Control System Design 1st. Edition by Graham C. Goodwin, Stefan F. Graebe, Mario E. Salgado | 9780139586538 | 139586539 | Holooly

Mechanical Engineering Design

Control System Design

50 SOLVED PROBLEMS

Question: 7.4

A plant has a nominal model given by Go(s) = 2/(s + 1)(s + 2) (7.3.9) Synthesize a PID controller which yields a closed loop with dynamics dominated by the factor s² + 4s + 9. ...

Verified Answer:

The controller is synthesized by solving the pole ...

Question: 25.5

Consider a plant having a nominal model given by G(s) = 1/(s + 1)(s + 3)[s + 1 1 2 1] (25.5.29) Design a MIMO control loop with bandwidths of, approximately, 0.5 [rad/s] in both channels. ...

Verified Answer:

This is a stable and strictly proper plant. Howeve...

Question: 26.6

Consider the following MIMO system Go(s) = [1-s/(s+1)² s+3/(s+1)(s+2) 1_s/(s+1)(s+2) s+4/(s+2)²] = Gon(s)[GoD(s)]^-1I (26.7.1) where GoN(s) = [(1_s)(s+2)² (s+1)(s+2)(s+3) (1-s)(s+2)(s+3) (s+1)(s+4)] (26.7.2) GoD(s) = (s + 1)²(s + 2)² (26.7.3) (i) Determine the location of RHP zeros and their ...

Verified Answer:

(i) The zeros of the plant are the roots of [latex...

Question: 26.7

Consider a MIMO process having a nominal model given by Go(s) = 1/(s² + 2s +4)[-s+2 2s+1 -3 -s+2] with det(Go(s)) = s² + 2s + 7/(s² + 2s + 4)² (26.9.3) For this plant carry out the following For this plant carry out the following (a) Design a dynamically decoupling controller to achieve a closed ...

Verified Answer:

(a) Note that this model is stable and minimum pha...

Question: 26.1

Consider a stable 2 × 2 MIMO system having a nominal model given by Go = 1/(s + 1)²(s+ 2)[2(s + 1) -1 (s + 1)² (s + 1)(s + 2)] (26.2.10) Choose a suitable matrix Q(s) to control this plant, using the affine parameterization, in such a way that the MIMO control loop is able to track references of ...

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We will attempt a decoupled design, i.e. to obtain...

Question: 26.4

Consider the plant Go(s) = Gon(s)[GoD(s)]^-1 (26.4.2) where GoN(s) = [-5 s² 1 -0.0023]; GoD(s) = [25s + 1 0 0 s(s + 1)²] (26.4.3) (i) Convert to state space form and evaluate the zeros. (ii) Design a pre-stabilizing controller to give static decoupling for reference signals. (iii) Design a ...

Verified Answer:

(i) If we compute

det(G_{oN}(s))

we...

Question: 22.3

Consider a 2 × 2 multivariable system, having state space model given by Ao = [1 1 1 2 -1 0 3 -2 2]; Bo = [0 1 1 0 2 -1]; Co = [ 1 0 0 0 1 0]; Do = 0 (22.8.6) Find a state feedback gain matrix K such that the closed loop poles are all located in the disk with center at (−α; 0) and radius ρ, where ...

Verified Answer:

We use the approach proposed above, i.e. we tran...

Question: 22.4

Consider a system with state space description given by (A, B, C, 0), A = [-1 2 -5 -3]; B = [-1 0 -1 2]; C = [-1 0 1 1] (22.13.4) Synthesize a linear quadratic optimal regulator which forces integral action and where the cost function is built with Ψ and Φ, where Ψ = I and ...

Verified Answer:

We first choose the poles for the plant state obse...

Question: 22.2

Consider the scalar system x(t) = ax(t) + u(t) (22.5.3) and the cost function J = ψfx(tf )² + ∫0^tf (ψx(t)²+ u(t)²) dt (22.5.4) 22.2.1 Discuss this optimal control problem in the light of Lemma 22.1 and Lemma 22.2. 22.2.2 Discuss the convergence of the solutions of the CTDRE to P∞^s. ...

Verified Answer:

22.2.1 The associated CTDRE is

\dot{P}(t)=-...

Question: 13.10

Consider a continuous time plant with nominal transfer function Go(s) given by Go(s) = 2/(s + 1)(s + 2) (13.7.5) Assume that this plant has to be digitally controlled with sampling period ∆ = 0.2[s] in such a way that the plant output tracks a periodic reference, r[k], given by ...

Verified Answer:

From (13.7.6) we observe that the reference genera...

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