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Application Studies for Control System Design

1/21/13

DCS – DC Servomotor

Dynamics

θ^˙ = ω

ω˙ = −αω + βu y = θ

Let

       x₁ = y,        x₂ = ω,       α = 4,        β = 1

Application studies

1.    Simulate open-loop system

2.    Calculate plant transfer function G(s) = C(sI − A)⁻¹B

3.    Closed-loop proportional control: u = −G₁(θ − θ_r) (See block diagram above.)

(a)    Analytically show that closed-loop system is stable for all values of G₁.

(b)    Obtain Nyquist plot using Matlab.

(c)     Obtain Bode plot using Matlab.

(d)    Sketch Root locus and verify using Matlab.

(e)    Develop closed-loop simulation and simulate performance for several values of G₁.

PEN – Motor-Driven Pendulum

Dynamics

θ^˙ = ω

ω˙ = −αω + (g/L)sinθ + βu y = θ

Let

                                                          x₁ = y,        x₂ = ω,       α = 4,        β = 1,      g/L = 12

Application studies

1.    Simulate open-loop system

2.    Obtain the plant transfer functions G(s) = C(sI − A)⁻¹B for

•    θ = 0

•    θ = π

Do this analytically and using the Matlab function linmod

3.    Closed-loop proportional control for θ = π and u = −G₁(θ − θ_r)

(a)    Analytically show that closed-loop system is stable for all values of G₁.

(b)    Obtain Nyquist plot using Matlab

(c)     Obtain Bode plot using Matlab

(d)    Sketch Root locus and verify using Matlab.

(e)    Develop closed-loop simulation and simulate performance for several values of G₁.

4.    Closed-loop proportional control for θ = 0 and u = −G₁(θ − θ_r)

(a)    Find the range of G₁ for which the system is stable.

(b)    Obtain Nyquist plot.

(c)     Obtain Bode plot.

(d)    Sketch Root locus and verify using Matlab.

(e)    Develop closed-loop simulation and simulate performance for several values of G₁.

5.    Full-state feedback, θ_r = 0, Pole placement For each equilibrium state (θ = 0,θ = π determine the full state feedback gains that place the closed-loop poles at

(a)    s = −6 ± 4j

(b)    s = −8,−10

For each equilibrium state and set of control gains, simulate the closed loop behavior, and comment on the results.

6.    Full-state feedback with reference input. For θ_r = 0.5 compute the gain G₀ for the exogenous input for each equilibrium state in Part 5, and simulate the performance. Verify that the steady-state error goes to zero.

TH3 – Third-Order Heat Conduction

Dynamics

x˙₁ = −3x₁ + x₂ + u x˙₂ = x₁− 2x₂− x₃ x˙₃ = x₂− 3x₃ + x₀ y = x₃

Application Studies

1.    Simulate open-loop system

2.    Calculate plant transfer function G(s) = C(sI − A)⁻¹B

3.    Closed-loop proportional control: u = −G₃(x₃− x_r)

(a)    Find the range of G₃ for which the closed-loop system is stable.

(b)   Obtain Nyquist plot using Matlab(c) Obtain Bode plot using Matlab.

(d)   Sketch Root locus and verify using Matlab.

(e)    Develop a closed-loop simulation and simulate performance for several values of G₃.

(f)     Determine the steady-state error for a unit step reference input as a function of G₃. 4. Closed-loop proportional + integral control For the closed-loop control law

U(s) = −(G_p + G_I/s)(Y (s) − Y_r(s))

(a)    Show that the steady state error is zero for G_I > 0

(b)    Find the region in the G_p,G_I plane for which the system is stable.

(c)     Develop a closed-loop simulation and simulate performance for several values of G_P,G_I within the range of stability.

5.    Design a full-state feedback control law that places the closed-loop poles at s = −10 ± 5j,−15 and accounts for a non-zero nonzero reference value, and for a nonzero exogenous constant input. Simulate closed-loop performance.

6.    Design a full-order observer for x₀ = 0 ( and for y = x₃) with poles at s = −20 + 10j,−30. Using this observer with the gain matrix of Part 5, design the full-order compensator and simulate performance

7.    Augment the observer of Part 6 by the additional state variable to represent the reference and exogenous inputs.

8.    Design (and simulate performance) of a compensator using a reduced-order observer, with poles ats = −20 ± 10j

9.    Design (and simulate performance) of a compensator with gains designed for a linear-quadratic regulator with

                                                                                     Q = diag[0,0,q₃],        R = 1

and a full-order Kalman filter using

F = B, V = v,, W = 1

Choose q₃ and v to make the step response five times faster than open-loop with overshoot less than 5% . This is to be done by simulation.

PCA – Pendulum on Cart

Linearized Dynamics

For an inverted pendulum on a cart, driven by a d-c motor, the linearized dynamics are

x¨ = −αx˙ − (mg/M)θ + βu

θ^¨ = (α/L)x˙ + ((M + m)g/ML)θ − (β/L)u

The state and output variables are

                                                         x₁ = x, x₂ = θ, x₃ = x, x˙          ₄ = θ, y^˙      ₁ = x₁, y₂ = x₂

Numerical values of the parameters are:

α = 4, β = 1, L = 0.25M, m = 0.4Kg, M = 1Kg

Application Studies

1.        Develop a simulation of the system

2.        Calculate the transfer functions from the input u to the outputs y₁ and y₂. (Do this analytically–with the aid of Matlab’s Symbolic Toolbox, if necessary, and verify using the Control System Toolbox.)

3.        Find the open loop poles and the zeros of each transfer function.

4.        Determine whether the system can be stabilized by the control law

u = −G₁x₁

5.        Determine whether the system can be stabilized by the control law

u = −G₁x₁− G₂x₂

6.        Determine the full state feedback gains that place the closed-loop poles at

(a)    s = −10 ± 5j,−20 ± 10j

(b)    s = −10,−15,−20,−30

Simulate the transient response using this gain matrix. (Investigate whether the poles farther from the origin are very important.)

(a)    For an initial position offset (x₁ = .1), all other state variables initially zero, for both set of gains.

(b)    For an initial angular error (x₂ = .1), all other state variables zero, for both sets of gains.

7.        Design a full-order observer having poles at s = −40 ± 30j,−60 ± 20j. Simulate the performance with this observer in place and using the gain matrix determined in Part 6. Compare the actual state and the state estimate produced by the observer.

8.        Design a reduced-order observer with poles at s = −40 ± 30j. Simulate the performance with this observer in place and using the gain matrix determined in Part 6. Compare the actual state and the state estimate produced by the observer.

9.        Design a full-state feedback control law using quadratic optimization (LQR) with matrices given by

                                                                                   Q = diag[q₁,q₂,0,0],        R = 1

Choose q₁ and q₂ to meet the following design goal (if possible). For an initial cart position error of

.1 m, and with the pendulum vertical,

•    Reduce the position error to zero in about 2 sec with overshoot <.01 m

•    Keep the pendulum angle < .05 rad.

The values of q₁ and q₂ are to be determined by simulation.

10.    Design a compensator by use of the gains obtained in Part 9 and the reduced-order observer of Part 8,and simulate performance.

11.    Determine the gains of a Kalman-filter based full-order observer. Use F = B and W = I₂ (2 by 2 identity matrix). To pick the value of V , try to match the pole locations of Part 6.