2026 ECE 4550 — Control System Design Lab #6-Integral Control of a Positioning System Georgia instit

2026 ECE 4550 — Control System Design Lab #6-Integral Control of a Positioning System Georgia institute of technology GEORGIA INSTITUTE OF TECHNOLOGY SCHOOL of ELECTRICAL and COMPUTER ENGINEERING

ECE 4550 — Control System Design — Fall 2026 Lab #6: Integral Control of a Positioning System

Contents 1 Background Material 1 1.1 Plant Modeling .............................................................................................................................. 1 1.2 Position Controller Design ........................................................................................................... 2 1.2.1 Continuous-Time Controller ............................................................................................ 2 1.2.2 Compensation for Actuator Saturation ......................................................................... 3 1.2.3 Discrete-Time Approximation.......................................................................................... 4 2 Lab Assignment 5 2.1 Pre-Lab Preparation ..................................................................................................................... 5 2.2 Tasks .............................................................................................................................................. 5 2.2.1 Response of Position Control System to Step Input.................................................... 6 2.2.2 Response of Position Control System to Sinusoidal Input .......................................... 6

1

Background Material

The objective of this lab is to implement a state-space integral control algorithm on a microcontroller, in order to control the position of a moving object. We will leverage the reduced-order physics-based model of the DC motor plant; the coefficient matrices of this model will be needed for controller design and implementation, so we will use approximate values inferred from measured input-output data. We will exploit plant model knowledge and the computational capability of the microcontroller to perform the numerical processing required by the state-space integral control algorithm, including the real-time approximate solution of differential equations governing the estimator and regulator subsystems of the controller; integrator anti-windup compensation will be used to avoid the poor transient response that would otherwise result from actuator saturation.

1.1

Plant Modeling

The DC motor plant can be modeled with two fidelity levels; its simulation model has three state variables (position, speed and current) whereas its design model has only two state variables (position and speed). The simulation model has higher accuracy, but the design model has lower complexity and is being featured in this and the previous lab; we have this option since the dynamics being neglected by the design model (the voltage to current transient) is stable, well damped, and fast. The design model has state-space form ẋ (t) =

0 1 0 x(t) + u(t) 0 −α β

y(t) = 1 0 x(t).

(1) (2)

The input u(t) is voltage and the output y(t) is position. Parameters α and β are generally unknown because they depend on the inertia and damping of the load which varies with application.

1.2

Position Controller Design

The task of position control can be robustly achieved using state-space integral control. From (1)– (2), the design model has n = 2 state variables and A=

0 1 , 0 −α

B=

0 , β

C= 1 0 .

State-space integral control will use an estimator with n = 2 feedback gains and a regulator with n + 1 = 3 feedback gains to locate 2n + 1 = 5 overall system eigenvalues.

1.2.1

Continuous-Time Controller

The stability of the estimator subsystem is determined by the matrix A − LC =

0

1

0 −α

−

l1

1 0 =

l2

−l1

1

−l2

−α

.

If we want the actual estimator polynomial det (sI − (A − LC)) = s2 + (l1 + α) s + (l1α + l2) to match the desired estimator polynomial (s + λe)2 = s2 + 2λes + λ2e having two roots at s = −λe, then the estimator gains must be (verify this) l1 = 2λe − α,

l2 = λ2e − 2αλe + α2.

(3)

It is possible to solve the estimator eigenvalue assignment problem in this case because our sensor choice results in an observable (A, C). As expected, l1 and l2 depend on plant parameters (α, β) and design parameter λe. A large λe would lead to a fast estimator, but also to large l1 and l2 which could result in problems relating to sensor noise amplification; therefore, selection of λe involves a trade-off between rate of response and noise attenuation.

The regulator subsystem is modeled by the matrices     0 0 1 0 A 0 B   ,   = β K = K1 K2 = k1 k2 k3 = 0 −α 0 , B = A= 0 C 0 0 1 0 0 and its stability is determined by the matrix     0 0 1 0 A − BK =  0 −α 0 −  β k1 1 0 0 0



k2 k3

 0 1 0 =  −βk1 −α − βk2 −βk3 .

1 0 0

If we want the actual regulator polynomial det (sI − (A − BK)) = s3 + (α + βk2) s2 + (βk1) s + (βk3) to match the desired regulator polynomial (s + λr)3 = s3 + 3λrs2 + 3λ2s + λ3 r

r

having three roots at s = −λr, then the regulator gains must be (verify this) 1 1 1 k = 3λ2, k = λ3.

(4) — α), k (3λr 1 3 2 = β r β r β It is possible to solve the regulator eigenvalue assignment problem in this case because our actuator choice results in a controllable (A, B). As expected, k1, k2 and k3 depend on plant parameters (α, β) and design parameter λr. A large λr would lead to a fast regulator, but also to large k1, k2 and k3 which could result in problems relating to actuator command levels; therefore, selection of λr involves a trade-off between rate of response and control effort.

The low complexity of the design model has resulted in closed-form expressions for the feedback gains, (3) and (4), which provide useful insights. The continuous-time controller has the form

1.2.2

u(t) = − K 1 x̂(t) − K2σ(t) x̂˙ (t) = Ax̂(t) + Bu(t) − L (Cx̂(t) − y(t))

(5)

σ˙ (t) = y(t) − r(t).

(7)

(6)

Compensation for Actuator Saturation

The reference input r(t) guides the plant output y(t). If an application requires large transitions in y(t), then a smooth trajectory for r(t) should be planned so that the transition takes place over a reasonable (not too short) time interval. Large rapid changes of r(t) in (7) lead to growth of |σ(t)| via (7) and |u(t)| via (5); eventually |u(t)| reaches its limit Umax, at which point the actuator is said to be saturated. Even if r(t) has been designed to avoid actuator saturation, a disturbance event might still cause actuator saturation. If the actuator saturates, the benefits of feedback are lost, at least temporarily; in the worst case, actuator saturation could lead to instability. Hence, the control designer must always anticipate the possibility of actuator saturation.

The continuous-time controller (5)–(7) performs poorly if actuator saturation occurs, which is not surprising because the actuator saturation limit Umax has not been accounted for. There are two defects associated with this controller design. The regulator integrates output error at all times as indicated in (7), even when the actuator saturates, causing integrator windup problems (see notes posted to Canvas). The estimator fails to correctly imitate plant physics during intervals of actuator saturation, as indicated in (6). Both of these defects should be eliminated.

Since the actuator may saturate, it is essential to distinguish the signal feeding the plant from the signal computed by the controller; the same notation, u(t), has until now been used for both purposes in (1) and (5). To correct this notational oversimplification, we will continue to refer to the plant input in (1) as u(t) but we will now refer to the controller output in (5) as uc(t). These two signals are related by the saturation function, sat(·), defined as   +U max , if uc (t) > +U max −Umax , if uc(t) < −Umax u(t) = sat (uc(t)) =  uc(t) , otherwise.

Note that u(t) = uc(t) unless |uc(t)| exceeds Umax. Actuator saturation represents an undesirable mode of operation, so we seek a feedback mechanism to regulate any nonzero discrepancy uc(t)−u(t) to zero through the influence of regulator state σ(t). This thought process results in u(t) = sat (uc(t)) uc(t) = − K 1 x̂(t) − K2σ(t) x̂˙ (t) = Ax̂(t) + Bu(t) − L (Cx̂(t) − y(t)) σ˙ (t) = y(t) − r(t) + κ (uc(t) − u(t))

(8) (9) (10) (11)

where uc(t)−u(t) is seen to influence σ(t) through feedback gain κ in (11). A block diagram view of the overall system with controller (8)–(11) is provided in Figure 1. The saturation block represents embedded software, with Umax chosen to be smaller than the actuator’s physical limit.

Plant

Estimator

Figure 1: Architecture diagram for position controller with actuator saturation compensation.

With the redesigned controller, the actuator’s command u(t) satisfies the actuator’s physical limit at all times, and also (8)–(11) agrees with (5)–(7) whenever the actuator is not saturated. In other words, the additional complexity of (8)–(11) over (5)–(7) vanishes under normal operating conditions. On the other hand, if actuator saturation were to occur, then (8)–(11) features (i) a regulator that incorporates feedback to encourage resumption of normal operation and (ii) an estimator that accounts for the actuator’s true command when predicting the plant’s response.

The redesigned controller (8)–(11) has an additional feedback gain κ. During normal operation, κ has no effect since u(t) = uc(t). During saturated operation, κ influences σ(t), and ultimately uc(t), since u(t) /= uc(t). Considering (9) and (11), the κ feedback term will assist recovery from saturated operation by reducing the magnitude of σ(t), motivating the following choice of κ: λ κ= r ⇒ σ˙ (t) = − λrσ(t) + other terms.

(12) K2 1.2.3

Discrete-Time Approximation

Since the microcontroller is a discrete-time device, any differential equations appearing in the control algorithm will need to be solved in approximate fashion using calculations performed during time-periodic ISR visits. Such visits begin at time instants t = kT , where T is the ISR period and k = 0, 1, 2, . . . is an integer index. The simplest approximation, referred to as the forward Euler method, replaces any derivative ẋ(t) by the difference ratio (x(kT + T ) − x(kT ))/T . Using this approximation, the redesigned continuous-time controller (8)–(11) is approximated by u(kT ) = sat (uc(kT )) uc(kT ) = − K 1 x̂ (kT ) − K2σ(kT ) x̂ (kT + T ) = x̂ (k T ) + T (Ax̂ (kT ) + Bu(kT ) − L (C x̂(kT ) − y(kT ))) σ(kT + T ) = σ(kT ) + T (y(kT ) − r(kT ) + κ (uc(kT ) − u(kT )))

(13) (14) (15) (16)

with gain assignments (3), (4) and (12). The discrete-time controller (13)–(16) is not equivalent to the continuous-time controller (8)–(11). However, the discrete-time controller (13)–(16) will do a good job of approximating the continuous-time controller (8)–(11) if T is sufficiently small.

There are two ways to implement a controller in C code: a general-purpose approach in which the C code represents matrices as 2d arrays and vectors as 1d arrays, and a reusable (write once) function involving array multiplication and addition performs controller updates; or a case-by-case approach in which all controller equations are rewritten by hand, prior to writing C code, so that all variables are scalars and consequently only scalar math operations are required in C code. For example, the latter approach would leverage the following rewritten scalar-only equations: (14)

⇒

(15) ⇒ (15) ⇒

uc(kT ) = − k 1 x̂ 1 (kT ) − k 2 x̂ 2 (kT ) − k3σ(kT ) x̂ 1 (kT + T ) = x̂ 1 (kT ) + T (x̂ 2 (kT ) − l1 (x̂ 1 (kT ) − y(kT ))) x̂ 2 (kT + T ) = x̂ 2 (kT ) + T (− αx̂ 2 (kT ) + βu(kT ) − l2 (x̂ 1 (kT ) − y(kT ))) .

The argument kT indicates signal validity for ISR visit k whereas the argument kT + T = (k + 1)T indicates signal validity for ISR visit k + 1. Care must be taken to store both old and new versions of updating signals, to avoid unintended overwrites as your code performs updates line by line.

2

Lab Assignment

2.1

Pre-Lab Preparation

Each individual student must work through the pre-lab activity and prepare a pre-lab deliverable to be submitted by the beginning of the lab session. The pre-lab deliverable consists of a brief typed statement in response to the following pre-lab activity specification: 1. Read this entire document, and describe the overall purpose of this week’s project.

2. A control system design should be simulated before testing it experimentally, to identify design defects and to predict responses. Perform a Matlab simulation of the scenario specified in §2.2.1, and submit your results; use continuous-time controller (8)–(11) for simplicity.

Please note that it is not essential to write application code prior to the lab session; the point of the pre-lab preparation is for you to arrive at the lab session with firm ideas regarding the tasks assigned in §2.2 and the high-level algorithms supporting those tasks.

2.2

Tasks

This lab project uses the QMS as the plant (first used in Lab 4), with an inertia disk attached to the rotor of the DC motor as a load. A framework of low-level functions has been developed for this plant for two purposes: to simplify the process of setting actuator commands and getting sensor measurements; and to enable experiment management from within Matlab using mode assignment and data transfer through a back-channel UART connection. This framework is described in another document which you should study. Specifications for this project are listed in Table 1.

Table 1: Specifications for the Experimental Tasks Parameter

Symbol

Value

Units

Data Log Length Timer ISR Period Actuator Saturation Limit Regulator Bandwidth Estimator Bandwidth

N T Umax λr λe

2000 0.001 15.0 50.0 200.0

— s V rad/s rad/s

2.2.1

Response of Position Control System to Step Input

Develop code that implements the position controller defined by (13)–(16) with gain assignments (3), (4) and (12); use α and β values determined in Lab 5. To test the step response of the feedback control system, command the desired position by applying the 2 s reference input   π , if t ∈ [0.0, 0.5) or t ∈ [1.0, 1.5) r(t) = 0 , if t ∈ [0.5, 1.0) or t ∈ [1.5, 2.0)  0 , otherwise where r has units of rad and t has units of s. Perform data logging of r (in rad), y (in rad) and u (in V) over t ∈ [0, 2) (in s). Using subplot(21?), plot r versus t and y versus t in the top panel, plot u versus t in the bottom panel, use xlabel and ylabel to label the axes, and use legend to distinguish between the two signals plotted in the top panel.

Toggling r between 0 and π, as specified above, should not cause actuator saturation and thus should not trigger anti-windup compensation. To test anti-windup compensation, replace π by 4π in r. For on/off comparisons, you may disable anti-windup compensation by setting κ = 0.

2.2.2

Response of Position Control System to Sinusoidal Input

Develop code that implements the position controller defined by (13)–(16) with gain assignments (3), (4) and (12); use α and β values determined in Lab 5. To test the sinusoidal response of the feedback control system, command the desired position by applying the 2 s reference input   2π sin(4πt) , if t ∈ [0.0, 1.0) r(t) = 1π sin(8πt) , if t ∈ [1.0, 2.0)  0 , otherwise where r has units of rad and t has units of s. Perform data logging of r (in rad), y (in rad) and u (in V) over t ∈ [0, 2) (in s). Using subplot(21?), plot r versus t and y versus t in the top panel, plot u versus t in the bottom panel, use xlabel and ylabel to label the axes, and use legend to distinguish between the two signals plotted in the top panel.

Note from your plot that the steady-state response of y(t) does not match the commanded r(t).

The remaining goal is to establish agreement between your plot and a frequency response analysis of the overall system. Recall that y(t) and r(t) are related by Y (s) = H(s)R(s) for the case of zero initial conditions (as in this experiment), and for a plant with n = 2 and no zeros H(s) =

λr s + λr

3

s + λe s + λe

2

=

λr s + λr

3

.

The complete response of the overall system is the sum of its transient response and its steady-state response, and for sinusoidal input the latter is characterized by the input-output pair rss(t) = ar sin(ωrt + φr)

⇒

yss(t) = arao sin(ωrt + φr + φo)

where the overall system frequency response magnitude ao and phase φo are found at ωr from H(jωr) = aoejφo .

From §1.3 of the System Response notes, step response settling time for 3 poles at s = −50 is ts ≈

1.6441(3) + 3.2776 50

= 0.1642

1 s.

The specified r(t) has two distinct 1-second (“long” duration) time intervals to consider: t ∈ [0.0, 1.0)

⇒

(ωr, ar, φr) = (4π, 2π, 0)

t ∈ [1.0, 2.0)

⇒

(ωr, ar, φr) = (8π, 1π, 0).

For each of these time intervals, determine ao and φo and ultimately γ and τ such that yss(t) = γ sin (ωr (t − τ )) .

Validate your experimental plot using the computed values of γ and τ from your frequency response analysis; this can be done in your Matlab code or on the Matlab plot itself (via data tips).

GEORGIA INSTITUTE OF TECHNOLOGY SCHOOL of ELECTRICAL and COMPUTER ENGINEERING

ECE 4550 — Control System Design — Fall 2025 Lab #6: Integral Control of a Positioning System

INSTRUCTOR VERIFICATION PAGE LAB SECTION L01 L02 L03 L04

BEGIN DATE October 14 October 14 October 16 October 16

END DATE October 21 October 21 October 23 October 23

To be eligible for full credit, do the following (all students upload even if working in groups): (14) Upload your pre-lab deliverable to Canvas before lab session begins on begin date.

(15) Upload your C and Matlab files to Canvas before lab session ends on end date.

(16) Upload a scan of this verification page before lab session ends on end date.

Name #1: Name #2:

Checkpoint: Verify completion of the task assigned in §2.2.1.

Verified:

Date/Time:

Checkpoint: Verify completion of the task assigned in §2.2.2.

Verified:

Date/Time: