ROSE­HULMAN INSTITUTE OF TECHNOLOGY
Department of Mechanical Engineering
ME 506
Advanced Controls
Homework 7 Due 30 October 2015
A. LQR control of a Triple Integrator SISO plant.
The purpose of this problem is to provide you analytical insight into some of the properties of LQR controllers in SISO applications. There is a significant analytical component to this problem, requiring a moderate amount of algebra. The CAD software (matlab) should only be used to draw the Nyquist and Bode plots and the transient responses.
The state­space description of the plant is
ẋ 1 ( t ) =x 2 ( t )
ẋ 2 ( t ) =x3 ( t )
ẋ 3 ( t ) =u ( t )
with output equation
y ( t ) =x1 ( t )
Clearly,
y ( s )=
1
u(s)
s3
Physically this represents the motion of a unit mass, in the absence of any friction where the control u(t) is the jerk (defined as the derivative of the acceleration), x3(t) is the acceleration, x2(t) is the velocity, and x1(t) is the position.
Let us consider the following quadratic cost functional
∞
J=∫ [ y 2 ( t ) +ρu 2 ( t ) ] dt; ρ> 0
0
where  is the control weighting parameter. Notice that when  is large, then we are penalizing the control magnitude heavily, so we are telling the mathematics that “slow” motions are better; when  is small, then we are penalizing the control magnitude lightly, so we are telling the mathematics that ”fast” motions are better (but remember that the control must be integrated three times to impact position, so don’t be surprised if large “jerks” arise).
We know that the optimal control for this LQ regulator problem takes the form
u ( t )=−g T x ( t )
Where gT is the LQ gain (row) vector, i.e.
g= [ g 1
g2
g3 ]
1.Determine analytical expressions for the LQ gains g1, g2, and g3 as a function of the control weighting parameter . In order to do this you must solve analytically for some elements of the 3x3 matrix K from the associated control algebraic Riccati equation (CARE). (Hint: Recall that the diagonal elements of a positive definite matrix must be positive; this will help you resolve the ±sign ambiguities). Double­check
your algebra so that the rest of the problem can be done correctly. Although this part has some tedious algebra, it pays to appreciate how many different solutions can arise in solving Riccati equations because of sign ambiguities.
Plot the gains g1, g2, and g3 vs  for 10­6≤ ≤ 104; use log­log scales. Discuss the way the LQ gains change as a function of Homework 7
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ROSE­HULMAN INSTITUTE OF TECHNOLOGY
Department of Mechanical Engineering
ME 506
Advanced Controls
2. Evaluate analytically the closed­loop poles as a function of the control weighting  and sketch the locus of the closed­loop poles in the s­plane as  varies, 0 <  < ∞. (Hint: You should find that all three closed­loop poles are on a circle whose radius depends on . The discussion on Butterworth polynomials
on page 299 of Kwakernaak and Sivan should be useful).
3. Draw a detailed block diagram of the LQ regulator and evaluate the LQ loop transfer function
g LQ ( s ) =g T ( sI −A )−1 b
Determine the poles and zeros of gLQ(s). Discuss their dependence on . (Hint: You should find that both
zeros are on a circle whose radius depends on ).
B. LQ Root­Locus with Exponential Weights.
The purpose of this problem is to give you a feel for the variations of the closed­loop poles in LQR design with and without exponential weighting. Consider the SISO system with scalar control u(s) and scalar output y(s) related by the transfer function
g ( s) =
s2+ 9
s ( s2 +4 )
1. Write the system in the standard controllable state­space representation.
2. Consider the quadratic cost functional
∞
J=∫ e 2 at [ y 2 ( t ) +ρu 2 ( t ) ] dt; a≥0
0
Let a=0 and solve the standard LQR problem. Plot the poles of the closed­loop system as 0 <  < ∞. Note that as →0 the closed loop poles approach the plant complex zeros at s = ±j3.
3. Let a = 3 and solve the exponentially weighted LQR problem. Plot the poles of the closed­loop system as 0<<∞. Discuss the differences with the results found in part (b) and explain.
Homework 7
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