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1 Chap 4. State-Space Solutions and Realizations

2 Outlines 1. Introduction 2. Solution of LTI State Equation 3. Equivalent State Equations 4. Realizations 5. Solution of Linear Time-Varying (LTV) Equations 6. Equivalent Time-Varying Equations 7. Time-Varying Realizations 2

3 1.Introduction How to find the solution for a linear system Convolution Discrete version LTI systems Transfer function, involves Laplace transform, poles and zeros, inverse Laplace transform They are inconvenient and inaccurate

4 2. Solution of LTI State Equations Find the solution excited by x(0) and u(t) Solution Preliminary

5 2. Solution of LTI State Equations Verification Initial state General solution applying

6 2. Solution of LTI State Equations How to compute e At Alternatively,

7 2. Solution of LTI State Equations

8 2. Solution of LTI State Equations Discretization

9 2. Solution of LTI State Equations Discretization

10 2. Solution of LTI State Equations Computation of B d If A is nonsingular Matlab command c2d

11 2. Solution of LTI State Equations Convergence of zero-input response A k x[0] E.g.,

12 2. Solution of LTI State Equations Convergence of zero-input response A k x[0]

13 3. Equivalent State Equations Equivalent state

14 3. Equivalent State Equations The same eigenvalues The same transfer matrix Indeed equivalent state equations have the same Indeed, equivalent state equations have the same characteristic polynomial

15 3. Equivalent State Equations Zero-state equivalent The same transfer function matrix using Theorem Algebraic equivalence implies zero-state equivalence; however, zero-state equivalence does not imply Algebraic equivalence

16 3. Equivalent State Equations Canonical forms Matlab command: [ab, bb, cb, db, P] = canon(a, b, c, d, type ) Type = companion {A, b 1 } is controllable Type = model Diagonal with complex eigenvalues

17 3. Equivalent State Equations Canonical forms

18 4. Realization Transfer matrix and state-space equation Every LTI system has input-output description If it is lumped as well Unique transformation at o from SS to TM, How about the inverse problem Realization = from TM to SS

19 4. Realization Realizable Not every TM is realizable, e.g., a distributed system If TM is realizable, a e, it has infinitely many realizations, at not necessarily of the same dimension Theorem

20 4. Realization Proof: :

21 4. Realization :

22 4. Realization : See P102 for more details.

23 4. Realization A special case of p = 1 The controllable-canonical-form can be directly read out from y the coefficients of TM

24 4. Realization Example

25 4. Realization Example

26 5. Solution of LTV Equations LTV system LTI: scalar because vector because

27 5. Solution of LTV Equations LTV system LTV: scalar because vector? because but

28 5. Solution of LTV Equations Fundamental matrix For every initial state x i (t 0 ), there exists a unique solution x i (t). Let X(t) ( ) = [x 1 1( (t), x 2 2( (t),, x n( (t)]. If X(t( 0 ) is nonsingular, X(t) ( ) is the fundamental matrix, satisfying X(t 0 ) can be arbitrarily chosen, as long as it is nonsingular, X(t) is not unique X(t) is nonsingular for all t

29 5. Solution of LTV Equations State transition matrix Properties

30 5. Solution of LTV Equations Solution

31 5. Solution of LTV Equations Verification Initial condition Derivative

32 5. Solution of LTV Equations Output Zero-input Zero-state

33 5. Solution of LTV Equations Input-Output description Special case If A(t) has commutative property, such as diagonal or constant we have constant A

34 5. Solution of LTV Equations Discrete case System Transition matrix Response Input-output description

35 6. Equivalent Time-Varying Equations System (4.69) Let P(t) is nonsingular and both P(t) and its derivative are continuous for all t. Let Equivalence transformation (4.70) where

36 6. Equivalent Time-Varying Equations Relationship between fundamental matrices Proof Theorem

37 6. Equivalent Time-Varying Equations

38 6. Equivalent Time-Varying Equations Input-output t t description equivalence Block diagram

39 6. Equivalent Time-Varying Equations Conclusion

40 7. Time-Varying Realization Input-output description State t equation Impulse response matrix If such {A(t), B(t), C(t), D(t)} exists, is said realizable

41 7. Time-Varying Realization Theorem 4.5

42 7. Time-Varying Realization

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