Modern Control Theory (Digital Control) Lecture 1

Course Overview Analog and digital control systems MM 1 – introduction, discrete systems, sampling. MM 2 – discrete systems, specifications, frequency response methods. MM 3 – discrete equivalents, design by emulation. MM 4 – root locus design. MM 5 – root locus design.

Outline Short repetition of analog control methods Introduction to digital control Digitization Effect of sampling Sampling Spectrum of a sampled signals Sampling theorem Discrete Systems Z-transform Transfer function Pulse response Stability

Digitization Analog Control System + - For example, PID control continuous controller r(t) e(t) ctrl. filter D(s) u(t) plant G(s) y(t) + - sensor H(s)

System caracteristics Transfer function Characteristic equation 1+D(s)G(s)H(s) = 0 Poles are the roots of the characteristic equation

Time functions associated with poles

Second-order system Transfer function is the damping ratio is the undamped natural frequency

Rise time, overshoot and settling time

Response og second-order system versus

Bode-plot design Determin the open loop gain end phase as function of Evaluate the phase margin and gain margin Adjust the margins by use of poles, zeros and gain scheduling.

Bode plot

Digitization Analog Control System + - For example, PID control continuous controller r(t) e(t) ctrl. filter D(s) u(t) plant G(s) y(t) + - sensor 1

Digitization Digital Control System T is the sample time (s) Sampled signal : x(kT) = x(k) digital controller bit → voltage control: difference equations r(t) r(kT) e(kT) u(kT) D/A and hold u(t) y(t) plant G(s) T + - clock y(kT) sensor 1 A/D T voltage → bit

Digitization Continuous control vs. digital control Basically, we want to simulate the cont. filter D(s) D(s) contains differential equations (time domain) – must be translated into difference equations. Derivatives are approximated (Euler’s method)

Digitization Example (3.1) Using Euler’s method, find the difference equations. Differential equation Using Euler’s method

Digitization Significance of sampling time T Example controller D(s) and plant G(s) Compare – investigate using Matlab 1) Closed loop step response with continuous controller. 2) Closed loop step response with discrete controller. Sample rate = 20 Hz 3) Closed loop step response with discrete controller. Sample rate = 40 Hz

Digitization Matlab - continuous controller Controller D(s) numD = 70*[1 2]; denD = [1 10]; numG = 1; denG = [1 1 0]; sysOL = tf(numD,denD) * tf(numG,denG); sysCL = feedback(sysOL,1); step(sysCL); Controller D(s) and plant G(s) Matlab - discrete controller numD = 70*[1 2]; denD = [1 10]; sysDd = c2d(tf(numD,denD),T); numG = 1; denG = [1 1 0]; sysOL = sysDd * tf(numG,denG); sysCL = feedback(sysOL,1); step(sysCL);

Digitization Notice, high sample frequency (small sample time T ) gives a good approximation to the continuous controller

Effect of sampling D/A in output from controller The single most important impact of implementing a control digitally is the delay associated with the hold.

Effect of sampling Analysis Approximately 1/2 sample time delay Can be approx. by Padè (and cont. analysis as usual) r(t) e(t) ctrl. filter D(s) u(t) Padé P(s) y(t) plant G(s) + - sensor 1

Effect of sampling Example of phase lag by sampling Example from before with sample rate = 10 Hz Notice PM reduction

Spectrum of a Sampled Signal Consider a cont. signal r(t) with sampled signal r*(t) Laplace transform R*(s) can be calculated r(t) r*(t) T

Spectrum of a Sampled Signal

Spectrum of a Sampled Signal High frequency signal and low frequency signal – same digital representation.

Spectrum of a Sampled Signal Removing (unnecessary) high frequencies – anti-aliasing filter digital controller control: difference equations r(t) r(kT) e(kT) u(kT) D/A and hold u(t) y(t) plant G(s) T + - clock anti-aliasing filter y(kT) sensor 1 A/D T

Spectrum of a Sampled Signal

Sampling Theorem Nyquist sampling theorem In practice, we need One can recover a signal from its samples if the sampling frequency fs=1/T (ws=2p /T) is at least twice the highest frequency in the signal, i.e. ws > 2 wb (closed loop band-width) In practice, we need 20 wb < ws < 40 wb

Discrete Systems Discrete Systems Z-transform Transfer function Pulse response Stability