EE5102 - Multivariable Control Systems

Part II Homework II

Swee King PhangA0033585A

April 3, 2011

1 Motivation

In this project, a problem description on an unmanned helicopter flight control system is proposed. We arerequired to use either the H2 optimal or the H∞ suboptimal control methodology to design a measurementfeedback control law that meet the following specification:

1. The overall system is stabilized;

2. The dynamics of the helicopter will be driven to the hovering state, i.e. all state variables are to bedriven to 0, as quickly as possible from the given initial condition.

The proposed H2 and H∞ controller has been widely used in the past decades. In this assignment, H2

optimal controller is used as it has the following advantages over the H∞ controller:

1. We are able to get the optimal control law for H2 controller, but only a suboptimal control law for H∞controller;

2. The optimal value ∗ of H2 controller can be computed easily, while ∗ of H∞ controller has no closedform solution, can only be computed numerically;

3. By using the small perturbation method, the resultant H∞ control law will cause huge control signalto the system;

4. H2 controller is relatively easier to design and be computed by solving simple Riccati equation (ARE).

2 Unmanned Helicopter State Space Model

According to the given parameters, the unmanned aerial vehicle (UAV) platform codenamed HeLion has alinearized state space model as described below:

x = Ax+Bu

y = Cx

1

where

A =

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

−0.1778 0 0 0 0 −9.7807 −9.7807 0 0 0 00 −0.3104 0 0 9.7807 0 0 9.7807 0 0 0

−0.3326 −0.5353 0 0 0 0 75.764 343.86 0 0 0−0.1903 −0.2490 0 0 0 0 172.62 −59.958 0 0 0

0 0 1 0 0 0 0 0 0 0 00 0 0 1 0 0 0 0 0 0 00 0 0 −1 0 0 −8.1222 4.6535 0 0 00 0 −1 0 0 0 −0.0921 −8.1222 0 0 00 0 0 0 0 0 0 0 −0.6821 −0.0535 00 0 0 0 0 0 0 0 −0.2892 −5.5561 −36.6740 0 0 0 0 0 0 0 0 2.7492 −11.112

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠,

B =

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

0 0 0 00 0 0 00 0 0 00 0 0 00 0 0 00 0 0 0

0.0496 2.6224 0 02.4928 0.1741 0 0

0 0 7.8246 00 0 1.6349 −58.40530 0 0 0

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠, x =

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

UVpq��asbsWrrfb

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠, u =

⎛⎜⎜⎝�lat�lon�col�ped

⎞⎟⎟⎠

C =

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

1 0 0 0 0 0 0 0 0 0 00 1 0 0 0 0 0 0 0 0 00 0 1 0 0 0 0 0 0 0 00 0 0 1 0 0 0 0 0 0 00 0 0 0 1 0 0 0 0 0 00 0 0 0 0 1 0 0 0 0 00 0 0 0 0 0 0 0 1 0 00 0 0 0 0 0 0 0 0 1 0

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠.

Furthermore, a form of wind gust disturbance will be considered in the problem formulation. Thus thedynamical equation is modified as

x = Ax+Bu+ Ew

where

E = A

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

1 0 00 1 00 0 00 0 00 0 00 0 00 0 00 0 00 0 10 0 00 0 0

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠, w =

⎛⎝uwindvwindwwind

⎞⎠ =

⎛⎝10 cos(2�40 (t− 20)

)10 cos

(2�40 (t− 20)

)3 cos

(2�40 (t− 20)

)⎞⎠ , 10 ≤ t ≤ 30

2

3 H2 Controller Design

In our simulation, we are required to show that all state variables are able to be driven to 0 from an initialcondition of

x0 =(

1 0 0 0 0 −0.1 0 0 0 0 0)T

and the input to the system is limited as

∣�lat∣ < 0.35

∣�lon∣ < 0.35

∣�col∣ < 0.12

∣�ped∣ < 0.40

due to physical limitation and safety.

Now let us consider the UAV system written in

x = Ax+Bu+ Ew

y = C1x+D1w

z = C2x+D2u

where w is the disturbance to the system, y is the measured output of the system, and z is the output to becontrolled of the system. Also, we have the controller dynamic given by

v = Acv +Bcy

u = Ccv +Dcy

Here, Ac, Bc, Cc and Dc are the system matrices of the controller. In order to design the controller, we needto satisfy 4 conditions:

1. D2 is of maximal column rank;

2. The subsystem (A,B,C2, D2) has no invariant zeros on the imaginary axis;

3. D1 is of maximal row rank;

4. The subsystem (A,E,C1, D1) has no invariant zeros on the imaginary axis;

However, for our case in this assignment, we can easily see that the system does not satisfy the conditions,as both D1 and D2 are equal to 0 matrix. This is refer as a singular case.

In literature, for a general system which the regularity conditions are not satisfied, it can be solved by usinga perturbation approach. In this approach, we define a new controlled output:

z =

⎡⎣ z"x"u

⎤⎦ =

⎡⎣C2

"I0

⎤⎦x+

⎡⎣D2

0"I

⎤⎦uand new matrices associated with the disturbance inputs:

E =[E "I 0

]D1 =

[D1 0 "I

]Then, the singular output feedback system will then transform to the following perturbed regular systemwith sufficiently small ":

x = Ax+Bu+ Ew

y = C1x+ D1w

z = C2x+ D2u

3

Here, intuitively, we let w to be the combination of the disturbance w as the first 3 variables in w, andrandom measurement Gaussian noise as the last 8 variables. The Gaussian noise has a mean of 0 andstandard deviation of 0.1. The invariant zero of the systems can be checked by using the Matlab commandtzero(A,B,C,D). It turns out that there is no invariant zero for both the perturbed subsystems, thus as-sumption 2 and 4 above become valid.

Next, the controller is designed by first solving the Riccati equations

ATP + PA+ CT2 C2 − (PB + CT2 D2)(DT2 D2)−1(DT

2 C2 +BTP ) = 0

QAT +AQ+ EET − (QCT1 + EDT1 )(D1D

T1 )−1(D1E

T + C1Q) = 0

to obtain 2 positive definite P and Q matrices. Matlab command h2care from the MIMO toolkit developedby the course lecturer, Professor Ben M. Chen is used to compute the matrix P and Q. Positive definitenesscan be shown by finding the eigenvalue of the matrices using Matlab command eigP. Here, eigenvalues of Pare

4.92, 4.73, 0.23, 0.21, 0.22, 0.15, 0.01, 0.007, 0.05, 0.002, 0.013

and eigenvalues of Q are

0.13, 0.07, 0.05, 0.053, 0.004, 0.004, 0.0004, 0.0004, 0.06, 0.005, 0.0004

As shown here, all eigenvalues are positive for both the matrices.

Upon obtaining the matrix P and Q, the H2 optimal output feedback controller can be computed as follows:

v = (A+BF +KC1)v −Kyu = Fv

where F = −(DT2 D2)−1(DT

2 C2 +BTP ) and K = −(QCT1 + EDT1 )(D1D

T1 )−1. Here, we have

Ac = A+BF +KC1

Bc = −KCc = F

Dc = 0

and the closed-loop transfer function from w to z will then be given by

T (s) = Ccl(sI −Acl)−1Bcl +Dcl

where

Acl =

[A BCc

BcC1 Ac

]Bcl =

[E

BcD1

]Ccl =

[C2 D2Cc

]Dcl = 0

To check the stability of the closed-loop system, Matlab command eig is used to find eigenvalues of Acl:

−587,−78,−11,−67± 67j,−49± 47j,−10± 20j,−8± 14j,−2± 2j,−2.2± 2.2j,−3± 2j,−3± 1.8j,−9± 10j

Here, all eigenvalue has a negative real part, and hence the closed-loop system is asymptotically stable.

Lastly, the optimal value ∗2 is computed as

∗2 = {trace(ETPE) + trace[(ATP + PA+ CT2 C2)Q]}1/2 = 1.0125

4

4 Simulation Results

Since the UAV model parameters and controller gains have already derived in the previous section, simu-lation can be done by using Matlab and Simulink. In this section, simulation results will be provided forclosed-loop system with input and measurement noise, together with a wind gust disturbance in the form ofsine wave to the velocity channels.

Fig. 1 shows the Simulink block diagram of the controller design in this part. Here, the random noise addedto the input and the measurement output are in Normal distribution with mean 0 and standard deviation0.1. The simulation results in Fig. 2 to Fig. 4 has verified that this H2 controller is able to give goodresponses to the closed-loop system.

In detail, Fig. 2 shows the U, V,W and p, q, r of the UAV in simulation from time t = 0 to t = 40 secs.However, at time t = 10 to t = 30 sec, the UAV is undergoing wind gust disturbances. Based on the plottedgraphs, it seems that the controller designed here provide better disturbance rejection as compared to theprevious assignment, where LQG/LQR controller was used. Here, although there are still defection fromthe equilibrium state during the period of disturbance, it shows a much lower amplitude as compared to theprevious assignment.

Fig. 3 shows the �, � state variables of the system. Similarly, during the period of disturbance, the angledeviate with a small magnitude. In steady state, the system is able to achieve asymptotic stable again as allstate variables converges to 0.

Lastly, Fig. 4 shows the control input to the UAV. Note that here, the 3rd control input, �col is saturated(reached it’s limit) due to countering the effect of the disturbance. Here, the disturbance effect mainly causedby the wind gust on z direction as discussed in previous assignment. We can see that now with this H2

controller design, the system is trying to achieve the equilibrium state again by applying large control actionto the �col input, which directly alter the z-velocity, W in this case.

5 Conclusion

In conclusion, as can be seen from the results, H2 optimal controller is able to reject disturbance fairly well.Intuitively, H2 control law is a type of optimal control as it minimize the H2 norm of the transfer functionT . In other word, the H2 norm is the total energy corresponding to the impulse response of the transferfunction from w to z, and thus minimizing it is equivalent to the minimization of the total energy from thedisturbance w to the controlled output z. H2 controller is able to achieve this aim by reaching the optimalvalue, ∗2 = 1.0125, in this assignment.

5

Figure 1: Simulink block diagram of H2 controller design

6

Figure 2: Velocities and angular velocities of the UAV

7

Figure 3: Euler angles of the UAV

8

Figure 4: Control signal of the UAV

9