EE611 Deterministic Systems Examples and Discrete Systems Descriptions Kevin D. Donohue Electrical and Computer Engineering University of Kentucky

State-Space Description Example: Find the state-space descriptions for the following circuit:

State-Space Description A general approach: Identify the independent voltages and/or currents in the circuit. These are values that you need initial conditions for in order to solve for the unique complete solution. Find an equation for each independent value that relates its first derivative to zero-order derivatives of the other independent values and sources/inputs. Set up a system of first order differential equations in a matrix form. Express the particular output value in terms of a linear combination of the independent values. Use matrix notation.

State-Space Description Example: Find the state-space descriptions for the following circuit:

State-Space Description Example: Find the state-space descriptions for the following circuit: R f1 R f2 x1x1 x2x2 +y-+y- +u-+u- C1C1 C2C2 R1R1 R2R2

Example Find TF from SS Given SS description, for a relaxed system find the TF Show:

Example Find TF from SS Given SS description. For a relaxed system find the TF matrix

Discrete-Time Systems Dirac Delta (impulse) Function: Kronecker Delta Function:

Discrete-Time Convolution Let g[k, m] be the system impulse response. The output y[k] can be computed from the input u[m] with convolution summation: For a LTI system it reduces to: For a relaxed causal system it reduces to:

Z-Transform Analogous to the Laplace Transform for continuous-time systems is the Z-transform for discrete systems: For a LTI system it can be shown that discrete convolution in time is multiplication in the z-domain Delay Property:

Discrete Transfer Function Analogous to the differential equation for continuous-time systems, is the difference equation for discrete systems: For a relaxed LTI system the transfer function can be expressed as: Describe the relationship between causality and the orders of the numerator and denominator.

Discrete State-Space Description State-space description for a time-varying discrete linear system: for a time invariant system:

Transfer Functions and State Space For a MIMO system the relation between TF and the zero-state and zero-input responses of a state-space representation is given by: For the zero-state case the TF is given by:

Discrete-Time System Example For the SISO system, assume initial state x[0] = [-1 1] T and input u = 0, find output for the k = 0, 1, and 2. Assume the system is relaxed at k = 0. Find closed-form expression for y[k] for k  0, given u[k] is the unit impulse function.

Discrete-Time System Example Zero input response A = [ ; ]; % Define systems matrix c = [1 0]; % Define output matrix x(:,1) = [-1 1]'; % Define initial state ktot = 10; % Define number of outputs to compute for k=1:ktot x(:,k+1) = A*x(:,k); y(k) = c*x(:,k); end kaxis = [0:length(y)-1]; % Time index axis plot(kaxis, y) xlabel('Time') ylabel('Amplitude') Zero-State impulse response:

Lecture Note Homework U2.1 Derive state-space and output equations for the analog computer circuit below (voltage u is the input and y is the output): (Hint: Use input-output integrator relationships to come up with a series of state variables, also note that state equation should be third order). 20k  +y-+y- +u-+u- 1mF 1k  0.1mF 1mF 10k  1k  

Lecture Note Homework U2.2 Find the transfer function for the given the SISO system: Assume the system is relaxed at k = 0. Find closed-form expression for y[k] for k  0, given u[k] is the unit step function (Hint: Use z-transform tables or Matlab's iztrans() function from the symbolic toolbox).