Survey
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
Outline • Discrete-time (DT) state equation from solution of continuous-time state equation. • Expressions in terms of constituent matrices. • Solution of DT state equation. • Example. Discrete-Time State-Space Equations M. Sami Fadali Professor of Electrical Engineering UNR 1 Solution of State Equation • Analog systems with piecewise constant inputs over a sampling period: relate state variables at the end of each period by a difference equation. • Obtain difference equation from the solution of the analog state, over a sampling period . • Solution of state equation for initial time final time 2 Piece-wise Constant Input 1. Move input outside the integral. 2. Change the variable of integration , and Discrete-time state equation = state vector at time 3 4 State & Input Matrices Constituent Matrices discrete state matrix discrete input matrix (same orders as their continuous counterparts). • Discrete state matrix = state transition matrix of the analog system evaluated at the sampling period . • Properties of the matrix exponential: integral of the matrix exponential for invertible matrix • Use expansion of the matrix exponential in terms of the constituent matrices. • Eigenvalues of discrete state matrix related to those of the analog system. 5 Discrete-time State-space Representation Input Matrix • Discrete state & output equation. • Discrete-time state equation: approximately valid for a general input provided that the sampling vector period is sufficiently short. • Scalar integrands: easily evaluate integral. 1 , 1 6 0 , • Output equation evaluated at time 0 • Assume distinct eigenvalues (only one zero eigenvalue) 7 8 Example 7.15 Discrete state matrix . • Obtain the DT state equations for the system of Example 7.7 . . • for a sampling period T=0.01 s. • Solution: From Example 7.7, the state-transition matrix is 10 11 1 0 0 0 10 0 0 0 0 0 0 10 10 10 1 1 1 9 0 0 0 1 10 100 1 10 100 • Simplifies to 90 9 Discrete-time Input matrix . . 10 MATLAB . MATLAB command to obtain form » pd = c2d(p,0.01) Alternatively the matrices are obtained using the MATLAB commands » ad = expm(a * 0.05) » bd = a\ (ad-eye(3) )* b . • Simplifies to 11 12 Solution of DT State-Space Equation Solution by Induction 2 • DT State Equation: state at time in terms and the of the initial condition vector . input sequence • At we have 0 0 • State-transition matrix for the DT system . • State-transition matrix for time-varying DT system: – not a matrix power – dependent on both time k and initial time k0 . • Solution=zero-input response+ zero-state response 13 14 Z-Transform Solution of DT State Equation Output Solution • Substitute in output equation • z-transform the discrete-time state equation 15 16 Matrix Inversion Inverse z-transform • Evaluate using the Leverrier algorithm. • Partial fraction expansion then multiply by . • Inverse z-transform Z • Analogous to the scalar transform pair Z 17 18 DT State Matrix Zero-state Response • Parentheses: pertaining to the CT state matrix . • Equality for any sampling period and any matrix • Known inverse transform for • Multiplication by term. : delay by . Convolution theorem: inverse of product is the convolution summation • Same constituent matrices for DT state matrix & CT state matrix A • DT eigenvalues are exponential functions of the CT eigenvalues times the sampling period. 19 20 Example 7.16 Alternative Expression x ZS (t ) k 1 n i 0 x ZS (t ) n j 1 Z je Z j Bd e (a) Solve the state equation for a unit step input and the initial condition vector x(0) = [1 0]T (b) Use the solution to obtain the discrete-time state equations for a sampling period of 0.1s. (c) Solve the discrete-time state equations with the same initial conditions and input as in (a) and verify that the solution is the same as that of (a) evaluated at multiples of the sampling period T. j A k i 1 T j A k 1 T j 1 Bd u(i ) k 1 j A iT u(i ) e i 0 Useful when the summation over i can be obtained in closed form. 21 1 x1 0 x1 0 x 2 3 x 1u 2 2 22 State-transition Matrix Solution (a)The resolvent matrix • Partial fraction expansions 23 24 Zero-input Response Zero-state Response 1 t 1 1 2t 0 2 x ZS (t ) e 2 2 e 1 1t 2 1 1 1 e t 1t e 2t 1t 1 2 1 1 1 e 2t t x ZS (t ) 1 e 1 2 2 1 2 1 t 1 e 2t e 0 1 2 2 25 Total Response 26 (b) Discrete-time state equations . x(t ) x ZI (t ) x ZS (t ) 2 t 1 2t 1 2 1 t 1 e 2t x(t ) e e e 2 2 0 1 2 2 1 2 1 t 1 e 2t e 0 1 2 2 At the sampling points: t = multiples of 0.1s 27 . . . . . • CT system response to a step input of duration one sampling period, is the same as the response of a system due to a piecewise constant input 28 z-transform of the zero-state response Zero-input Response 2 1 0.1k 1 1 0.2 k Adk (01 . k) e e 2 2 2 1 x ZI ( k ) (01 . k )x(0) z z 1 1 2 1 ( z ) 0.1 0 . 2 2 2 ze 2 1 z e 2 1 0.1k 1 1 0.2 k 1 e e 2 2 2 1 0 X ZS ( z ) ( z ) z 1BdU ( z ) 1 2 e 0.1k e 0.2 k 2 2 Same as the zero-input response of the continuous-time system at all sampling points k = 0, 1, 2, ... 2 1 x ZI (t ) e t e 2t 2 2 29 Partial Fractions z e z 1 30 1 0.5 1 x zs (k ) e 0.1k 0.5 e 0.2 k 2 0 1 1z z 10.5083 z 1 z 0.9048 0 .5 z 1 z 1 z 0 .5 X zs ( z ) 0.1 0.2 0 z 1 1 z e 2 z e 1z 1 z 0.1 0.1 z e z 1 1 e z 1 z e 0.1 z Expand zero-state response z 0.2 1 2 z z 0.0045 z 1 z 1 1 0.1 2 2 z e 0.2 0.0861 z 1 2 1 z e z 1 9.5163 10 2 0.1 z 1 1 z e 1 z 9.0635 10 2 0.2 z 1 2 z e Identical to the zero-state response for the continuous system at time t = 0.1 k, k = 0,1,2,... 1z 1 z 0.2 1 e z 1 z e 0.2 1 e 2t 1 2 1 x ZS (t ) e t 2 2 0 1 1z z 5.5167 z 1 z 0.8187 31 32 Zero-state response n x ZS (t ) Z j Bd e j A k 1 T j 1 k 1 ai i0 x ZS ( k ) n Z j Bd e k 1 j A iT u(i ) e i 0 1 ak 1 a j A k 1 T j 1 z-Transfer Function • For zero initial conditions 1 e j A kT j AT 1 e • Substitute 1 e j A kT Z j Bd j AT j 1 1 e n 33 Impulse Response & Modes Poles and Stability • Inverse transform of the transfer function Z ↔ , 1 , 0 • Substitute in terms of constituent matrices 1 Z ↔ , , 34 1 0 35 • poles= eigenvalues of discrete-time state matrix = exponential functions of (continuous-time state matrix ). have negative real parts • For stable , and have magnitude less than unity. • Discretization yields a stable DT system for a stable CT system. 36 Minimal Realizations • • • • • Decoupling Modes Product can vanish & eliminate eigenvalues from the transfer function: if , , or both. If cancellation occurs, the system is said to have an – : output-decoupling zero at – : an input-decoupling zero at – , & : an input-output-decoupling zero at Poles of the reduced transfer function are a subset of the eigenvalues of the state matrix . A state-space realization that leads to pole-zero cancellation is said to be reducible or nonminimal. If no cancellation occurs, the realization is said to be irreducible or minimal. • • • • Output-decoupling zero at : the forced system response does not include the mode . Input-decoupling zero: the mode is decoupled from or unaffected by the input. Input-output-decoupling zero: the mode is decoupled both from the input and the output. These properties are related to the concepts of controllability and observability discussed later in this chapter. 37 38 Example 7.17 Solution z z 2 1 1 1 ( z ) 0.1 0 . 2 2 2 ze 2 1 z e • Obtain the z-transfer function for the position control system of Example 7.16 (a) With x1 as output. (b) 1 2 1 1 e 0.2 0.0045 Bd e 0.1 0 1 2 2 0.0861 With x1 + x2 as output. The resolvent matrix and input matrix were obtained in Example 7.16. (a) Output : 1 1 2 1 1 1 0.0045 G ( z ) 1 0 0.1 0.2 2 2 z e 0.0861 2 1 z e 9.5163 10 2 9.0635 10 2 z e 0.1 z e 0.2 39 40 (c) Output x1+x2 Step Response 1 1 2 1 1 1 0.0045 G ( z ) 1 1 0.1 0.2 2 2 z e 0.0861 2 1 z e 0 9.0635 10 2 z e 0.1 z e 0.2 . 1. Output-decoupling zero at since 2. System response to any input does not include the decoupling term. 9.0635 10 2 z Y ( z) z e 0.2 z 1 1z 0.5 z 1z 9.0635 10 2 z 0 .2 0. 2 z 1 z 0.8187 1 e z 1 z e Z-transform inverse 41 z-Transfer Function: MATLAB • Let T =0.05s » P = ss(Ad, Bd, C, D, 0.05) » g = tf(P) % Obtain z-domain transfer function » zpk(g) % Obtain transfer function poles and zeros • The command reveals that the system has a zero at 0.9048 and poles at (0.9048, 0.8187) with a gain of 0.09035. • With pole-zero cancellation, the transfer function is the same as that of Example 7.17(b). » minreal(g) % Cancel poles and zeros 43 . 42