RST_ct.sq

Continuous-time two-degrees-of-freedom linear controller

RST controllers, or two-degrees-of-freedom linear single-input single-output controllers, are a more general form of linear controller than the popular PID controller. Their name comes from the three polynomials which characterize them. In addition to the feedback S/R, which permits to reduce the sensitivity to model uncertainties and to disturbances, the reference signal is filtered by T/R, which permits better tracking. The two degrees of freedom refer to the independence between those two filters (their common denominator does not constitute a constraint). While many linear single-input single-output controllers (including PID) can be expressed as RST, the SQ file RST_ct.sq preferred design method is the direct manipulation of closed-loop poles. Taking into account other quantities, such as step responses and sensitivities, enables easy and robust design.

First contact

When you open RST_ct.sq from Sysquake (menu File/Open), four figures are displayed: the closed-loop poles, the Bode magnitude, the step response, and the Nyquist diagram. They correspond to a first-order system and a second-order controller with a scalar feedforward calculated to remove steady-state error. You can move the closed-loop poles by dragging them with the mouse. You can also change the gain and the cut-off frequency of the feedback by dragging the Bode magnitude. Observe what happens when you drag the poles to the left of the imaginary axis: the closed-loop system becomes unstable, the step response becomes very large, and the Nyquist diagram crosses the critical point -1.

Figures

Step Response Y/U

Open-loop step response, useful to get an idea of the dynamics of the system.

Impulse Response Y/U

Open-loop impulse response. Depending on the system and the preferences of the user, the impulse response may be better to represent the dynamics of the system. The presence of an integrator, for instance, may make the step response more difficult to understand.

Step Response Y/R

Tracking closed-loop step response. This step response shows important transient behavior of the controlled system, such as the overshoot, the rise time, the settling time. The tracking steady-state error (or lack of it) is also visible. The input/output stability is usually immediately visible, unless a very slow unstable mode is hidden by the limited range of time. Beware of potential internal unstable modes.

Step Response U/R

Tracking closed-loop step response between the reference signal and the system input. Risks of saturation, high-frequency modes (ringing), and slow or unstable internal modes are clearly visible and complete the time-domain information obtained with the step response of Y/R.

Step Response Y/W

Input disturbance rejection step response. The disturbance step is added to the input of the system. This response may be very different from the tracking step response, especially with two-degrees-of-freedom controllers where the prefilter polynomial T(z) is tuned to cancel closed-loop poles or to make the tracking faster.

Step Response Y/D

Output disturbance rejection step response. The disturbance step is added to the output of the system. This response may be very different from the tracking step response, especially with two-degrees-of-freedom controllers where the prefilter polynomial T(z) is tuned to cancel closed-loop poles or to make the tracking faster.

Step Response U/D

Step response between an output disturbance and the system input.

Ramp Response Y/R

Tracking closed-loop ramp response. This response may be better suited to the study of transient behavior and permanent error than the step response if the real set-point changes with a fixed rate.

Ramp Response Y/D

Tracking closed-loop ramp response between a disturbance and the system output.

Bode Magnitude and Phase

Open-loop frequency response, displayed as functions of the frequency expressed in radians per time unit. The cross-over slope of the magnitude, and the low- and high-frequency open-loop gains give important insights about the robustness of the controller.

The Bode magnitude can be dragged up and down to change the gain of the controller.

Nyquist

Open-loop frequency response, displayed in the complex plane. The phase is expressed in radians. The gain and phase margins are clearly visible. With high-order systems and controllers, make sure that the system is stable by inspecting the closed-loop poles, the robustness margins (where the stability is explicitly checked) or at least a time-domain response.

Nichols

Logarithm of the frequency response, displayed in the complex plane. The phase is expressed in radians. The gain and phase margins are clearly visible.

The Nichols diagram can be dragged up and down to change the gain of the controller.

Sensitivity

Closed-loop frequency response between an output disturbance and the output. Only the amplitude is displayed, which is enough to give important information about the robustness of the design. Its supremum is the inverse of the modulus margin, which is defined as the distance between the Nyquist diagram and the critical point -1 in the complex plane. Peaks and large values of the sensitivity should be avoided. The sensitivity should be small at low frequency, to make the behavior of the system insensitive with respect to model uncertainties in the bandwidth.

Clicking in any sensitivity diagram highlights the corresponding frequency in all the sensitivity diagrams, the Nyquist diagram, the Nichols diagram, the Bode diagrams, and the open-loop and close-loop poles plots.

Complementary Sensitivity

Closed-loop frequency response between measurement noise and the output. Its name comes from the fact that the sum of the sensitivity and the complementary sensitivity is 1 for any frequency (however, this does not apply to their amplitude). In the case of a one-degree-of-freedom controller, the complementary sensitivity is also the frequency response between the set-point and the output. It should be close to 1 at low frequency, and small at high frequency.

Perturbation-Input Sensitivity

Closed-loop frequency response between output disturbance and the system input. Small values at high frequency reduce the excitation of the actuators in presence of measurement noise.

Open-Loop Zeros/Poles

All the open-loop zeros and poles are represented. The zeros and poles of the system are represented by black circles and crosses, respectively. The zeros and poles of the free part of the feedback are represented by red circles and crosses; the zeros and poles of the fixed part of the feedback are represented by green circles and crosses; the fixed part of the feedforward polynomial is represented by green squares. All the zeros and poles of the controller can be manipulated with the mouse. The system cannot be changed. As an help to cancel some of the dynamic of the closed-loop system with the feedforward zeros, the closed-loop poles are displayed as magenta (pink) dots.

Closed-Loop Poles

The closed-loop poles are displayed as black crosses. If there are as many closed-loop poles as free coefficients in the feedback, they can be moved; a new controller is calculated by pole placement.

Root Locus

The root locus is the locus of the closed-loop poles when the gain of the feedback is a positive real number. The zeros and poles of the feedback are preserved. The open-loop zeros and poles are represented by black circles and crosses for the system, and red circles and crosses for the feedback. Feedback zeros and poles can be dragged to change the controller. The closed-loop poles are represented by triangles. They can be moved on the root locus to change the feedback gain. If they move beyond open-loop zeros and poles, the sign of the feedback changes, and the root locus is inverted.

Robustness Margins

The gain margin (in dB) and phase margin (in degrees) are displayed with the corresponding frequencies (in radians per time unit). For unstable open-loop systems, the gain margin can be negative and is a lower stability limit for the feedback gain. If the closed-loop system is unstable, no margin is displayed. If the open-loop gain is smaller or larger than 1 at all frequencies, the phase margin is not displayed.

Discrete-Time Step Resp. Y/R and Y/D

Comparison between the step responses of the closed-loop system with an analog controller (in light blue) and with a digital controller (in black). The sampling period can be set by choosing "Sampling Period" in the Settings menu, or adjusted interactively in the figure "Nyquist Frequency" (see below).

Nyquist Frequency

Once an analog RST controller has been designed, it is possible to choose a sampling frequency for a digital implementation. Then the dynamic behavior of the closed-loop system will differ from the initial design. The figure "Nyquist Frequency" displays a Bode diagram of various transfer functions: the continuous-time system is displayed in blue, the continuous-time open-loop response in black, and the discrete-time response of the sampled system (obtained with a zero-order hold) in red. The Nyquist frequency is displayed as a vertical line in red, and can be manipulated interactively. If the closed-loop system is stable, the cross-over frequency is displayed in light blue. Typically, the Nyquist frequency should be 5-10 times larger.

Settings

System

A continuous-time model can be given as two row vectors which contain the numerator and denominator of a transfer function. Multiple models can be provided; each model corresponds to a row.

Feedback Coefficients

The coefficients of the numerator and denominator of the feedback are given as two row vectors. If they are not factors of the feedback fixed parts (see below), the user is asked whether he wants to modify them.

Feedback Fixed Parts

The coefficients of the fixed parts of the numerator and denominator of the feedback are given as two row vectors. The fixed parts can be used to impose some poles and zeros in the feedback (for instance an integrator with [1,0] in the denominator); they are enforced during pole placement. The gain is meaningless; only the zeros are used. When the fixed parts are changed, a new controller is computed such that the closed-loop poles are preserved. If this is not possible because there are not enough closed-loop poles to permit pole placement, the variable part of the controller is preserved. If the resulting controller (product of fixed and variable parts) is non-causal, fast poles are added.

Two DOFs

The Two DOFs setting is a binary value which enables an arbitrary feedforward polynomial. Otherwise, the feedforward is set to the same value as the feedback numerator; this means that the error between the system output and the set-point is used as a whole to compute the system input. When Two DOFs is enabled, the feedforward contains the zeros of its fixed part (see below), and its gain is calculated to have a unit gain between the set-point and the system output.

Feedforward Fixed Part

The feedforward fixed part is given as a row vector. It provides all the zeros of the feedforward; its gain is ignored. The Feedforward Fixed Part setting is enabled only for two-degrees-of-freedom controllers.

Characteristic Polynomial

The controller can be calculated by specifying directly the characteristic polynomial, i.e. the denominator of all the closed-loop transfer functions which can be defined, whose roots are the closed-loop poles. To enter the closed-loop poles, use the poly function (e.g. poly([-0.8,-1.3+0.3j,-1.3-0.3j])).

In order to obtain a solution for any value of the coefficients, the degree of the characteristic polynomial must be larger than or equal to 2 deg A + deg Rf + deg Sf - 1, where A, Rf and Sf are respectively the system denominator, the fixed part of the feedback denominator and the fixed part of its numerator. This lower limit is displayed in the dialog box. There is no upper limit (from a mathematical point of view).

Sampling Period

The sampling period is given as a positive pure number. It is used for the discrete-time step response to show the difference between the responses with purely continuous-time elements and a digital implementation with a zero-order hold D/A converter.

Bilinear/Back Rect/For Rect Method

Method used for converting the controller from continuous time to discrete time. Usually, the bilinear method is the best one and permits lower sampling frequencies.

Damping Specifications

Absolute and relative damping can be specified; they are represented in the complex plane of the closed-loop poles and the root locus by red lines. For a stable system, the absolute damping is the time constant of the envelope of the slowest mode; the relative damping is the absolute damping divided by the oscillation time.

Display Frequency Line

When selected, moving the mouse above a frequency response (Bode or sensitivity) will display a corresponding line in other frequency responses, Nyquist diagrams, and zero/pole diagrams.