Bu/ D.w T x/ or regardless of the input u.t/! w T x.t/ D e t.w T x.// PBH RANK TESTS: The following two tests are often easier to perform h i. fa; Bg controllable iff rank si A B D n for all s C. h i ) If rank si A B h D n for all s C then there can be no nonzero i h i vector v such that v T si A B D v T.sI A/ v T B D : Lecture notes prepared by Dr. Gregory L. Plett. Copyright c 5,, 9,, 5, 3,,, Gregory L. Plett

32 ECE45/55, OBSERVABILITY AND CONTROLLABILITY 5 3 Consequently, there is no nonzero vector v such that v T s D v T A and v T B D. BythePBHeigenvectortest,thesystemwill therefore be controllable ( If the system is controllable, then there is no nonzero vector v such that v T s D v T A and v T B D by the PBH eigenvector test. i Therefore, rank hsi A B D n for all s C. ". fc;ag observable iff rank C si A D n for all s C. Proofsimilar. h i si A B D n for all s not eigenvalues of A, sothe h i test is really fa; Bg controllable iff rank i I A B D n for i, i D ; : : : ; n the eigenvalues of A. (Dualargumentforobservability). i If hsi A B drops rank at s D then there is an uncontrollable mode with exponent (frequency). " C If drops rank at s D then there is an unobservable si A mode with exponent (frequency). COMMENTS: rank Summary Therefore, we can label individual modes of a system as either controllable or not, or observable or not. The overall picture is: Lecture notes prepared by Dr. Gregory L. Plett. Copyright c 5,, 9,, 5, 3,,, Gregory L. Plett

33 ECE45/55, OBSERVABILITY AND CONTROLLABILITY 5 33 u.t/ Controllable and observable y.t/ Controllable but not observable Observable but not controllable Some other definitions: Neither observable nor controllable STABILIZABLE: Asystemwhoseunstablemodesarecontrollable. DETECTABLE: Asystemwhoseunstablemodesareobservable. Lecture notes prepared by Dr. Gregory L. Plett. Copyright c 5,, 9,, 5, 3,,, Gregory L. Plett

34 ECE45/55, OBSERVABILITY AND CONTROLLABILITY : Minimal realizations Why a system isn t control/observ-able To realize H.s/ D s C we could use. Px.t/ D x.t/ C u.t/, y.t/ D x.t/. Thisgives A D Œ, B D Œ, C D Œ, D D Œ. C adj.si A/B det.si A/ D s C : This realization is both controllable and observable. s. Observer realization of s C s D s s.thisgives " " h i A D, B D, C D, D D Œ. C adj.si A/B D s det.si A/ s D s C : " " C O D D.Observable. CA " h i C D B AB D.Notcontrollable. s C 3. Controller realization of s C s C D s C s C s C. " " h i A D, B D, C D, D D Œ. C D C adj.si A/B s C D det.si A/ s C s C D s C : " h i B AB D.Controllable. Lecture notes prepared by Dr. Gregory L. Plett. Copyright c 5,, 9,, 5, 3,,, Gregory L. Plett

35 ECE45/55, OBSERVABILITY AND CONTROLLABILITY 5 35 O D " C CA D ".Notobservable. TREND: Non-minimal realizations of transfer functions will either be uncontrollable, unobservable, or both. Four equivalent statements: I: There exist common roots of C adj.si A/B and det.si A/. II: There exist eigenvalues of A which are not poles of G.s/, counting multiplicities. III: The system is either unobservable or uncontrollable. IV: There exist extra (unnecessary) states non minimal. DEFINITION: We say a system is minimal if no system with the same transfer function has fewer states. PROOF: I II IH) II: The transfer function G.s/ D C.sI A/ B C D D C adj.si A/B C D det.si A/ : det.si A/ If there are common roots in C adj.si A/B and det.si A/ they will cancel out of the transfer function. But, eigenvalues of A are det.si A/ D, sopolesofg.s/ will not contain all eigenvalues of A. IIH) I: Eigenvalues of A are det.si A/ D. PolesofG.s/, fromabove, are det.si A/ D unless canceled. The only way to cancel a pole is to have common root in C adj.si A/B. Lecture notes prepared by Dr. Gregory L. Plett. Copyright c 5,, 9,, 5, 3,,, Gregory L. Plett

36 ECE45/55, OBSERVABILITY AND CONTROLLABILITY 5 3 PROOF: I IV fa; B; C; Dg is minimal iff C adj.si A/B and det.si A/ are coprime (have no common roots). IH) IV: Suppose C adj.si A/B and det.si A/ have common roots. Cancel to get G.s/ D C adj.si A/B det.si A/ C D D b r.s/ a r.s/ where b r.s/ and a r.s/ are coprime ( r means reduced ). Because of cancellation, k D deg.ar /<deg.det.si A// D n. Consider controller canonical form realization of br.s/=a r.s/, for example. It has k states, but same transfer function as fa; B; C; Dg, contradicting that fa; B; C; Dg minimal. IVH) I: If there are extra states (non-minimal) then n>k(n D states, k D polesing.s/). G.s/ D C.sI A/ B C D D C adj.si A/B C D det.si A/ det.si A/ G.s/ has n poles unless some cancel with C adj.si A/B. Therefore, if n>k, C adj.si A/B and det.si A/ are not coprime. PROOF: III IV Controllable and observable iff minimal. IIIH) IV: Uncontrollable or unobservable H) not minimal. Perform Kalman decomposition to split system into co, c No, Nco and Nc No parts. Lecture notes prepared by Dr. Gregory L. Plett. Copyright c 5,, 9,, 5, 3,,, Gregory L. Plett

37 ECE45/55, OBSERVABILITY AND CONTROLLABILITY 5 3 u.t/ B c R x c.t/ C c A c Uncontrollable part of realization A R x c.t/ C Nc y.t/ N A D " A c A, NB D " B c, NC D A Nc h C c C Nc i, ND D Œ. A Nc Same transfer function using A c, B c, C c as A, N NB, NC.Therefore uncontrollable and/or unobservable means not minimal. IVH) III: Non-minimal means uncontrollable or unobservable. Suppose fa; B; C; Dg is non-minimal. C.sI A/ B C D ƒ n states D NC.sI N A/ NB C ND ƒ r<nstates CB C CAB s s Consider OC D 4 C CA B s 3 C CA CA : CA n 3 CD N C NB s C C N AN NB s C C N AN NB s 3 h i B AB A B A n B 5 C Lecture notes prepared by Dr. Gregory L. Plett. Copyright c 5,, 9,, 5, 3,,, Gregory L. Plett

38 ECE45/55, OBSERVABILITY AND CONTROLLABILITY CB CA n B D 4 : : CA n B CA n B x 3 NC NC N A D n : 4? NC AN n 5 4 y NC AN n!! r n r 5 D NB 4 NC NB NC N A n NB : NC AN n NB AN NB AN n NB! n : NC AN n NB 3 5 x r? y x n r? y Therefore det.o/ det.c/ D, sothesystemiseitherunobservable, or uncontrollable, or both. The four equivalences have now been proven. FACT: All minimal realization of G.s/ are related by a unique change of coordinates T.Canyouprovethis? Where to from here? Have seen important topics of observability and controllability. Now understand how a system could be unobservable or uncontrollable. If physically true, may need to add sensors or actuators. Important to work with minimal realizations, when possible. Now, time to consider: How do I build a controller? Very powerful tools in next chapter. 3 5 Lecture notes prepared by Dr. Gregory L. Plett. Copyright c 5,, 9,, 5, 3,,, Gregory L. Plett