Sami Fadali Professor of Electrical Engineering UNR Outline Controllability Observability Stabilizability Detectability Identical tests for CT and DT systems Controllability Definition 84 An LTI system is controllable if for any initia ID: 27926 Download Pdf

Sami Fadali Professor of Electrical Engineering UNR Outline Controllability Observability Stabilizability Detectability Identical tests for CT and DT systems Controllability Definition 84 An LTI system is controllable if for any initia

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Contollability & Observability M. Sami Fadali Professor of Electrical Engineering UNR Outline • Controllability. • Observability. • Stabilizability. • Detectability. • Identical tests for CT and DT systems. Controllability Definition 8.4 • An LTI system is controllable if for any initial state there exists a control sequence such that an arbitrary final state can be reached in finite time. Uncontrollable State is uncontrollable if it is orthogonal to the zero state response for all and all inputs • Inputs can only drive the system in directions orthogonal to the uncontrollable

states. • Integrand identically zero 2 1 Controllable Subspace Uncontrollable Subspace

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Theorem 8.4: Controllability • An LTI system is controllable if and only if the products left eigenvector, input matrix. mode is uncontrollable. Proof: Necessity • Zero-input response • Can only decay to zero asymptotically, not in finite time. • Each mode must be influenced by the input to go to zero in finite time. • We need Proof: Necessity (Cont.) only if • If the the mode is uncontrollable. Cayley-Hamilton Theorem • Every matrix satisfies its own characteristic equation. • By induction,

all higher powers can be written in terms of can be expressed in terms of

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Proof: Sufficiency Proof: Sufficiency (Cont.) is full rank given that is full rank and are nonzero. are all rank 1 and are linearly independent provided if • Can solve the following equation for the input sequence that drives the system to a specified final state, but nonuniquely 10 11 Controllability Rank Condition Theorem 8.5: A LTI system is completely controllable if and only if the controllability matrix has rank Proof • Use the Cayley-Hamilton Theorem to write Solution for exists if and only if 12

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Example Determine the controllability of the system 13 Solution • The controllability matrix of the system is • Controllability matrix has rank 3: controllable. • First 3 columns of matrix linearly independent: sufficient to conclude controllability. • In general, compute more columns until linearly independent columns are obtained. 14 15 Theorem 8.6: Controllability of Systems in Normal Form A system in normal form is controllable if and only if its input matrix has no zero rows. Furthermore, if the input matrix has a zero row then the corresponding mode in uncontrollable.

Proof:Necessity • The diagonal form is equivalent to: Necessity: • If then the system can only converge to zero asymptotically. • For controllability we must have convergence in finite time. • If the mode is not controllable. 16

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Proof: Sufficiency • If has no zero rows the matrix is full rank • We can find a control to go to any 17 MATLAB • Controllability matrix: same for CT and DT >> A=[[0;0],eye(2);-6,-11,-6]; B=[1;-1;1]; >> Cc=ctrb(A,B) >> rank(Cc) ans = (2 uncontrollable modes) • For diagonal form use ss2ss • For the eigenvectors use eig with A’ (rows) 18 Transfer

Function (not reduced) >> g=zpk(ss(A,B,C,0)) Zero/pole/gain from input to output... 12 (s+3) (s+2) #1: ----------------- (s+3) (s+2) (s+1) (s+3) (s+2) #2: ----------------- (s+3) (s+2) (s+1) 19 Cancel Poles and Zeros >> minreal(g) Transfer function from input to output... 12 #1: ----- s + 1 #2: ----- s + 1 20

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Stabilizability A system is stabilizable if all its uncontrollable modes decay to zero asymptotically. Stabilizable: all unstable modes are controllable. • Stability and controllability: independent properties • Physical systems are often stabilizable but not

controllable: not a problem if the uncontrollable dynamics decay to zero sufficiently fast. 21 22 Observability A system is said to be observable if any initial state can be estimated from the control sequence and the measurements 23 Unobservable States (orthogonal) • Unobservable state • All vectors are indistinguishable • Unobservable state eigenvector so that for all the response remains zero (along ). 2 1 Observable Subspace Unobservable Subspace 24 Observability: Eigenvector Theorem 8.8: A system is observable if and only if is nonzero for where is the eigenvector of the state matrix.

Furthermore, if the product is zero then the mode is unobservable.

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Proof Sufficiency is nonzero and the matrix is full rank. For a full rank matrix we can solve for uniquely. 25 Proof: Necessity • Let be in the direction of indistinguishable 26 27 Observability Rank Test Theorem 8.9 A LTI system is completely observable iff the observability matrix has rank 28 Proof of Necessity • Assume observable with rank deficient matrix unobservable state ( indistinguishable) and system cannot be observable: contradicts observability assumption. • Rank deficit= number of eigenvectors s.t.

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29 Proof of Sufficiency • Assume full rank observability matrix then • No unobservable states hence observable. 30 Theorem 8.10: Observability for Normal Form A system in normal form is observable if and only if its output matrix has no zero columns. Furthermore, if the output matrix has a zero row column the corresponding mode in unobservable. Proof • Recall that the column of the output matrix of the diagonal form is given by • The result follows from the eigenvector test. • Recall: For normal form each state variable associated with a different mode. 31 32 Example 8.10

Determine the observability of the discrete-time system using two different tests. If the system is not completely observable, determine the unobservable modes.

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33 Solution • State matrix in companion form. • Characteristic equation and modal matrix output-decoupling zero at zero, i.e. one unobservable mode. The unobservable mode is stable (inside the unit circle): detectable system 34 MATLAB Commands % Calculate observability matrix >>o = obsv(A, C) » rank(o) % Find the rank of the matrix. 35 Example: Eigenvector Test • Check the controllability and observability of the CT

system 10 11 36 Example Continued: Controllability le controllab WB 0841 4361 1429 75 1.0841 1.4361 4.3084 1.1429 0.25 25 11

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37 Example Continued: Observability le unobservab CV 0.9225 0.9225 2.4371 10 10 0.9225 0.3075 0.2321 0.1757 0.9284 0.1537 0.2901 11 38 Example: Rank Test >@ >@ le controllab rank AB 1331 402 96 169 121 32 11 39 Detectability • A system is detectable if all its unobservable modes decay to zero asymptotically. • Detectable: all unstable modes are observable. • Observability and stability: independent properties. • Physical systems are typically detectable

but not observable: not a problem if the unobservable modes decay to zero sufficiently fast. 40 Example • Normal form: zero rows in uncontrollable, unobservable. not stabilizable. not detectable. >@

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41 Example: Diagonal Form >@ le unobservab ble uncontrola 11 100 11 42 Important Relations • Internally stable systems are stabilizable and detectable: no unstable modes. • Observable systems are detectable: no unobservable modes. • Controllable systems are stabilizable: no uncontrollable modes. • For minimal realizations, BIBO stability and internal stability are equivalent.

Kalman Decomposition • Any system can be decomposed into four subsystems as shown in the figure: Controllable Observable Uncontrollable Observable Controllable Unobservable Uncontrollable Unobservable 43 Duality 1. controllable (stabilizable) is observable (detectable). 2. is observable (detectable) is controllable (stabilizable). • Transposing the controllability matrix of gives the observability matrix of and both have the same rank. Use the rank tests. • Duality used in controller/observer design. 44

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Duality Example Determine the controllability of the dual of the

discrete-time system of Example 8.10. Also determine the uncontrollable modes of the dual system. 45 46 Dual Pair • The dual system is 47 Rank Test • The controllability matrix • Transpose of the observability matrix of Example 8.10 • Rank = 2: One uncontrollable mode. Uncontrollable Modes • From Example 8.10 for the original system • Left eigenvectors of • The first mode is not controllable but is inside the unit circle: stabilizable system. 48

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