lmi control design with input covariance constraint for a … · 2018-05-16 · lmi control design...

LMI CONTROL DESIGN WITH INPUT COVARIANCE CONSTRAINT FOR ATENSEGRITY SIMPLEX STRUCTURE

Ali Khudhair Al-JibooryMechanical Engineering

Michigan State UniversityEast Lansing, Michigan 48824

Email: [email protected]

Guoming ZhuMechanical Engineering

Michigan State UniversityEast Lansing, Michigan 48824


Cornel SultanAerospace and Ocean Engineering

Virginia Technology UniversityBlacksburg, Virginia 24061


ABSTRACTThe Input Covariance Constraint (ICC) control problem

is an optimal control problem that minimizes the trace of aweighted output covariance matrix subject to multiple con-straints on the input (control) covariance matrix. ICC control de-sign using the Linear Matrix Inequality (LMI) approach was pro-posed and applied to a tensegrity simplex structure in this paper.Since it has been demonstrated that the system control variancesare directly associated with the actuator sizes for a given set ofL2 disturbances, the tensegrity simplex design example is usedto demonstrate the capability of using the ICC controller to op-timize the system performance in the sense of output covariancewith a given set of actuator constraints. The ICC control designwas compared with two other control design approaches, poleplacement and Output Covariance Constraint (OCC) control de-signs. Simulation results show that the proposed ICC controllersoptimize the system performance (the trace of a weighted outputcovariance matrix) for the given control covariance constraintswhereas the other two control design methods cannot guaranteethe feasibility of the designed controllers. Both, state feedbackand full-order dynamic output feedback controllers have beenconsidered in this work.

INTRODUCTIONThe ICC control problem is an optimal control problem that

minimizes the trace of the weighted output covariance matrixsubject to multiple constraints on the input (control) covariancematrix. The constraints on the input covariance can be inter-

preted physically as constraints on actuator sizes of the closedloop system [1,2]. The physical interpretation of the ICC controlproblem is that for a set of actuators with a given size, the ICCcontroller provides the best performance in terms of the outputcovariance matrix. Therefore, it is an optimization problem thatminimizes the output performance subject to multiple constraintson the control input covariance matrix U of the form Ui ≤U i,where U i is given and it represents the upper bound on the con-trol covariance of the input channel i.

The ICC control problem has two interesting interpretations:stochastic and deterministic. For the stochastic interpretationthe exogenous inputs are assumed to be uncorrelated zero-meanwhite noises with a given intensity. With the exogenous inputdefined in this way, the ICC control problem minimizes the traceof the weighted output covariance while satisfying constraints onactuators size. These constraints can be interpreted as constraintson the variance of the control inputs covariance matrix. On theother hand, deterministic interpretation assumes that the exoge-nous inputs are unknown disturbances that belong to a boundedL2 energy set. Since the output variance is the L2 to L∞ gainof the closed loop system from L2 disturbance input to the as-sociated output channel, it is critical to design a controller thatguarantees L∞ output performance as long as this performance isdirectly related to the actuators capabilities. Therefore, the ICCcontrol problem is to design a stabilizing controller that mini-mizes the weighted sum of worst-case peak values (L∞ norm) onthe performance outputs subject to constraints on the worst-casepeak values of the control inputs. This interpretation is importantin practical applications where hard constraints on the actuating

Proceedings of the ASME 2014 Dynamic Systems and Control Conference DSCC2014

October 22-24, 2014, San Antonio, TX, USA

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signals are present, such as space telescope pointing control [3],system identification, and machine tool control.

Covariance controllers have been widely studied in controlliterature for the last two decades [4, 5]. Chen et al [6] consid-ered the constrained LQ control problem that minimizes the con-trol energy while satisfying output covariance constraints. Zhu etal [7] developed algorithm that solves the problem in [6] with op-timal selection of the output weight matrix and guaranteed con-vergence. The constrained LQ control that minimizes quadraticperformance index have been considered in [8] with LMI formu-lation. Therefore, this paper will consider the dual problem con-sidered in [6, 7] of design control policy that minimizes the out-put performance while satisfying control covariance constraintsusing LMI approach.

For a linear system, it is assumed that the system actuatorsare linear with infinite magnitude. The resulting closed loop con-trol system could drive control to the level that a physical actu-ator can not provide, which violates the assumption of linearity.In this case, the closed loop system stability and/or performancecannot be guaranteed. Hence, it is important to guarantee thatthe system remains in the linear range. Also in many spacecraftcontrol problems, the actuator size is critical since it is related tothe spacecraft payload. Therefore, it is critical to design a con-troller that optimizes the closed loop system performance sub-ject to the constraints on the actuator sizes (or the magnitude ofcontrol signals). Additionally, when there are multiple actuatorsavailable to control a multi-input physical system, it can be diffi-cult to know how to obtain the best performance with the givenactuation resources. The ICC controller not only guarantees thatthe control signals will stay within the actuator capabilities butalso utilizes all the actuator resources to provide the best pos-sible performance. These concepts will be demonstrated in thispaper through the design example.

The outline of this paper is as follow. Section II describesthe necessary theorems for the ICC control problem using LMIformulation for state feedback control and full-order dynamicoutput feedback control in two subsections respectively. TheLMI solution of the dual problem, OCC control problem is alsoprovided in this section. A realistic tensegrity simplex structuremodel (which has been used as an application design example)with its mathematical model are presented in Section III. To showthe effectiveness of the ICC control design proposed in this pa-per, the closed loop system performance of the ICC control de-sign is compared with the pole placement and OCC designs inSection IV. Conclusion remarks are given in the last section.

LMI FORMULATION OF ICC PROBLEMLinear matrix inequalities (LMI) have received substantial

attention as a powerful formulation and design tool for a vari-ety of linear control problems [9,10]. Regarding the ICC controlproblem, this section describes the formal statement of the prob-

lem and gives the theoretical foundation to design ICC controllerusing LMI formulation. Consider the LTI continuous-time sys-tem:

xp(t) = Apxp(t)+Bpu(t)+Dpwp(t)

yp(t) =Cpxp(t)

z(t) = Mpxp + v(t)(1)

where xp(t), u(t), yp(t), wp(t), z(t), and v(t)represent the states,control, controlled variables, process noise, measured outputs,and measurement noise, respectively.

State Feedback ControlSuppose that we apply to the open-loop (1) a full state feed-

back stabilizing controller of the form:

u(t) = K xp(t) (2)

Then the resulting closed-loop system is

x(t) = Ax(t)+Dw(t)

y(t) =[

yp(t)u(t)

]=

[CpK

]x(t) =Cx(t)

A = Ap +BpK

(3)

where

DW 1/2 = DpW 1/2p

x = xp, w = wp(4)

and W = W T > 0 with dimensions compatible to the noise w.Considering the closed-loop system (3), let X represent theclosed-loop controllability Gramian from the (weighted) distur-bance input W−1/2w. Since A is stable, X satisfies

0 = AX +X AT +DWDT (5)

in other words, X represents state covariance matrix. The con-trol input u(t) in (3) is partitioned into

u = [uT1 ,u

T2 , · · · ,uT

m]T, (6)

such that each ui for i = 1,2, · · · ,m is given by

ui = Ki x = Φi K x ∈ R, (7)



where Φi is an appropriately selected projection matrix for eachinput ui. Our goal is to find controller of the form (2) that mini-mizes the (weighted) output performance trace(Wy CpX CT

p ) andsatisfies the constraints

Ui = Φi U ΦTi ≤U i, i = 1,2, · · · ,m

U , KX KT(8)

where U is the control covariance, X solves (5), U i > 0 (i =1,2, · · · ,m) are given (upper bounds), and Wy > 0 is the outputweighting matrix. This problem, which is called the input co-variance constraint (ICC) problem, is defined as follows. Find astatic state feedback controller for the system (1) to:

1. Stabilize the closed loop system (3), and2. Minimize the ICC cost

JICC = trace(Wy Y )

Y ,CpX CTp

(9)

subject to (5) and (8), where Y represents the output covari-ance matrix.

In this paper, we consider a convex optimization solution tothe ICC problem using LMIs. The next two theorems are impor-tant in such formulation (See [11] for proof).

Theorem 1. Consider the closed-loop system (3). Given theinput constraints U i for i= 1,2, · · · ,m, if there exists a symmetricmatrix P such that the following LMIs are satisfied

[AP +PAT DW 1/2

W 1/2DT −I

]< 0, (10)

WyCyPCTy > 0, (11)[

U i ΦiKPPKTΦT

i P

]> 0, (12)

then the closed-loop system (3) is asymptotically stable with aninput covariance bounded by

U i > ΦiKPKTΦ

Ti > ΦiKX KT

ΦTi =Ui

∀ i = 1,2, · · · ,m(13)

and an ICC cost bounded by

JICC = trace(WyCpPCT

p)

≥ trace(WyCpX CT

p)= JICC

(14)

�

Theorem 2. There exists a controller in the form (2), given by

K = LP−1, (15)

that minimizes JICC (9) and satisfies the input constraints (8) ifthere exists a matrix L ∈ Rm×n and a symmetric positive definitematrix P ∈ Rn×n that minimize the upper bound of the ICC cost

JICC = minP,L

trace(WyCpPCT

p)

> trace(WyCpX CT

p)= JICC

(16)

subject to the LMIs

[PAT

p +ApP+LTBTp +BpL DW 1/2

∗ −I

]< 0, (17)

WyCpPCTp > 0, (18)[

U i ΦiL∗ P

]> 0, (19)

for i = 1,2, · · · ,m. �

On the other hand, the OCC control problem can be definedas following: find a static state feedback controller for the system(1) to minimize the OCC cost

JOCC = trace(Wu U ) , Wu > 0 (20)

subject to (5) and

Yi =Cp,iX CTp,i ≤ Y i, i = 1,2, · · · ,m

CTp = [CT

p,1,CTp,2, ...,C

Tp,m]

(21)

Y i > 0 are given bounds on the output performance and Wu isweighting matrix on control input channels. In other words, theOCC control problem minimizes the control effort while ensur-ing that the maximum singular value of the regulated outputs areless than the corresponding output performance constraints. Thenext two theorems illustrates the LMI formulation of the OCCcontroller (see [2] for proof).


K = LP−1, (22)

that minimizes JOCC (20) and satisfies the output constraints (21)if there exists a matrix L ∈ Rm×n and a symmetric positive def-inite matrix P ∈ Rn×n and Z ∈ Rp×p that minimize the upper



bound of the OCC cost

JOCC = minP,L,Z

trace(WuCpPCT

p)> trace(WuU ) = JOCC (23)

subject to the LMIs

[PAT

p +ApP+LTBTp +BpL DW 1/2

∗ −I

]< 0, (24)[

Z W 1/2u L

∗ P

]> 0, (25)

Y i−WuCp,iPCTp,i ≥ 0 (26)

for i = 1,2, · · · ,m. �

Stochastic analysis of the ICC controller provides informa-tion to bound the deterministic behavior (L∞ norm) of the systemas illustrated in the next theorem [2].

Theorem 4. For the asymptotically stable system (1), the out-put covariance matrix is Y , CpX Cp , where X is defined by(5). Let w in (1) be any L2 disturbance such that

‖ w ‖22,∫

∞

0wT (τ)W−1w(τ)dτ < ∞ (27)

then

‖ y ‖2∞= sup

t≥0‖ y ‖2≤ σ(Y ) ‖ w ‖2

2 (28)

where ‖ y ‖∞ is the L∞ norm of the output y, and σ(Y ) is themaximum singular value of the output covariance matrix Y . �

Theorem 4 has the following interpretation. When any L2input is applied to a stable linear system, its L∞ response (usedas a measure of the actuators size in this paper) is bounded by theproduct of the L2 norm of the actual input and the square root ofthe maximum singular value (spectral norm) of the controllabil-ity gramian with input weight W .

Output Feedback ControlConsider now the full-order dynamic output feedback con-

troller given by:

xc(t) = Acxc(t)+Bcz(t),

u(t) =Ccxc(t).(29)

Then augmenting this controller with (1), results the followingclosed-loop system:

x(t) = Ax(t)+Dw(t)

y(t) =[

yp(t)u(t)

]=

[CyCu

]x(t) =Cx(t)

(30)

with the closed-loop matrices:

x(t) =[

xp(t)xc(t)

], A =

[Ap BpCc

BcMp Ac

]Cy =

[Cp 0

], Cu =

[0 Cc

]Bw =

[DpW 1/2

p 0]

, DW 1/2 =

[Bw

BcDyw

]Dyw = [0 W 1/2

v ] , W = diag[Wp Wv]

(31)

The following theorem determine the controller matrices that sat-isfy covariance constraints [11].


Ac =V−1 (Q−YApX−Y BpL−FMpX)U−1

Bc =V−1F

Cc = LU−1

(32)

that minimizes JICC (9) and satisfies the input constraints (8) ifthere exists matrices L ∈ Rm×n, F ∈ Rn×q, and Q ∈ Rn×n andsymmetric matrices X ∈ Rn×n and Y ∈ Rn×n that minimize theupper bound of the ICC cost

JICC = minX ,Y,Q,L,F

trace(WyCpXCT

p)> trace

(WyCyPCT

y)= JICC

(33)subject to the LMIs

β Ap +QT Bw? α Y Bw +FDyw? ? −I

< 0,

WyCpXCTp > 0,U i ΦiL 0

? X I? ? Y

> 0,

(34)

for i = 1,2, · · · ,m.where



β , ApX +XATp +BpL+LTBT

p

α , YAp +ATpY +FMp +MT

p FT

�

TENSEGRITY SIMPLEX STRUCTURETensegrity structures are extremely flexible prestressed as-

semblies of tendons and bars, with few rigid to rigid joints. Thesefeatures make them ideal for control applications because theirshape can be modified with low energy consumption.1

In this paper we consider, as a representative example, atensegrity simplex. This structure, depicted in Fig.1, is composedof 6 tendons labeled AiB j and BiB j and 3 bars labeled AiBi. Thebars are attached to a fixed, rigid base, via rotational frictionlessjoints at Ai. Bars are rigid, identical, of length l and negligiblethickness, and the tendons are massless, viscoelastic Voigt ele-ments, which means that the force in tendon i is

Fi = di li + ki(liri−1)

where di is the tendon damping coefficient, li the length, ki itsstiffness, and ri its rest-length. Triangle A1A2A3 is equilateral ofside length b. A right handed inertial reference frame {b1,b2,b3}is introduced with origin at the centroid of triangle A1A2A3, b1 isparallel to A1A3, and b3 perpendicular onto A1A2A3. The vectorof generalized coordinates is

q =[δ1 δ2 δ3 δ4 δ5 δ6

]Twhere δi is the angle between b3 and AiBi. The controls aretorques acting on the bars. Lagrange equations applied to thissystem yield six nonlinear second order ordinary differentialequations, which are easily linearized around an arbitrary equi-librium solution [13], [14]. The resulting linear equations are

M q+D q+K q = u (35)

where M > 0, D ≥ 0, K > 0, are mass, damping, and stiffnesssymmetric matrices, while q and u ∈ Rn are small perturbationsfrom the equilibrium values of the generalized coordinates andcontrols, respectively (see [13] for details).

In this example we consider an equilibrium configurationobtained under no external actions applied to the structure.Such equilibria are called prestressable configurations and are

1Their dynamics can also be described accurately enough using finite sets ofordinary differential equations for relatively large ranges of parameter values,which is a major benefit for control design, validation and verification; see [12]for a recent review.

A2A1 A3

B2

B1

B3

b1

b3

FIGURE 1. TENSEGRITY SIMPLEX

quintessential to tensegrity. Effectively the structure achieves theequilibrium configuration only due to the internal interactions be-tween bars and tendons [12]. In addition the prestressable con-figuration considered here belongs to the class of symmetricalconfigurations defined by

q0 =[δ α δ α +240 δ α +120

]Twhere αi is the angle between b1 and the projection of AiBi ontoA1A2A3. The structure is symmetrical with respect to the verticalaxis b3. The advantage of symmetrical configurations is that an-alytical solutions to the equilibrium problem under zero externalactions exist (see [13] where it was also proved that these con-figurations are exponentially stable and detailed formulas for theM ,D ,K matrices are given). Here a particular element of thisclass characterized by l = 1, b = 0.67,α = 15, and δ = 48.04is considered (distances are in meters and angles in degrees).The mass of each bar is m = 0.5kg and its central transversalmoment of inertia, J = 0.042 kg.m2. In addition the exampleanalyzed herein corresponds to an optimal design presented indetails in [14], for which the stiffness and damping coefficientshave the following values

k1 = 4.32, k2 = 5.33, k3 = 3.94,k4 = 2.30, k5 = 3.69, k6 = 1.52,

k7 = 13.62, c1 = 0.85, c2 = 0.71,c3 = 0.40, c4 = 0.008, c5 = 5e−9, c6 = 0.01

(36)

Note that in (36) the tendons are indexed as follows: 1 =A1B2,2 = A2B3,3 = A3B1,4 = B1B2,5 = B2B3,6 = B3B1, and allnumbers in (36) are physically achievable. For example tendonscan be made of elastomers, which have a wide range of stiff-ness and damping properties, or metals (like in Kenneth Snel-son’s tensegrity structures) or other materials for which dampingcan be ignored.



The second-order model matrices of the tensegrity simplexstructure (see Eq. (35)) are given by:

M =

0.1667 0 0 0 0 00 0.0921 0 0 0 00 0 0.1667 0 0 00 0 0 0.0921 0 00 0 0 0 0.1667 00 0 0 0 0 0.0921

,

D =

0.1579 −0.0469 0.0024 −0.0027 0.0031 0.0009−0.0469 0.0166 0.0007 −0.0008 −0.0034 −0.00100.0024 0.0007 0.3255 −0.0984 0.0000 −0.0000−0.0027 −0.0008 −0.0984 0.0310 0.0000 −0.00000.0031 −0.0034 0.0000 0.0000 0.2746 −0.07930.0009 −0.0010 −0.0000 −0.0000 −0.0793 0.0244

,

K =

15.4947 −0.7788 −0.8229 −3.6248 −1.1802 2.6607−0.7788 7.2493 2.7672 1.2260 −3.2275 1.3444−0.8229 2.7672 16.8334 −0.8723 −0.1862 −4.3329−3.6248 1.2260 −0.8723 7.6684 2.9569 1.0150−1.1802 −3.2275 −0.1862 2.9569 16.4930 −1.95952.6607 1.3444 −4.3329 1.0150 −1.9595 8.1732

The state space model is :

x(t) = Ax(t)+Bu(t), y(t) =Cx(t),

x(t) =[

q(t)q(t)

],

(37)

where the corresponding matrices are easily constructed usingthe relations [15]:

A =

[0 I

−M−1K −M−1D

], B =

[0

M−1uT

](38)

where A ∈ R2n×2n , B ∈ R2n×m , n = 6 (number of generalizedcoordinates), and m = 6 (number of inputs), respectively.

CONTROL DESIGN AND SIMULATION RESULTSInitially, pole placement controller (Kpp) has been designed

with the square root of the maximum diagonal elements of thecontrol covariance matrix has been used as constraint on thetorque actuators of the tensegrity simplex structure as follows

U i =√

max(diag(U )) =√

76.1127 = 8.7243f or i = 1,2, . . . ,6. (39)

where U is the control covariance associated with Kpp definedin (8) and i represents the ith control channel. Using Theorem2, the set of LMIs (17)-(19) has been implemented and solved inMATLAB using YALMIP [16] to design ICC controller (KICC1).By comparing the ICC cost (JICC) of the both controllers Kpp

TABLE 1. COMPARISON OF ICC WITH POLE PLACEMENTCONTROLLER

Kpp KICC1 KICC2

‖U1 ‖2∞ 50.3831 76.0996 76.0690

‖U2 ‖2∞ 15.4330 76.1087 76.0982

‖U3 ‖2∞ 76.1127 76.1019 76.0683

‖U4 ‖2∞ 24.2909 76.1087 38.6161

‖U5 ‖2∞ 18.9388 76.0997 38.6130

‖U6 ‖2∞ 38.6172 76.1087 38.6161

JICC 1.9743 4.1842e-004 0.0019

and KICC1, the ICC controller (KICC1) achieves a much lower cost(JICC1 = 0.0004184) than that associated with the pole placementcontroller (JICCpp = 1.9743). In other words, the ICC controllerprovides better output performance (measured in terms of ICCcost) than the pole placement controller using the same availableactuators size (control covariance constraints).

For the tensegrity simplex structure with multiple control in-puts, the constraints on the torque actuators could be differentfrom each other. Since the first and last three control channelsof the Kpp controller have the peak control covariances equal to76.1127 and 38.6172, respectively, ICC controller (KICC2) hasbeen designed to achieve different constraints on different torqueactuators as follow

U i =√

76.1127 = 8.7243 i = 1,2,3

U i =√

38.6172 = 6.2143 i = 4,5,6(40)

In this set of constraints, actuators associated with the con-trol channels i= 4,5,6 designed to be more conservative than thefirst three control channels. Table 1 shows the L∞ norm of thecontrol energy and the resulting output performance (ICC costfor KICC2 is JICC2 = 0.0019). By comparing the ICC costs, Itis clear that KICC2 has a slightly larger cost than that of KICC1(JICC1 = 0.0004184) because the hard constraints in (40). How-ever, the output performance is still much better than the costassociated with the pole placement controller by a factor of morethan 1000 for the same actuators size.

For comparison purposes, two OCC controllers (KOCC1 andKOCC2) have been designed to achieve the same output perfor-mances associated with the ICC controllers, respectively. Theresults given in Tab. 2 show that both OCC controllers main-tain the same output performance, but fail to satisfy the controlconstrains defined by (39) and (40), respectively. This indicatesthat the actuators would be saturated under certain L2 distur-bance belonging to the given L2 set, which results in a deviation



TABLE 2. COMPARISON OF ICC WITH OCC CONTROLLERS

KOCC1 KOCC2

‖U1 ‖2∞ 61.3518 35.4943

‖U2 ‖2∞ 84.5545 50.3619

‖U3 ‖2∞ 60.2525 34.3456

‖U4 ‖2∞ 84.3535 50.1353

‖U5 ‖2∞ 60.5794 34.6815

‖U6 ‖2∞ 84.3641 50.1095

JICC 4.1824e-004 0.0019

from the linearity assumption of the system, thus, even stabilitywill no longer be guaranteed. For KOCC1, the control covariancesfor channels 2,4, and 6 are 84.5545, 84.3535, and 84.3641, re-spectively, which violate the upper bound in (39). On the otherhand, the control covariances associated with KOCC2 for chan-nels 4 and 6 are 50.1353 and 50.1095, respectively, which in-dicate that their upper bounds in (40) have also been violated.Thus, both OCC controllers require larger actuator sizes (on theviolated input channels) to achieve the same output performancecompared with the ICC controllers. To demonstrate that thesecontrol constraints are tight, a worst case L2 disturbance signalwp has been designed such that

‖ wp ‖22= 1

in order to guarantee the L∞ norm of the controller response bebounded by the square root of the maximum singular value ofthe control covariance. This disturbance signal used to simulatethe closed loop systems with KOCC1 and KICC1 controllers, re-spectively. The L∞ norm (peak) of the sixth (actuator) controlchannel associated with the worst case L2 disturbance signal forboth controllers and their responses are shown in Fig. 2. It caneasily be shown that the KOCC1 violates the constraints on the ac-tuators size to achieve the same output performance as the KICC1does. The L∞ norm of the response of KICC1 is bounded by thesquare root of the control covariance:

‖ u6 ‖∞= 8.7234 <√

76.1087 = 8.7240 < 8.7243 =U

while the L∞ norm of the response associated with the KOCC1 is

‖ u6 ‖∞= 9.1842 >U

violates the constraints (39) on the actuator size. The unit stepresponse of both controllers and their outputs are shown in Fig. 3.

0 0.2 0.4 0.6 0.8 1

0

5

10

U6

OCC Controller (KOCC1

)

0 0.2 0.4 0.6 0.8 1−10

−5

0

5x 10

−3

y 6

The response (y6) (K

OCC1)

0 0.2 0.4 0.6 0.8 1

0

5

10

U6

ICC Controller (KICC1

)

0 0.2 0.4 0.6 0.8 1−10

−5

0

5x 10

−3

y 6The response (y

6) (K

ICC1)

FIGURE 2. THE RESPONSES OF THE KOCC1 AND KICC1 CON-TROLLERS AND ASSOCIATED OUTPUTS

Another comparison has been made between the KOCC2 andKICC2 controllers. Figure 4 shows the control effort requirementof KOCC2 and KICC2 for the fourth control channel to maintain thesame performance output (JICC = 0.0019). The controller designrequirement constraint on the 4th control channel is defined in(40) to be U = 6.2143 for both controllers. For the same L2input signal wp, the L∞ norm of the KICC2 controller is

‖ u4 ‖∞= 6.2140 <√

38.6161 = 6.2142 < 6.2143 =U

which is bounded by the square root of the control variance cor-responding to this control channel. On the other hand, the L∞

norm of the KOCC2 controller is

‖ u4 ‖∞= 7.0786 <√

50.1353 = 7.0806 > 6.2143 =U

which indicates that this controller will provide a control signalthat will breach the (limit) constraint (40) of this actuator.



0 0.1 0.2 0.3 0.4 0.50

2

4x 10

−3

U6


)

0 0.1 0.2 0.3 0.4 0.50

2

4x 10

−3

y 6


OCC1)

0 0.1 0.2 0.3 0.4 0.50

1

2

3x 10

−3

U6


)

0 0.1 0.2 0.3 0.4 0.50

1

2

3x 10

−3

y 6


ICC1)

FIGURE 3. STEP RESPONSES OF KOCC1 AND KICC1 CON-TROLLERS AND THE ASSOCIATED OUTPUTS

Using Theorem 5, the LMIs (34) are implemented into MAT-LAB using the LMI parser YALMIP and solved with SDPT3[16]- [17] solver to design a full-order dynamic output feedbackcontroller matrices that stabilize the closed loop system (30) andsatisfy different actuators constraints while minimizing the ICCcost. Fig. 5 illustrates the response of the output feedback con-troller with different actuators constraints

√Ui =√

76.1127 = 8.7243 i = 1,2,3√

Ui =√

38.6172 = 6.2143 i = 4,5,6,

respectively. As can be easily seen from this figure, this con-troller optimizes the physical resources (actuators) efficiently andall control signals provided by this controller satisfy (tightly) thedifferent actuator limitations with JICC = 0.0141.

These control design and simulation results show the bene-fits of the ICC controllers proposed in this paper (state feedbackand dynamic output feedback) over the pole placement and OCCcontrollers when (different) multiple hard constraints (for MIMOsystems) are posted on the actuators.

0 0.2 0.4 0.6 0.8 1

0

5

10

U4


)

0 0.2 0.4 0.6 0.8 1−5

0

5x 10

−5

y 4


OCC2)

0 0.2 0.4 0.6 0.8 1

0

5

10

U4


)

0 0.2 0.4 0.6 0.8 1−5

0

5x 10

−5

y 4


ICC2)

FIGURE 4. THE RESPONSES OF KOCC2 AND KICC2 CON-TROLLERS AND THE ASSOCIATED OUTPUTS

CONCLUSIONSICC controllers have been proposed and used to design

closed loop controllers for a tensegrity simplex structure in thispaper using the LMI formulation. The ICC control problem min-imizes the system output performance subject to multiple con-straints on the control input covariances. The physical inter-pretation of this problem has significant importance in practicalapplications when the outputs of the physical system actuatorshave limits. For instance, a 10 foot-pound torque actuator cannot produce output torque more than its limit (10 foot-pounds).Thus, the hardware constraints in terms of control variances havebeen considered in the ICC control designs to achieve the bestpossible performance with the given actuator set. This theorysuccessfully applied to a realistic tensegrity simplex structuremodel in this paper. The proposed ICC control designs werecompared with the other two control design methods, pole place-ment and OCC control designs. The control design results showthat pole placement controller, for a given actuator set, producesthe worst output performance compared with the ICC and OCCcontrollers. On the other hand, the designed OCC controllersachieve the same output performance as ICC ones, but the actu-ator constraints are violated or larger actuator sizes are required.This indicated the importance of utilizing the ICC control designmethod when the system actuator constraints need to be satis-



0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 105

101st actuator response with its constraint

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 105

102nd actuator response with its constraint

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 105

103rd actuator response with its constraint

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 105

104th actuator response with its constraint

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 105


0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 105


FIGURE 5. OUTPUT FEEDBACK CONTROL RESPONSE ANDACTUATORS CONSTRAINTS

fied. Both, state-feedback and full-order dynamic output feed-back controllers have been successfully designed in this paper.

ACKNOWLEDGMENTAli Khudhair Al-Jiboory would like to thank the Higher

Committee for Education Development (HCED) and Universityof Diyala in Iraq for their financial support during his graduatestudy at MSU. C. Sultan acknowledges support by the NationalScience Foundation under grant CMMI-0952558.

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