2026 ECE 4550 — Control System Design Fall Lab #5- Identification of a Positioning System Georgia in

2026 ECE 4550 — Control System Design Fall Lab #5- Identification 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 #5: Identification of a Positioning System

Contents 1 Background Material 1 1.1 Plant Modeling............................................................................................................................1 1.1.1 Full-Order Model............................................................................................................1 1.1.2 Reduced-Order Model....................................................................................................2 1.2 Parameter Identification..............................................................................................................2 2 Lab Assignment 5 2.1 Pre-Lab Preparation....................................................................................................................5 2.2 Tasks............................................................................................................................................5 2.2.1 Collect Input-Output Data............................................................................................5 2.2.2 Identify Plant Parameters..............................................................................................6

1

Background Material

This lab project investigates the use of parameter identification to approximate the parameters of an electromechanical plant, thereby taking the first step towards model-based controller design.

To prepare, review posted notes on Parameter Identification and DC Machine Systems.

The plant is the Quanser Motion System (QMS) with an inertia disk as load. Its base houses a motor, a power converter, a microcontroller, and a connector for two-way serial communication.

This internal microcontroller enables the serial communication, interfaces with the power converter to impose motor voltage and measure motor current, and interfaces with an optical encoder to mea- sure angular position; operation of the system is the responsibility of our external microcontroller, with commands and measurements conveyed as serial data packets.

1.1

Plant Modeling

1.1.1

Full-Order Model

The state-space model of a rotary positioning system driven by a DC motor is dθ dt dω dt di dt L

= ω(t) =

1

J (Ki(t) − Dω(t) − τ (t)) 1 = (v(t) − Ri(t) − Kω(t))

where the state variables are angular position θ(t), angular speed ω(t) and current i(t), the control input is voltage v(t), and the disturbance input is torque τ (t) which is assumed to be zero here since there is no shaft-connected load torque in this project. The electromechanical coupling is represented by induced torque Ki(t) and induced voltage Kω(t). The mechanical parameters are

inertia J and damping D, and the electrical parameters are inductance L and resistance R. Note that J and D are application dependent, whereas K, L and R are fixed for a given motor. Neglecting disturbance torque, the coefficient matrices of the state equation are     0 1 0 0 A = 0 −D/J K/J  , B= 0 . −K/L −R/L 0 1/ L 1.1.2

Reduced-Order Model

For DC motors with small inductance and large resistance, the voltage-to-current dynamics L

di dt

= v(t) − Ri(t) − Kω(t)

are fast and well damped, and thus the approximations L≈0

⇒

i(t)

v(t) − Kω(t) R

≈

lead to a simplified model involving only two state variables dθ dt dω dt

= ω(t) = −αω(t) + βv(t)

where disturbance torque is neglected and the two load-dependent parameters are α=

DR + K2 JR

,

β=

K . JR

A consequence of this modeling approximation is that the induced voltage contributes an additive effective damping coefficient of K2/R. The coefficient matrices of the state equation are 0 1 A = 0 −α , B=

1.2

0 β

.

Parameter Identification In this project, we will use the reduced-order model which has input-output form y¨(t) + αy˙(t) = βu(t)

(1)

and state-space form x˙ (t) =

0 1 0 − α

0 x(t) + β u(t)

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

(2) (3)

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.

The values of command u(t) and measurement y(t) will be recorded during an experiment that begins with zero initial conditions, so it follows that (1) may be written as s2Y (s) = − αsY (s) + βU (s).

(4)

The recorded u(t) and y(t) will be filtered to enable parameter identification while avoiding differentiation of noise. Filtering is done by multiplying both sides of (4) by λi3/(s + λi)3 to obtain λ3i s2

3 1 Y (s) = − α λ i s

((s) s + λi)3 ` ˛ ¸ Y

2(

(s 3+ λ) `i ˛ ¸

x s)

Y

λ3i s0

Y (s) +β

1(

U

(s + λ )3

(5)

i

x

`

s)

˛ ¸

U

0(

x

s)

where λi is the assigned identifier bandwidth. The result of filtering is summarized by Y2(s) = − αY1(s) + βU0(s).

(6)

Y2(s) is an acceleration signal, in rad/s2, that is obtained from position Y (s), in rad, using Y2(s) = H2(s)Y (s).

(7)

Y1(s) is a velocity signal, in rad/s, that is obtained from position Y (s), in rad, using Y1(s) = H1(s)Y (s).

(8)

U0(s) is a voltage signal, in V, that is obtained from voltage U (s), in V, using U0(s) = H0(s)U (s).

(9)

The three filters used in (7)–(9) for data processing have transfer functions Hj(s) =

λi3

(s + λ i )3

sj ,

j = 0, 1, 2.

(10)

Since Hj(s) ≈ sj for low frequencies ( λi rad/s) and Hj(s) ≈ 0 for high frequencies ( λi rad/s), λi can be adjusted to limit the influence of noise. There are infinitely many ways to represent the filter transfer functions of (10) in state-space form, but the preferred canonical form        u˙0(t) 0 1 0 u0(t) 0 u˙1(t) =  0      0 1 u1(t) + 0 u(t) (11) i i −λ3 −3λ2 −3λ λ3i u˙ 2 ( u2(t) t)       y˙0(t) 0 1 i 0 y0(t) 0 y˙1(t) =  0      0 1 y1(t) + 0 y(t) (12) −λ3i −3λ2i −3λ λ3i y˙2(t y2(t) ) i is very convenient because the filter state variables are the desired filter output variables.

The s-domain expression (6) is equivalent to the t-domain expression y2(t) = − αy1(t) + βu0(t)

(13)

if filtering is performed using zero initial conditions. The signals appearing in (13) are accurately

approximated by computing the zero-state response of (11)–(12) using the forward Euler method with small time increment equal to the sampling period used for data logging. If N samples of

the filtered signals are computed at t = t1, . . . , tN , then the resulting N linear constraints on the unknown α and β generated by (13) are     −y1(t1) u0(t1) y2(t1)   α   =  (14)   . . . . β y2(tN ) −y1(tN ) u0(tN ) We cannot expect to find values for α and β that exactly satisfy all N constraints, so we wish to compute values for α and β that minimize the sum of squared errors E defined by N Σ E = (− αy1(tk) + βu0(tk) − y2(tk))2 .

(15)

k=1

By imposing the extremum conditions ∂E/∂α = 0 and ∂E/∂β = 0, it can be shown that the values of α and β that minimize E in (15) are found by solving the square system of equations  r    r   −y (t ) −y (t ) −y (t ) u (t ) u (t ) u (t ) y (t ) 1





`

1

0

1

.N −y1.(tN ) u0(t )

1

1

0

1

α

     . N ) β −y1.(tN ) u0(t ˛ x 2

¸

=



1

1

0

1

2

1

   . . N   y2(t. N )  −y1(tN ) u0(t ) x

(16)

` ˛ ¸

2 × 1

× 2

obtained by multiplying both sides of (14) by the transpose of its N × 2 coefficient matrix. If the excitation u(t) used to produce y(t) has sufficient frequency content, then (16) will have a unique solution for α and β (because its 2 × 2 coefficient matrix will have a non-zero determinant). Since the true values of α and β are not known, the values computed from (16) cannot be directly used to infer success or failure of the identification process; therefore, the result is indirectly judged by comparing experimental y(t) data with simulation y(t) data. Quality of fit is quantified by yexp − ysim Q = 100 1 − , − ∞ < Q ≤ 100 (17) yexp − mean{yexp}1vec where yexp and ysim are length-N arrays of experimental and simulation data, 1vec is a length-N array of ones, and · denotes 2-norm (square root of sum of components squared). Figure 1 summarizes the parameter identification workflow just described. actual plant

yexp

xa : Aa, Ba, Ca

excitation signal

u

filter

solver

identified plant

xi : Ai, Bi, Ci

filter

qualit y of fit

ysim

Q

Figure 1: Workflow diagram for parameter identification experiment and post-processing.

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. Prove that the filter outputs defined in (7)–(9), which use transfer functions defined in (10), are equal to the (exact) zero-state response of the filter state equations (11)–(12).

3. Prove that the sum of squared errors defined by function E(α, β) in (15) will be minimized if α and β are the solutions of the system of linear algebraic equations in (16).

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 operating mode logic 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 Data Log Length Timer ISR Period Identifier Bandwidth 2.2.1

Symbol N T λi

Value 2000 0.001 50.0

Units — s rad/s

Collect Input-Output Data

Develop microcontroller code that enables collection of input-output data during operation of the plant. For the data collection experiment, apply the 2 s actuator input  +5 , if t ∈ [0.00, 0.25) or t ∈ [1.00, 1.25)   0 , if t ∈ [0.25, 0.50) or t ∈ [1.25, 1.50) u(t) = −5 , if t ∈ [0.50, 0.75) or t ∈ [1.50, 1.75)  0 , if t ∈ [0.75, 1.00) or t ∈ [1.75, 2.00) 0 , otherwise  where u has units of V and t has units of s. Perform data logging of y (in rad) and u (in V) over t ∈ [0, 2) (in s). Using subplot(21?), plot yexp versus t in the top panel, plot u versus t in the bottom panel, and use xlabel and ylabel to label the axes.

2.2.2

Identify Plant Parameters

Develop Matlab code that implements the parameter identification method of §1.2 to approximate plant parameters α and β. Using subplot(21?), plot yexp versus t and ysim versus t in the top panel, plot u versus t in the bottom panel, use xlabel and ylabel to label the axes, use legend to distinguish between the two signals plotted in the top panel, and use title on the top panel to report the numerical values obtained for α and β in SI units, and for the dimensionless Q. These plant parameters will be used in a future lab to design a model-based position controller.

GEORGIA INSTITUTE OF TECHNOLOGY

SCHOOL of ELECTRICAL and COMPUTER ENGINEERING

ECE 4550 — Control System Design — Fall 2025 Lab #5: Identification of a Positioning System

INSTRUCTOR VERIFICATION PAGE LAB SECTION L01 L02 L03 L04

BEGIN DATE September 30 September 30 October 14 October 14

END DATE October 7 October 7 October 16 October 16

To be eligible for full credit, do the following (all students upload even if working in groups): 1. Upload your pre-lab deliverable to Canvas before lab session begins on begin date.

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

3. 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: