06.Interconnections and feedback

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Exercises

Answer the following questions.

E6.1

Let (S, G) be a signal flow graph with GS,G its corresponding matrix. Show that the

(a)

Use Theorem 6.38 and the Routh/Hurwitz criteria to ascertain for which values

of a and b the interconnected system is IBIBO stable.

gains in the ith column correspond to branches that originate from the ith node, and

(b)

For which values of a and b can IBIBO stability of the interconnected system

that the gains in the jth row correspond to branches that terminate at the jth node.

be inferred from looking only at the determinant of the graph without having to

E6.2 Consider the block diagram of Exercise E3.2.

resort to using the characteristic polynomial?

(a)

Draw the corresponding signal flow graph.

E6.6 In this exercise, you will investigate the matter of relating BIBO stability of individual

(b)

Write the signal flow graph as (S, G) as we describe in the text—thus you should

branch gains in a signal flow graph (S, G) to the IBIBO stability of the interconnection.

identify the nodes and how the nodes are connected.

(c) Write the structure matrix GS,G.

(a)

Is it possible for (S, G) to be IBIBO stable, and yet have branch gains that are

(d)

Determine the pair (AS,G, BS,G) as per Procedure 6.22.

not BIBO stable? If it is not possible, explain why not. If it is possible, give an

(e)

Write all simple paths through the signal flow graph which connect the input rˆ

example.

with the output yˆ.

(b)

Is it true that BIBO stability of all branch gains for (S, G) implies IBIBO sta-

(f)

Identify all loops in the

graph

by writing their gains—that

is, determine

bility of the

interconnection?

If it is true, prove it.

If it is not true, give a

Loop(S, G).

counterexample.

(g)

For k ≥ 1 determine Loopk(S, G).

E6.7

Prove Proposition 6.45 directly, without reference to Theorem 6.38.

(h)

Find

S,G.

finish

E6.8

Well-posedness and existence and uniqueness of solutions.

(i)

Find TS,G.

E6.9

For the feedback interconnection of Figure E6.2, let ΣC

= (A1, b1, c1t , D1) be the

(j)

Write each of the rational functions R1, . . . , R6 in Figure E3.1 as a numerator

polynomial over a denominator polynomial and then determine PS,G.

E6.3 Let Σ = (A, b, ct, D) be a SISO linear system.

rˆ(s)

RC (s)

RP (s)

yˆ(s)

(a)

Show

that there exists P

Σ R

[s]n×n so that x˙ (t) = Ax(t)

if and only if

−

(b)

P Σ

d

x(t) = 0.

dt

Show that Theorem 5.2 then follows from Theorem 6.33. That is, show that the

hypotheses of Theorem 5.2 imply the hypotheses of Theorem 6.33.

E6.4 For the block diagram of Figure E6.1, determine the values of a, b, and c for which

Figure E6.2 Plant/controller feedback loop

rˆ(s)

1

1

yˆ(s)

s − b

−

(s + a)(s + c)

canonical minimal realisation for RC and let ΣP = (A2, b2, c2t , D2) be the canonical

minimal realisation for RP . Show that the interconnection is well-posed if and only if

D1D2 6= [−1].

s + a

E6.10 For the block diagram configuration of Figure E6.3, show that as K → 0 the poles

Figure E6.1 A block diagram for determining IBIBO stability

the interconnected system is IBIBO stable. For which values of a and b can stability be inferred from the zeros of the determinant, without having to resort to looking at the characteristic polynomial.

E6.5 For the block diagram of Exercise E3.2 (which you investigated as a signal flow graph in Exercise E6.2), consider the following assignment of specific rational functions to the block transfer functions:

1							1
R1(s) =			, R2(s) = s − 1, R3(s) =						,
		s						s + b
1					1
R4(s) =				,	R5(2) =		, R6(s) = s + a.
	s + 2					s + a

rˆ(s)	K	RL(s)	yˆ(s)
	−

Figure E6.3 Unity gain feedback loop with variable gain

and zeros of the closed-loop transfer function approach those of RL.

E6.11 Let RP R(s) be a proper plant. For RC R(s) define RL = RCRP , as usual, and let TL be the corresponding closed-loop transfer function. Suppose that RP has zeros

z1, . . . , z` C+ {∞} and poles p1, . . . , pk C+ (there may be other poles and zeros of the plant, but we do not care about these). Prove the following result.

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Proposition RC S(RP ) if and only if the following four statements hold:

(i)TL RH+∞;

(ii)the zeros of 1 − TL contain {p1, . . . , pk}, including multiplicities;

(iii)the zeros of TL contain {z1, . . . , z`}, including multiplicities;

(iv)lims→∞ RL(s) 6= −1.

E6.12 Let RP R(s) be a proper plant. For RC R(s) define RL = RCRP , as usual, and let SL be the corresponding sensitivity function. Suppose that RP has zeros z1, . . . , z` C+ {∞} and poles p1, . . . , pk C+ (there may be other poles and zeros of the plant, but we do not care about these). Prove the following result.

Proposition RC S(RP ) if and only if the following four statements hold:

(i)SL RH+∞;

(ii)the zeros of SL contain {p1, . . . , pk}, including multiplicities;

(iii)the zeros of 1 − SL contain {z1, . . . , z`}, including multiplicities;

(iv)lims→∞ RL(s) 6= −1.

E6.13	Let RP R(s) be a BIBO stable plant. Show that there exists a controller RC R(s)
	for which the interconnection of Figure E6.2 is IBIBO stable with closed-loop transfer
				TL	+
	function TL if and only if TL,				RH∞.
				RP
E6.14	Let RP R(s) be a proper plant transfer function, and let (S, G) be an interconnected
	SISO linear system with the property that every forward path from the input to the
	output passes through the plant. Show that IBIBO stability of (S, G) implies that
		TS,G	+
	TS,G,		RH∞.
		RP
E6.15	In this exercise you will show that by feedback it is possible to move into C− the
	poles of a closed-loop transfer function, even when the poles of the plant are in C+.
	Consider the closed-loop system as depicted in Figure 6.21 with

1

RC(s) = 1, RP (s) = (s + 1)(s − a),

with a > 0. Determine for which values of the gain K the closed-loop system has all poles in C−. Is the system IBIBO stable when all poles of the closed-loop system are in C−?

In this exercise we explore the relationship between performing static state feedback for SISO linear systems, and performing design for controller rational functions for input/output systems.

E6.16 Consider a SISO linear system Σ = (A, b, ct, D) with (A, b) controllable and (A, c) observable. Let RP = TΣ. For f = (f0, f1, . . . , fn−1) Rn define the polynomial

F (s) = fn−1sn−1 + · · · + f1s + f0 R[s].

Suppose that (A − bft, c) is observable.

(a)Show that there exists a controller rational function RC with the property that the poles of the two transfer functions

RCRP

TΣf , T = 1 + RCRP

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agree if and only if the polynomial NP divides the polynomial F over R[s]. In particular, show that if NP is a constant polynomial, then it is always possible to find a controller rational function RC with the property that the poles of TΣf and T agree.

Hint: Without loss of generality suppose that (A, b) is in controller canonical form, and look at the proof of Proposition 10.13.

(b)Suppose that NP divides F over R[s] and by part (a) choose a controller rational function RC with the property that the poles of TΣf and T agree. What is the di erence of the numerators polynomials for TΣf and T .

Thus the problem of placement of poles in feedback design for SISO linear systems can sometimes be realised as feedback design for input/output systems. The following parts of the problem show that there are some important cases where controller rational function design cannot be realised as design of a state feedback vector.

Let (N, D) be a strictly proper SISO linear system in input/output form with Σ = (A, b, ct, 01) the canonical minimal realisation. Suppose that N has no root at s = 0. Let RP = TN,D.

(c)Is it possible, if RC is the controller rational function for a PID controller, to find f Rn so that the poles of the transfer functions of part (a) agree?

E6.17 Consider the SISO linear system Σ = (A, b, ct, 01) with

A =	0	0	,	b =	1	,	c =	0 .
	0	1			0			1

For this system, answer the following.

(a)Show that there is no continuous function u(x1) with the property that for every solution x(t) of the di erential equation

x˙ (t) = Ax(t) + bu(x1(t))

(E6.1)

satisfies limt→∞ kxk (t) = 0.

Hint: First prove that the function

V(x) = 2x22			− Z0	x1
				u(ξ) dξ
	1

is constant along solutions of the di erential equation (E6.1).

(b)If f R2 is a state feedback vector for which A − bft is Hurwitz, what can be said about the form of f from part (a).

E6.18 In this exercise we generalise Exercise E6.17 for linear feedback. We let Σ = (A, b, ct, 01) be a SISO linear system with tr(A) = 0 and ctb = 0.

(a)Show that there is no output feedback number F with the property that the closed-loop system is internally asymptotically stable.

Hint: Show that if

PA(s) = sn + pn−1sn−1 + · · · + p1s + p0

is the characteristic polynomial of A, then p0 = tr(A) (think of putting the matrix in complex Jordan canonical form and recall that trace is invariant under similarity transformations).

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sh	(b) Something with Liapunov
	E6.19 Let Σ = (A, b, ct, 01) be a complete SISO linear system and suppose that z C
	is a zero of TΣ. Show that by static output feedback it is not possible to obtain a
	closed-loop system with a pole at z.
	One of the potential problems with derivative control is that di erentiation can magnify
	high frequency noise, as the following exercise points out.
	E6.20 For a time signal
	y(t) = As sin(ωst) + An sin(ωnt + φn),
	consisting of a sinusoidal signal (the first term) along with sinusoidal noise (the second
	term), the signal-to-noise ratio is defined by S/N =	|As|	.
		|An|

(a)Show that for any such signal, the signal-to-noise ratio for y˙ tends to zero as ωn tends to infinity.

(b)Indicate in terms of Bode plots why di erentiation is bad in terms of amplifying high frequency noise.

E6.21 In this exercise we will investigate in detail the DC servo motor example that was used in Section 1.2 to provide an illustration of some control concepts.

We begin by making sure we know how to put the model in a form we can deal with. We model the system as a SISO linear system whose single state is the angular velocity of the motor. The output is the angular velocity of the motor (i.e., the value of the system’s only state), and the input is the voltage to the motor.

(a)Determine Σ = (A, b, ct, D). Your model should incorporate the time-constant τ and motor gain kE as in Section 1.2, but do not include any e ects from external disturbances.

(b)Determine TΣ.

We let the plant transfer function RP be TΣ whose c.f.r. we write as (NP , DP ). For the reasons we discussed in Section 1.2, an open-loop control scheme, while fine in an idealised environment, lacks robustness. Thus we employ a closed-loop control scheme like that depicted in Figure 6.21. For proportional control we use RC(s) = 1.

(c)Determine the closed-loop transfer function with gain K.

(d)Assuming that τ and kE are both positive, determine the range of gains K for which the closed-loop system has all poles in C−. That is, determine

Kmin and Kmax so that the closed-loop system has poles in C− if and only if

K (Kmin, Kmax).

Now we will see how the closed-loop system’s frequency response represents its ability to track sinusoidal inputs.

(e)Determine the frequency response for the closed-loop transfer function.

(f)Determine the output response to system when the reference signal is r(t) = cos ωt for some ω > 0. Assuming that K (Kmin, Kmax), what is the steady-state response, yss(t).

(g)Show that limK→∞ yss(t) = cos ωt, and so as we boost the gain higher, we can in principle exactly track a sinusoidal reference signal. Can you see this behaviour reflected in the frequency response of the system?

E6.22 Consider the coupled masses of Exercise E1.4 (assume no friction). As input take the situation in Exercise E2.19 with α = 0. Thus the input is a force applied only to the leftmost mass.

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We wish to investigate the e ect of choosing an output on our ability to manipulate the poles of the closed-loop transfer function. We first consider the case when the output is the displacement of the rightmost mass.

(a)Determine the transfer function TΣ for the system with this output.

(b)What are the poles for the transfer function? What does this imply about the uncontrolled behaviour of the coupled mass system?

First we look at an open-loop controller as represented by Figure 6.20. We seek a controller rational function RC whose c.f.r. we denote (NC, DC). We also let RP = TΣ, and denote by (NP , DP ) its c.f.r.

(c)Can you find a controller transfer function RC so that

1.DP and NC are coprime and

2.the open-loop transfer function has all poles in C−? Why do we impose the condition 1?

Now we look for a closed-loop controller as represented by Figure 6.21. For simplicity, we begin using a proportional control.

(d)For proportional control, suppose that RC(s) = 1, and derive the closed-loop transfer function with gain K.

(e)Show that it is impossible to design a proportional control law for the system with the properties

1.DP and NC are coprime and

2.the closed-loop transfer function has all poles in C−?

Hint: Show that for a polynomial s4 + as2 + b R[s], if s0 = σ0 + iω0 is a root, then so are σ0 − iω0, −σ0 + iω0, and −σ0 − iω0.

It turns out, in fact, that introducing proportional and/or derivative control into the problem described to this point does not help. The di culty is with our plant transfer function. To change it around, we change what we measure.

Thus, for the remainder of the problem, suppose that the output is the velocity of the leftmost mass (make sure you use the correct output).

(f)Determine the transfer function TΣ for the system with this output.

(g)What are the poles for the transfer function?

The open-loop control problem here is “the same” as for the previous case where the output was displacement of the rightmost mass. So now we look for a closed-loop proportional controller for this transfer function.

(h)For proportional control, suppose that RC(s) = 1, and derive the closed-loop transfer function with gain K.

(i)Choose m = 1 and k = 1, and show numerically that there exists K > 0 so that the poles of the closed-loop transfer function all lie in C−.

E6.23 Refer to Exercise E6.21. Set the time constant τ = 1 and the motor constant kE = 1. Produce the Bode plots for plant transfer function, and for the closed-loop system with the proportional controller with gains K = {1, 10, 100}. Describe the essential di erences in the Bode plots. How is your discovery of Exercise E6.21(g) reflected in your Bode plots?

E6.24 Refer to Exercise E6.22, taking m = 1 and k = 1, and use the second output (i.e., the velocity of the leftmost mass). Produce the Bode plots for plant transfer function, and for the closed-loop system with the proportional controller with gains

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K = {1, 10, 100}. Describe the essential di erences in the Bode plots. In what way does the open-loop transfer function di er from the rest?

E6.25 Consider the controller transfer function RC(s) = K 1 + TDs + 1 .

TI s

(a)Can you find a SISO linear system Σ = (A, b, ct, D) so that TΣ = RC? (Assume that K, TD, and TI are finite and nonzero.)

(b)What does this tell you about the nature of the relationship between the Problems 6.41 and 6.57?

PID control is widely used in many industrial settings, due to its easily predictable behaviour, at least when used with “simple” plants. In the next exercise you will see what one might mean by simple.

E6.26 Consider the interconnection in Figure E6.4 with RP a proper plant. Suppose that if

rˆ(s)	RC (s)	RP (s)	yˆ(s)
	−

Figure E6.4 Feedback loop for studying properties of PID control

RC = 1 then the interconnection is IBIBO stable. Show that there exists K0 > 1 and TD,0, TI,0 > 0 so that the controller

1

RC(s) = K 1 + TDs + TI s

IBIBO stabilises the interconnection for all K [1, K0], TD [0, TD,0], and TI [TI,0, ∞).

Hint: Use the Nyquist criterion of Section 7.1 to show that the number of encirclements of −1+i0 does not change for a PID controller with the parameters satisfying

K [1, K0], TD [0, TD,0], and TI [TI,0, ∞).

In the next exercise we will consider a “di cult” plant; one that is unstable and nonminimum phase. For this plant you will see that any “conventional” strategies for designing a PID controller, based on the intuitive ideas about PID control as discussed in Section 6.5, are unlikely to meet with success.

E6.27 Consider, still using the interconnection of Figure E6.4, the plant

1 − s RP (s) = s(s − 2).

Answer the following questions.

(a)Show that it is not possible to IBIBO stabilise the system using a PID controller with positive parameters K, TD, and TI .

Hint: One can use one of the several polynomial stability tests of Section 5.5. However, it turns out that the Routh test provides the simplest way of getting at what we want here.

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Thus we must take at least one of the PID parameters to be negative. Let us consider the simplest situation where we take K < 0, so that perhaps some of our intuition about PID controllers persists.

(b)Show that if K < 0 then it is necessary for IBIBO stability that TI > 1.

(c)Show that if K < 0 and TI > 1 then it is necessary for IBIBO stability that

TD < 0.

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