This problem is intended to give you more insight into controllability and

Why is it convenient to write equations of motion in state-variable form?

Give an expression for the transfer function of this system.

Give two expressions for the poles of the transfer function of the system.

Give an expression for the zeros of the system transfer function.

Under what condition will the state of the system be controllable?

Under what conditions will the system be observable from the output y?

Give an expression for the closed-loop poles if state feedback of the form u = Kx is used.

Under what conditions can the feedback matrix K be selected so that the roots of c(s) are arbitrary?

What is the advantage of using the LQR or SRL in designing the feedback matrix K?

What is the main reason for using an estimator in feedback control?

If the estimator gain L is used, give an expression for the closed-loop poles due to the estimator.

Under what conditions can the estimator gain L be selected so that the roots of e(s) = 0 are arbitrary?

If the reference input is arranged so that the input to the estimator is identical to the input to the process, what will the overall closed-loop transfer function be?

If the reference input is introduced in such a way as to permit the zeros to be assigned as the roots of (s), what will the overall closed-loop transfer function be?

What are the three standard techniques for introducing integral control in the state feedback design method?

Write the dynamic equations describing the circuit in Fig. 7.82. Write the equations as a second-order differential equation in y(t). Assuming a zero input, solve the differential equation for y(t) using Laplace transform methods for the parameter values and initial conditions shown in the figure. Verify your answer using the initial command in MATLAB. Figure 7.82 Circuit for Problem 7.1

A schematic for the satellite and scientific probe for the Gravity Probe-B (GP-B) experiment that was launched on April 30, 2004 is sketched in Fig. 7.83. Assume that the mass of the spacecraft plus helium tank, m1 , is 2000 kg and the mass of the probe, m2 , is 1000 kg. A rotor will float inside the probe and will be forced to follow the probe with a capacitive forcing mechanism. The spring constant of the coupling k is 3.2 10 6 . The viscous damping b is 4.6 10 3 . (a) Write the equations of motion for the system consisting of masses m 1 and m 2 using the inertial position variables, y1 and y2 . (b) The actual disturbance u is a micrometeorite, and the resulting motion is very small. Therefore, rewrite your equations with the scaled variables Z1 = 10 6y1 , Z2 = 10 6y2 , and v = 1000u. (c) Put the equations in state-variable form using the state x = [Z1 1 Z2 2] T , the output y = Z2 , and the input an impulse, u = 10 -3 (t) N-sec on mass m1 . (d) Using the numerical values, enter the equations of motion into MATLAB in the form and define the MATLAB system: sysGPB = ss(F,G,H,J). Plot the response of y caused by the impulse with the MATLAB command impulse(sysGPB). This is the signal the rotor must follow. (e) Use the MATLAB commands p = eig(F) to find the poles (or roots) of the system and z = tzero(F,G,H, J) to find the zeros of the system. Figure 7.83 Schematic diagram of the GP-B satellite and probe

Give the state description matrices in control-canonical form for the following transfer functions:

Use the MATLAB function tf2ss to obtain the state matrices called for in Problem 7.3.

Give the state description matrices in normal-mode form for the transfer functions of Problem 7.3. Make sure that all entries in the state matrices are real valued by keeping any pairs of complex conjugate poles together, and realize them as a separate subblock in control canonical form.

A certain system with state x is described by the state matrices Find the transformation T so that if x = Tz, the state matrices describing the dynamics of z are in control canonical form. Compute the new matrices A, B, C, and D.

Show that the transfer function is not changed by a linear transformation of state.

Use block-diagram reduction or Masons rule to find the transfer function for the system in observer canonical form depicted by Fig. 7.31.

Suppose we are given a system with state matrices F, G, H (J = 0 in this case). Find the transformation T so that, under Eqs. (7.24) and (7.25), the new state description matrices will be in observer canonical form.

Find the state transformation that takes the observer canonical form of Eq. (7.35) to the modal canonical form.

(a) Find the transformation T that will keep the description of the tape-drive system of Example 7.11 in modal canonical form but will convert each element of the input matrix Bm to unity. (b) Use MATLAB to verify that your transformation does the job.

(a) Find the state transformation that will keep the description of the tape-drive system of Example 7.11 in modal canonical form but will cause the poles to be displayed in Am in order of increasing magnitude. (b) Use MATLAB to verify your result in part (a), and give the complete new set of state matrices as A, B, C, and D.

Find the characteristic equation for the modal-form matrix Am of Eq. (7.17a) using Eq. (7.58).

Given the system with zero initial conditions, find the steady-state value of x for a step input u

Consider the system shown in Fig. 7.84: (a) Find the transfer function from U to Y . (b) Write state equations for the system using the state-variables indicated. Figure 7.84 A block diagram for Problem 7.16

Using the indicated state-variables, write the state equations for each of the systems shown in Fig. 7.85. Find the transfer function for each system using both block-diagram manipulation and matrix algebra [as in Eq. (7.48)].

For each of the listed transfer functions, write the state equations in both control and observer canonical form. In each case, draw a block diagram and give the appropriate expressions for F, G, and H. (control of an inverted pendulum by a force on the cart).

Consider the transfer function (a) By rewriting Eq. (7.263) in the form find a series realization of G(s) as a cascade of two first-order systems. (b) Using a partial-fraction expansion of G(s), find a parallel realization of G(s). (c) Realize G(s) in control canonical form.

For the system design a state feedback controller that satisfies the following specifications: (a) Closed-loop poles have a damping coefficient = 0.707. (b) Step-response peak time is under 3.14 sec. Verify your design with MATLAB.

(a) Design a state feedback controller for the following system so that the closed-loop step response has an overshoot of less than 25% and a 1% settling time under 0.115 sec: (b) Use the step command in MATLAB to verify that your design meets the specifications. If it does not, modify your feedback gains accordingly

Consider the system (a) Design a state feedback controller for the system so that the closed-loop step response has an overshoot of less than 5% and a 1% settling time under 4.6 sec. (b) Use the step command in MATLAB to verify that your design meets the specifications. If it does not, modify your feedback gains accordingly.

Consider the system in Fig. 7.86. (a) Write a set of equations that describes this system in the standard canonical control form as = Fx + Gu and y = Hx. (b) Design a control law of the form that will place the closed-loop poles at s = 2 2j. Figure 7.86 System for Problem 7.24

Output Controllability. In many situations a control engineer may be interested in controlling the output y rather than the state x. A system is said to be output controllable if at any time you are able to transfer the output from zero to any desired output y* in a finite time using an appropriate control signal u*. Derive necessary and sufficient conditions for a continuous system (F, G, H) to be output controllable. Are output and state controllability related? If so, how?

Consider the system (a) Find the eigenvalues of this system.(Hint: Note the block-triangular structure.) (b) Find the controllable and uncontrollable modes of this system. (c) For each of the uncontrollable modes, find a vector v such that v T G = 0, v T F = V T . (d) Show that there are an infinite number of feedback gains K that will relocate the modes of the system to 5, 3, 2, and 2. (e) Find the unique matrix K that achieves these pole locations and prevents initial conditions on the uncontrollable part of the system from ever affecting the controllable part.

Two pendulums, coupled by a spring, are to be controlled by two equal and opposite forces u, which are applied to the pendulum bobs as shown in Fig. 7.87. The equations of motion are (a) Show that the system is uncontrollable. Can you associate a physical meaning with the controllable and uncontrollable modes? (b) Is there any way that the system can be made controllable? Figure 7.87 Coupled pendulums for Problem 7.27

The state-space model for a certain application has been given to us with the following state description matrices: (a) Draw a block diagram of the realization with an integrator for each state-variable. (b) A student has computed det C = 2.3 10 -7 and claims that the system is uncontrollable. Is the student right or wrong? Why? (c) Is the realization observable?

Staircase Algorithm (Van Dooren et al., 1978): Any realization (F, G, H) can be transformed by an orthogonal similarity transformation to where is an upper Hessenberg matrix (having one nonzero diagonal above the main diagonal) given by where g1 0, and Orthogonal transformations correspond to a rotation of the vectors (represented by the matrix columns) being transformed with no change in length. (a) Prove that if i = 0 and i+ 1,..., n-1 0 for some i, then the controllable and uncontrollable modes of the system can be identified after this transformation has been done. (b) How would you use this technique to identify the observable and unobservable modes of (F, G, H)? (c) What advantage does this approach for determining the controllable and uncontrollable modes have over transforming the system to any other form? (d) How can we use this approach to determine a basis for the controllable and uncontrollable subspaces, as in Problem 7.14? This algorithm can also be used to design a numerically stable algorithm for pole placement [see Minimis and Paige (1982)]. The name of the algorithm comes from the multi-input version in which the i are the blocks that make resemble a staircase. Refer to ctrbf, obsvf commands in MATLAB.

Consider the feedback system in Fig. 7.88. Find the relationship between K, T, and such that the closed-loop transfer function minimizes the integral of the time multiplied by the absolute value of the error (ITAE) criterion, for a step input. Figure 7.88 Control system for Problem 7.31

Prove that the Nyquist plot for LQR design avoids a circle of radius one centered at the 1 point, as shown in Fig. 7.89. Show that this implies that < GM < , the upward gain margin is GM= , and there is a downward GM = , and the phase margin is at least PM = 60. Hence the LQR gain matrix, K, can be multiplied by a large scalar or reduced by half with guaranteed closed-loop system stability. Figure 7.89 Nyquist plot for an optimal regulator

Consider the system and assume that you are using feedback of the form u = Kx + r, where r is a reference input signal. (a) Show that (F, H) is observable. (b) Show that there exists a K such that (F GK, H) is unobservable. (c) Compute a K of the form K = [1, K2] that will make the system unobservable as in part (b); that is, find K2 so that the closed-loop system is not observable. (d) Compare the open-loop transfer function with the transfer function of the closed-loop system of part (c). What is the unobservability due to?

Consider a system with the transfer function (a) Find (Fo , Go , Ho ) for this system in observer canonical form. (b) Is (Fo , Go ) controllable? (c) Compute K so that the closed-loop poles are assigned to s = 3 3j. (d) Is the closed-loop system of part (c) observable? (e) Design a full-order estimator with estimator error poles at s = 12 12j. (f) Suppose the system is modified to have a zero: Prove that if u = Kx + r, there is a feedback gain K that makes the closed-loop system unobservable. [Again assume an observer canonical realization for G1 (s).]

Explain how the controllability, observability, and stability properties of a linear system are related.

Consider the electric circuit shown in Fig. 7.90. (a) Write the internal (state) equations for the circuit. The input u(t) is a current, and the output y is a voltage. Let X1 = iL and X2 = vc . (b) What condition(s) on R, L, and C will guarantee that the system is controllable? (c) What condition(s) on R, L, and C will guarantee that the system is observable?

The block diagram of a feedback system is shown in Fig. 7.91. The system state is Figure 7.91 Block diagram for Problem 7.37 and the dimensions of the matrices are as follows: (a) Write state equations for the system. (b) Let x = Tz, where Show that the system is not controllable. (c) Find the transfer function of the system from r to y.

This problem is intended to give you more insight into controllability and observability. Consider the circuit in Fig. 7.92, with an input voltage source u(t) and an output current y(t). (a) Using the capacitor voltage and inductor current as state-variables, write state and output equations for the system. (b) Find the conditions relating R 1, R 2, C, and L that render the system uncontrollable. Find a similar set of conditions that result in an unobservable system. (c) Interpret the conditions found in part (b) physically in terms of the time constants of the system. (d) Find the transfer function of the system. Show that there is a pole-zero cancellation for the conditions derived in part (b) (that is, when the system is uncontrollable or unobservable). Figure 7.92 Electric circuit for Problem 7.38

The linearized equations of motion for a satellite are = Fx + Gu, y = Hx, where The inputs u1 and u2 are the radial and tangential thrusts, the state-variables X1 and X3 are the radial and angular deviations from the reference (circular) orbit, and the outputs y1 and y2 are the radial and angular measurements, respectively. (a) Show that the system is controllable using both control inputs. (b) Show that the system is controllable using only a single input. Which one is it? (c) Show that the system is observable using both measurements. (d) Show that the system is observable using only one measurement. Which one is it?

A certain fifth-order system is found to have a characteristic equation with roots at 0, 1, 2, and 1 1j. A decomposition into controllable and uncontrollable parts discloses that the controllable part has a characteristic equation with roots 0, and 1 1j. A decomposition into observable and nonobservable parts discloses that the observable modes are at0, 1, and 2. (a) Where are the zeros of b(s) = Hadj(sI F)G for this system? (b) What are the poles of the reduced-order transfer function that includes only controllable and observable modes?

Consider the systems shown in Fig. 7.94, employing series, parallel, and feedback configurations. (a) Suppose we have controllable-observable realizations for each subsystem: i = Fi xi + Gi ui , yi = Hixi , where i = 1, 2. Give a set of state equations for the combined systems in Fig. 7.93. (b) For each case, determine what condition(s) on the roots of the polynomials Ni and Di is necessary for each system to be controllable and observable. Give a brief reason for your answer in terms of pole-zero cancellations.

Consider the system + 3 + 2y = + u. (a) Find the state matrices Fc , Gc , and Hc in control canonical form that correspond to the given differential equation. (b) Sketch the eigenvectors of Fc in the (x1 , X2 ) plane, and draw vectors that correspond to the completely observable (x0 ) and the completely unobservable (x ) state-variables. (c) Express x0 and x in terms of the observability matrix O. (d) Give the state matrices in observer canonical form and repeat parts (b) and (c) in terms of controllability instead of observability.

The equations of motion for a station-keeping satellite (such as a weather satellite) are where X = radial perturbation, y = longitudinal position perturbation, u = engine thrust in the y-direction, as depicted in Fig. 7.95. If the orbit is synchronous with the earths rotation, then = 2/(3600 24) rad/sec. (a) Is the state x = [x y ] T observable? (b) Choose x = [ x y ] T as the state vector and y as the measurement, and design a full-order observer with poles placed at s = 2, 3, and 3 3j.

The linearized equations of motion of the simple pendulum in Fig. 7.96 are + 2 = u. (a) Write the equations of motion in state-space form. (b) Design an estimator (observer) that reconstructs the state of the pendulum given measurements of . Assume = 5 rad/sec, and pick the estimator roots to be at s = 10 10j. Figure 7.96 Pendulum diagram for Problem 7.45 (c) Write the transfer function of the estimator between the measured value of and the estimated value of . (d) Design a controller (that is, determine the state feedback gain K so that the roots of the closed-loop characteristic equation are at s = 4 4j.

An error analysis of an inertial navigator leads to the set of normalized state equations where X1 = eastvelocity error, X2 = platform tilt about the north axis, X3 = northgyro drift, u = gyro drift rate of change. Design a reduced-order estimator with y = X1 as the measurement, and place the observer-error poles at 0.1 and 0.1. Be sure to provide all the relevant estimator equations.

A certain process has the transfer function . (a) Find F, G, and H for this system in observer canonical form. (b) If u = Kx, compute K so that the closed-loop control poles are located at s = 2 2j. (c) Compute L so that the estimator error poles are located at s = 10 10j. (d) Give the transfer function of the resulting controller [for example, using Eq. (7.177)]. (e) What are the gain and phase margins of the controller and the given open-loop system?

The linearized longitudinal motion of a helicopter near hover (see Fig. 7.97) can be modeled by the normalized third-order system Figure 7.97 Helicopter for Problem 7.48 Suppose our sensor measures the horizontal velocity u as the output; that is, y = u. (a) Find the open-loop pole locations. (b) Is the system controllable? (c) Find the feedback gain that places the poles of the system at s = 1 1j and s = 2. (d) Design a full-order estimator for the system, and place the estimator poles at 8 and . (e) Design a reduced-order estimator with both poles at 4. What are the advantages and disadvantages of the reduced-order estimator compared with the full-order case? (f) Compute the compensator transfer function using the control gain and the full-order estimator designed in part (d), and plot its frequency response using MATLAB. Draw a Bode plot for the closed-loop design, and indicate the corresponding gain and phase margins. (g) Repeat part (f) with the reduced-order estimator. (h) Draw the SRL and select roots for a control law that will give a control bandwidth matching the design of part (c), and select roots for a full-order estimator that will result in an estimator error bandwidth comparable to the design of part (d). Draw the corresponding Bode plot and compare the pole placement and SRL designs with respect to bandwidth, stability margins, step response, and control effort for a unit-step rotor-angle input. Use MATLAB for the computations.

Suppose a DC drive motor with motor current u is connected to the wheels of a cart in order to control the movement of an inverted pendulum mounted on the cart. The linearized and normalized equations of motion corresponding to this system can be put in the form where = angle of the pendulum, = velocity of the cart. (a) We wish to control by feedback to of the form Find the feedback gains so that the resulting closed-loop poles are located at (b) Assume that and v are measured. Construct an estimator for and of the form where x = [ ] T and y = Treat both v and u as known. Select L so that the estimator poles are at 2 and 2. (c) Give the transfer function of the controller, and draw the Bode plot of the closed-loop system, indicating the corresponding gain and phase margins. (d) Using MATLAB, plot the response of the system to an initial condition on, and give a physical explanation for the initial motion of the cart.

Unstable equations of motion of the form arise in situations where the motion of an upside-down pendulum (such as a rocket) must be controlled. (a) Let u = Kx (position feedback alone), and sketch the root locus with respect to the scalar gain K. (b) Consider a lead compensator of the form Select a and K so that the system will display a rise time of about 2 sec and no more than 25% overshoot. Sketch the root locus with respect to K. (c) Sketch the Bode plot (both magnitude and phase) of the uncompensated plant. (d) Sketch the Bode plot of the compensated design, and estimate the phase margin. (e) Design state feedback so that the closed-loop poles are at the same locations as those of the design in part (b). (f) Design an estimator for x and using the measurement of x = y, and select the observer gain L so that the equation for x has characteristic roots with a damping ratio = 0.5 and a natural frequency = 8. (g) Draw a block diagram of your combined estimator and control law, and indicate where and appear. Draw a Bode plot for the closed-loop system, and compare the resulting bandwidth and stability margins with those obtained using the design of part (b).

A simplified model for the control of a flexible robotic arm is shown in Fig. 7.98, where k/M = 900 rad/sec 2 , y = output, the mass position, u = input, the position of the end of the spring. (a) Write the equations of motion in state-space form. (b) Design an estimator with roots as s = 100 100 j. (c) Could both state-variables of the system be estimated if only a measurement of was available? (d) Design a full-state feedback controller with roots at s = 20 20j. (e) Would it be reasonable to design a control law for the system with roots at s = 200 200j? State your reasons. (f) Write equations for the compensator, including a command input for y. Draw a Bode plot for the closed-loop system and give the gain and phase margins for the design. Figure 7.98 Simple robotic arm for Problem 7.52

The linearized differential equations governing the fluid-flow dynamics for the two cascaded tanks in Fig. 7.99 are where h1 = deviation of depth in tank 1 from the nominal level, h2 = deviation of depth in tank 2 from the nominal level, h = deviation in fluid inflow rate to tank 1 (control). (a)Level Controller for Two Cascaded Tanks: Using state feedback of the form u = K1h1 K2h2 , Figure 7.99 Coupled tanks for Problem 7.53 choose values of K1 and K2 that will place the closed-loop eigenvalues at s = 2(1j). (b) Level Estimator for Two Cascaded Tanks: Suppose that only the deviation in the level of tank 2 is measured (that is, y = h2 ). Using this measurement, design an estimator that will give continuous, smooth estimates of the deviation in levels of tank 1 and tank 2, with estimator error poles at 8(1 j). (c) Estimator/Controller for Two Cascaded Tanks: Sketch a block diagram (showing individual integrators) of the closed-loop system obtained by combining the estimator of part (b) with the controller of part (a). (d) Using MATLAB, compute and plot the response at y to an initial offset in Sh1. Assume = 1 for the plot.

The lateral motions of a ship that is 100 m long, moving at a constant velocity of 10 m/sec, are described by where = sideslip angle (deg), heading angle (deg), = rudder angle (deg), r = yaw rate (see Fig. 7.99). (a) Determine the transfer function from to and the characteristic roots of the uncontrolled ship. (b) Using complete state feedback of the form = K1 K2 r K3 ( d ), where d is the desired heading, determine values of K1 , K2 , and K3 that will place the closed-loop roots at s = 0.2, 0.2 0.2j. (c) Design a state estimator based on the measurement of (obtained from a gyrocompass, for example). Place the roots of the estimator error equation at s = 0.8 and 0.8 0.8j. (d) Give the state equations and transfer function for the compensator Dc(s) in Fig. 7.100, and plot its frequency response. (e) Draw the Bode plot for the closed-loop system, and compute the corresponding gain and phase margins. (f) Compute the feed-forward gains for a reference input, and plot the step response of the system to a change in heading of 5. Figure 7.100 View of ship from above for Problem 7.54 Figure 7.101 Ship control block diagram for Problem 7.54

As mentioned in footnote 11 in Section 7.9.2, a reasonable approach for selecting the feed-forward gain in Eq. (7.205) is to choose such that when r and are both unchanging, the DC gain from r to u is the negative of the DC gain from y to u. Derive a formula for based on this selection rule. Show that if the plant is Type 1, this choice is the same as that given by Eq. (7.205)

Assume that the linearized and time-scaled equation of motion for the ball-bearing levitation device is x x = u + w. Here w is a constant bias due to the power amplifier. Introduce integral error control, and select three control gains K = [K1 K2 K3 ] so that the closed-loop poles are at 1 and 1 j and the steady-state error to w and to a (step) position command will be zero. Let y = x and the reference input be a constant. Draw a block diagram of your design showing the locations of the feedback gains Ki . Assume that both X and x can be measured. Plot the response of the closed-loop system to a step command input and the response to a step change in the bias input. Verify that the system is Type 1. Use MATLAB (SIMULINK) software to simulate the system responses.

Consider a system with state matrices (a) Use feedback of the form u(t) = Kx(t) + r(t), where is a nonzero scalar, to move the poles to 3 3j. (b) Choose so that if r is a constant, the system has zero steady-state error; that is y() = r. (c) Show that if F changes to F + F, where F is an arbitrary 2 2 matrix, then your choice of in part (b) will no longer make y() = r. Therefore, the system is not robust under changes to the system parameters in F. (d) The system steady-state error performance can be made robust by augmenting the system with an integrator and using unity feedbackthat is, by setting I = r y, where xI is the state of the integrator. To see this, first use state feedback of the form u = Kx K1x1 so that the poles of the augmented system are at (e) Show that the resulting system will yield y() = r no matter how the matrices F and G are changed, as long as the closed-loop system remains stable. (f) For part (d), use MATLAB (SIMULINK) software to plot the time response of the system to a constant input. Draw Bode plots of the controller, as well as the sensitivity function (S) and the complementary sensitivity function (T).

Consider a servomechanism for following the data track on a computer-disk memory system. Because of various unavoidable mechanical imperfections, the data track is not exactly a centered circle, and thus the radial servo must follow a sinusoidal input of radian frequency 0 (the spin rate of the disk). The state matrices for a linearized model of such a system are The sinusoidal reference input satisfies r = 2 0 r. (a) Let 0 = 1, and place the poles of the error system for an internal model design at c(s) = (s + 2 j2)(s + 1 1 j) and the pole of the reduced-order estimator at e(s) = (s + 6). (b) Draw a block diagram of the system, and clearly show the presence of the oscillator with frequency 0 (the internal model) in the controller. Also verify the presence of the blocking zeros at j0. (c) Use MATLAB (SIMULINK) software to plot the time response of the system to a sinusoidal input at frequency 0= 1. (d) Draw a Bode plot to show how this system will respond to sinusoidal inputs at frequencies different from but near 0.

Compute the controller transfer function [from y (s) to u(s)] in Example 7.38. What is the prominent feature of the controller that allows tracking and disturbance rejection?

Consider the system with the transfer function e TsG(s), where The Smith compensator for this system is given by, Plot the frequency response of the compensator for T = 5 and Dc = 1, and draw a Bode plot that shows the gain and phase margins of the system. 22