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					     INTERNATIONAL JOURNAL OF INTELLIGENT CONTROL AND SYSTEMS
     VOL. 12, NO. 3, SEPTEMBER 2007, 237-244




           A Model Reference-Based Adaptive PID Controller for Robot
               Motion Control of Not Explicitly Known Systems

                                                                   Wei SU
    Abstract - This paper proposes a model reference tracking based      parameter variations is discussed for ASPR systems with a
adaptive PID controller with adaptive mechanisms in both
                                                                         feedforward compensator [20, 21] and for parabolic and
feedforward and feedback paths. The objective is to force the outputs
of a not explicitly known multiple input multiple output (MIMO)          hyperbolic systems [22]. The non-ASPR direct model
linear time-invariant system to track the outputs of a known reference   reference adaptive control algorithms have also been
model. A PID controller is inserted to the feedback path. The            extended to discrete-time cases [17], and systems with
parameters of the PID controller are computed adaptively by
                                                                         unknown nonlinear functions [18]. The direct model
eliminating output tracking errors. This approach allows us to
manipulate the multiple motions of a complicated, unstable, or high-     reference adaptive control method has been applied to many
order robot by operating on a simpler, stable, or lower-order known      practical problems [19] such as large flexible structures,
reference model. The mathematic description of the robot is not          robotic manipulators, drug infusion, and aircraft.
required. Output matching and tracking conditions are derived and
                                                                             In this paper, a direct model reference output tracking
analyzed.
                                                                         (DMROT) based adaptive PID controller is proposed using
    Index Terms – model reference adaptive control, PID control,         both feedforward and feedback adaptive mechanisms to
output feedback, output tracking, MIMO systems.                          stabilize the closed-loop system and, at the same time, force
                                                                         the outputs of a multi-input multi-output (MIMO) robot
                     1. INTRODUCTION                                     system to track the outputs of a known reference model. The
                                                                         concept of the direct model reference adaptive control of a
    Robust robot motion control of not explicitly known                  linear MIMO system is applied by deriving new sufficient
systems is a challenging subject in both military and                    conditions for perfect output matching and asymptotic
commercial applications such as unmanned aerial vehicles                 output tracking. The paper is arranged as follows: Firstly it
(UAVs), bomb disposal mechanics, remotely-operated                       shows the conditions under which the ideal states exist such
weapons, and satellites where the faithful mathematic                    that the system outputs match the desired trajectories
description of robots is not available. Recently, many                   provided by a reference model. Secondly it discusses the
adaptive PID methods have been developed [1-12]                          conditions under which an ideal control raw exists so that
including Back-Stepping (BS) based adaptive proportional-                the system states track the ideal states. Then it shows that
integral-derivative controller (PID) control which adds the              the output tracking can be implemented using a PID
integral action and nonlinear damping term to the basic                  controller with an adaptive mechanism. Finally, the
back-stepping algorithm to guarantee robustness and                      sufficient conditions of stability are discussed and an
bounded errors [1, 2], Generalized Predictive Control (GPC)              example is presented.
based one that uses a PID controller to be equivalent to a
GPC controller and incorporates the advantages of both PID                     2. FORMULATION OF THE SYSTEM
and GPC [3], Heuristic Rule (HR) based one that employs
rule-based switching and adaptive PID controller to perform                  The proposed approach controls the motions of both
autonomous functions [6], and many neural network based                  stable and unstable robots to follow the ideal trajectory
PID controllers used in various applications [7-9]. Robot                provided by a known reference model using a DMROT
motion control based on adaptive PID control has also been               based PID controller. It allows a single controller to
studied by applying an adaptive PID feedback control                     manipulate multiple motions of an unstable and not exactly
schemes and a feedforward input learning scheme to learn                 known robot by simply driving the stable known reference
robot motions [4, 5].                                                    model with multiple input commands as shown in Fig. 1. An
    The direct model reference adaptive control of MIMO                  adaptive PID controller is inserted into the feedback path of
systems [13, 14] uses an adaptive control structure that                 the DMROT structure which adds additional zeros and poles
consists of command inputs, reference model states, and the              to the closed loop system to improve the stability of the
feedback of the output errors. In an ideal situation, the                robot. This PID controller also coordinates with the adaptive
outputs of the system track the outputs of the known                     feedforward mechanism to control the multiple robot
reference model. Asymptotic stability of this algorithm is               motions for output tracking. Since the structure of the robot
guaranteed for both the class of almost strictly positive real           is not exactly known, adaptive mechanisms are used for
(ASPR) systems and the class of non-ASPR systems [15, 16]                self-adjustment of the PID gains for achieving the best
with supplementary dynamics. The robustness of system                    performance.
 Manuscript received September 18, 2006; revised March 15, 2007.
 W. Su is with U.S. Army RDECOM, Ft Monmouth, NJ 07703, USA
 (e-mail: Wei.Su@us.army.mil).
                                  SU: A Model Reference-Based Adaptive PID Controller for Robot Motion Control of Not Explicitly Known Systems 238




    The multi-input multi-output (MIMO) time invariant
linear system with input and output disturbances is                                                                              bf                               (6)
described as                                                                                                      H L (s) =
                                                                                                                               s − af
             x p (t ) = A p x p (t ) + B p u p (t ) − E p d ip (t ) (1)
                               y p (t ) = C a y p (t )                   (2)       The filtered error signals are feedback via an adaptive
                                                                                PID controller in order to drive the tracking output yp(t) to
                       y p (t ) = C p x p (t ) + d op (t )
                                                                                approach the reference model output ym(t).

where x p (t ) ∈ R n , y p (t ) ∈ R r , and u p (t ) ∈ R m are state,              The feedback mechanism is described by an adaptive
output, and input vectors, respectively, d ip (t ) ∈ R q and                    PID controller as shown below

d op (t ) ∈ R r are bounded input and output disturbances, and                                                             KI
                                                                                                     H C (s) = K P +          + KDs                               (7)
y p (t ) ∈ R rm ( rm   ≤ r ) is the tracking output vector of the                                                          s
unknown system. Matrices Ap, Bp , Cp, and Ep are not
explicitly known but their entries are assumed to be                               where the parameters KP, KI,, and KD are unknown and
bounded and the dynamics {Ap, Bp , Cp} is assumed to be                         are updated adaptively. The adaptive PID controller not
controllable and observable. Ca is a transfer matrix which                      only provides a quick feedback response but also eliminates
converts the output vector y p (t ) to tracking output vector                   the tracking error between unknown system and reference
                                                                                model outputs. The transform function of the adaptive PID
y p (t ) . When Ca is a unity matrix, y p (t ) = y p (t ) , and when            controller together with the low-pass filter is
rank (C a ) < r , dim( y p (t ) )<dim( y p (t ) ).
                                                                                                                            KDs2 + KPs + KI                       (8)
                                                              Desired                H (s) = H C (s) H L (s) =                              bf
                                                              Output                                                           s(s − a f )
     Multiple                                  Reference
      Input                                     Model
                                                                                  Define

                                                                                                                  K D = K pe ,                                    (9)
                                                                                                                K I = K pf 1a f , and                           (10)
                                                              Multiple
                                                              Output +                             K P = − K pf 1 − K pf 2 − K pe a f ,                         (11)
   Adaptive                   +                                     _
                                                   Robot
  Feedforward                   _                                               to obtain the transfer function in a parallel form

                                                                                                             Z f ( s)    ⎛        K pf 1    K pf 2      ⎞
                                                                                                 H (s) =               = ⎜ K pe −        −              ⎟b f , (12)
                                                                                                             E yp ( s ) ⎝⎜         s       s − af       ⎟
                                                                                                                                                        ⎠
                                               Adaptive
                                               Feedback
                                                               Output           where Zf(s) is the feedback signal obtained from the output
                                                                Error           of the filter H(s). The state-variable representation of H(s)
    Fig. 1. Robot motion control with a reference model                         can be described as

The tracking output vector tracks the output of a known                                                      y f 1 (t ) = e yp (t )                             (13)
reference model described by a stable MIMO time-invariant                                                                                                       (14)
                                                                                                             y f 2 (t ) = a f y f 2 (t ) + e yp (t )
system
                                                                                 z f (t ) = − K pf 1 (t ) y f 1 (t ) − K pf 2 (t ) y f 2 (t ) + K pe (t )e yp (t ) (15)
                 x m (t ) = Am x m (t ) + Bm u m (t )                    (3)
                                    y m (t ) = C m x m (t )              (4)    where zf(t) is the inverse Fourier transform of Zf(s) and eyp(t)
                                                                                is the inverse Fourier transform of Eyp(s). Eqs. (13)-(15) can
                                                                                be further represented by
where x m (t ) ∈ R nm , y m (t ) ∈ R rm , and u m (t ) ∈ R m are state,
input, and output vectors of the reference model. Matrices                                      y f (t ) = A f y f (t ) + B f e yp (t )                         (16)
Am, Bm, and Cm are known. The output error signals,
represented by vector                                                                           z f (t ) = − K pf (t ) y f (t ) + K pe (t )e yp (t )            (17)
                 e yp (t ) = y m (t ) − C a y p (t )                (5)
are filtered by a low-pass filter                                               where
 239 INTERNATIONAL JOURNAL OF INTELLIGENT CONTROL AND SYSTEMS, VOL. 12, NO. 3, SEPTEMBER 2007




                                   ⎡ y f 1 (t ) ⎤                                                                      3. OUTPUT MATCHING CONDITIONS
                        y f (t ) = ⎢              ∈ R 2 r , z f (t ) ∈ R ,
                                                                        m

                                   ⎣ y f 2 (t )⎥⎦                                                                  Since the structures of system and reference model are
         ⎡0   0 ⎤,
    Af = ⎢
                            ⎡b f
                    ⎥ B f = ⎢b
                                                       ⎤ , and
                                                       ⎥       K pf = K pf 1 [            K pf 2 .  ]          quite different, the output of the system matches the output
         ⎣0 a f I r ⎦       ⎣ f                        ⎦                                                       of the reference model only under certain conditions. In this
                                                                                                               section, we show that there exists an ideal state vector
The adaptive control law is described by                                                                       denoted by x * (t ) such that the tracking outputs of an
                                                                                                                              p

                                                                                                               unknown system match the desired outputs of a reference
                       u p (t ) = K pe (t )e yp (t ) − K pf (t ) y f (t )                                      model. Then, we demonstrate in the next section that the
                                                                                                        (18)
                       + K x (t ) x m (t ) + K u (t )u m (t )                                                  state vector of the unknown system xp(t) approaches the
                                                                                                               ideal state x * (t ) with an appropriate control input applied to
                                                                                                                             p

where the first and second terms are adaptive feedback                                                         the unknown system. When Ca is a full rank matrix, such as
signals contributed by the PID controller and the third and                                                    unity matrix, it yields r=rm. In this case, the multiple output
forth terms are adaptive feedforward signals contributed by                                                    matching is very restrictive since it requires all outputs of
the reference model. A block diagram is shown in Fig. 2.                                                       the unknown system track all outputs of the reference model.
                                                                                                               Mathematically
 Multiple
                                                              xm (t )
 Input Signal
                                                  Multiple                                                                  e yp (t ) = y p (t ) − y m (t ) → 0                 (19)
                          Bm
                                                Integrators             Cm                ym (t )
 u m (t )
                                                                                                               To do this, we need to resolve an n-dimensional vector xp(t)
                         Reference
                          Model                        Am
                                                                                                               from r number of time-varying linear equations represented
            Ku
                                                                                                               by
                 + +
                           Kx
                                                                                                                                       C p x p (t ) = y m (t )                  (20)
             +             Kpe
                  +                                                          e yp (t )
                  _
                                                 Multiple
                                                                                         +
                                                                                          Output
                                                                                                               This is difficult to achieve when the order of the output is
                           Kpf
                                  y f 1 (t )   Integrators                   bf
                                                                                           Error               high. However, in some applications, it is not necessary to
                                                                                         _
                                                                                                               match all outputs of the unknown system to the reference
                                                 Multiple                                y p (t )              model and the matching condition will be relaxed. When a
                                y f 2 (t )     Integrators          +
                                                                +                                              lower dimensional reference model output is used, we have
                                                  af
                                                                                    Ca
                                                                                                                r ≤ r for the r × r matrix Ca and the rank condition
                                                        Adaptive PID Controller

                                                                                                                                                                                (21)
                                                                                         Multiple
                                                                                                                            rank [C a C p C a C m ] = rank [C a C p ]
                                  u p (t )                      y p (t )                 Output
                                                                                         Signal
                                                  MIMO
                                                  System
                                                                                                               is easier to satisfy. The time-varying vector xp(t) can be
                                                                                                               resolved to yield
  Fig. 2. Output tracking with an adaptive PID controller
                                                                                                                                       C a C p x p (t ) = C m x m (t )          (22)
A reference model {Am , Bm , C m } is chosen to provide rm
                                                                                                                                                  or
desired output trajectories, represented by vector ym(t), when                                                                         y p (t ) = y m ( t )                     (23)
driven by m reference input signals, represented by vector
um(t). The not explicitly known system {Ap , B p , C p } is
                                                                                                               An extreme case is to have Ca = [1 0r-1] if C p11 ≠ 0 , where
manipulated by m input signals, represented by vector up(t),
and generated r output signals, represented by y p (t ) . In                                                   0r-1 is a 1 by (r-1) row vector of all zero entries. In this case,

the perfect output tracking case, the rm number of output
                                                                                                                                                     (                )
                                                                                                               we always have x p1 (t ) = C m11 / C p11 x m1 (t ) so that
error signals, represented by the vector eyp(t) approaches                                                     y p (t ) = C p11 x p (t ) = C m xm (t ) = y m (t ) . Let C p = C a C p
zero so that the tracking output yp(t) approaches the
                                                                                                               and define C p as pseudo-inverse matrix of C p , that is
                                                                                                                            +
reference output ym(t). Therefore, the objective of adaptive
control is to compute, without any explicit knowledge of                                                       C p C p = I n , Eq. (5) becomes
                                                                                                                 +

system parameter matrices Ap, Bp and C p , the adaptive gains:
KP, KI, and KD such that the tracking output vector yp(t) of                                                                e yp (t ) = C m x m (t ) − C p x p (t )             (24)
the unknown system follows the output vector ym(t) of a
stable known linear reference model.
                                    SU: A Model Reference-Based Adaptive PID Controller for Robot Motion Control of Not Explicitly Known Systems 240




If the matching condition is satisfied, we obtain an ideal                                  Defining the error dynamic equation of the PID controller as
state
                                                                                                           exf (t ) = y* (t ) − y f (t )
                                                                                                                       f
                   *            +
                 x (t ) = C C m x m (t )                                           (25)                                                                           (32)
                   p            p                                                                                  = Af y* (t ) − Af y f (t ) − B pC p exp (t )
                                                                                                                         f

                                                                                                                   = Af exf (t ) − B pC p exp (t ),
such that

                                                                                            we have the dynamic function for the feedback structure as
    y p (t ) = C p x * (t ) = C p C p C m x m (t ) = y m (t )
                                    +
                                                                                   (26)
                     p
                                                                                            below

The existence of output matching is determined by                                           ⎡exp (t )⎤ ⎡ Ap − B p K pe (t )C p − B p K pf (t )⎤ ⎡exp (t )⎤ ⎡Q (t )⎤ (33)
matrices C a , C m , and C p . Although the values of C a , and                             ⎢         ⎥=⎢                                     ⎥⎢          ⎥+⎢     ⎥
                                                                                            ⎢exf (t ) ⎥ ⎣
                                                                                            ⎣         ⎦        − Bf Cp              Af        ⎦ ⎢exf (t ) ⎥ ⎣ 0 ⎦
                                                                                                                                                ⎣         ⎦
C m are known, the ideal state                 x * (t ) cannot be calculated
                                                 p                                          Since the unknown system is assumed to be controllable and
                                                                                                                                                                   ~
since the value of C p is unknown. We have shown that                                       observable, there exist constant matrices K pe (t ) = K pe ,
                                                                                                        ~
such x * (t ) exists for many systems since the conditions
       p
                                                                                            K pf (t ) = K pf , Af, and Bf such that the meta-system is stable.
stated in (21) are not restrictive. Next, we show that the                                  Therefore, if Q(t)=0, e xp (t ) → 0 . A solution for Q(t)=0 is to
vector xp(t) approaches the ideal state vector x * (t ) . That is,
                                                 p                                          assume that y * (t ) is the linear combination of xm(t) and
                                                                                                          f

the state error vector                                                                      um(t). That is y * (t ) = S x xm (t ) + S u u m (t ) , where the values of
                                                                                                               f

                                                                                            constant matrices Sx and Su are introduced only for the
                e xp (t ) = x * (t ) − x p (t )
                              p
                                                                                   (27)     convenience of theoretic discussion and are not needed in
                                                                                            implementation. The condition for the existence of Sx and Su
approaches zero with appropriate inputs, and the PID                                        are discussed as follows: If the meta-system is stable with
controller state error vector                                                                K pf (t ) = 0 and

                e xf (t ) = e yf (t ) = y * (t ) − y f (t )
                                          f
                                                                                   (28)
                                                                                                 rank[ B p ] = rank[ B p C p Cm Am − Ap C p Cm C p Cm Bm ] , (34)
                                                                                                                           +              +      +



vanishes while the controller output reaches the ideal value.                                                                        ~                  ~
                                                                                            there exist constant matrices K x (t ) = K x and K u (t ) = K u
          4. OUTPUT TRACKING CONDITIONS                                                     such that

   The existence of the ideal state does not always yield a                                       (C   +
                                                                                                     C m Am − A p C p C m − B p K x x m (t )
                                                                                                       p
                                                                                                                     +             ~
                                                                                                                                           ) or Q(t ) = 0 . (35)
perfect tracking. This section derives the condition under                                             +              ~
                                                                                                  + (C p C m Bm − B p K u )u m (t ) = 0
which an ideal input exists to drive the system state to
approach the ideal state asymptotically.
                                                                                            In this case, the integrator does not contribute in the PID
    The derivative of (27) gives                                                            controller. Otherwise, the integrator is activated to balance
                                                                                            and eliminate the term Q(t). In general, if
                e xp (t ) = x * (t ) − x p (t )                                    (29)                    ~                  ~      +                +
                              p
                                                                                                 rank[ B p K pf ] = rank[ B p K pf C p C m Am − A p C p C m
                                                                                                                                                            , (36)
                                                                                                       ~      +              ~
Inserting (1), (3), and (25) into (29), we have                                                  − B p K x C p C m Bm − B p K u ]

  e xp (t ) = C p C m ( Am x m (t ) + B m u m (t ) ) − (A p x p (t ) + B p u p (t ) )
                +                                                                                                      ~           ~                ~
                                                                                            there exist matrices S x = S x , S u = S u , K x (t ) = K x , and
  = C p C m ( Am x m (t ) + B m u m (t ) ) − A p x p (t )
      +                                                                                                 ~
                                                                                             K u (t ) = K u such that
  − B p ( K pe (t )e yp (t ) − K pf (t ) y f (t ) + K x (t ) x m (t ) + K u (t )u m (t ))
                                                                                                               ~ ~                                       ~
    (                          )
  = A p − B p K pe (t )C p e xp (t ) + B p K pf (t )e xf + Q(t )
                                                                                                                              +                +
                                                                                                           B p K pf S x = C p C m Am − A p C p C m − B p K x (37)
                                                                                   (30)                        ~ ~                         ~
                                                                                                                            +
                                                                                                           B p K pf S u = C p C m Bm − B p K u               (38)
where
                                                                                            Therefore, Q(t ) = 0 .
          Q(t ) = B p K pf (t ) y * (t ) + (C p C m Am − A p C p C m
                                              +                +
                                  f
                                                                                   (31)
                                         +
           − B p K x (t )) x m (t ) + (C p C m Bm − B p K u (t ))u m (t )
 241 INTERNATIONAL JOURNAL OF INTELLIGENT CONTROL AND SYSTEMS, VOL. 12, NO. 3, SEPTEMBER 2007




    Before discussing the adaptive control, it shows that if                output tracking errors. Those adaptive gain matrices are
the system is known, we can satisfy the tracking conditions                                                          ~ ~ ~                 ~
                                                                            used to replace the theoretical matrices K pe , K pf , K x and K u
                                           ~ ~            ~
in (37) and (38) by manipulating matrices S x , S u , and K pf .
                                                                            in order to stabilize the adaptive control system and to track
This is very important since it proves that the ideal inputs                the desired output. The adaptive gain matrices are defined as
exist so that the tracking of the ideal states and desired
trajectories is achievable. In practice, the system parameters                           K (t ) = [ K e (t ) K px (t ) K pu (t )]        (43)
are not known and the system is subject to input and output
disturbances. An adaptive mechanism is used to adjust the
                                                                            and
matrices, K x (t ) , K u (t ) , and K pf (t ) so that the state xp(t) of
                                                                                                            ⎡e yv (t )⎤
the not explicitly known system matches the ideal state
 x * (t ) and the tracking output of the unknown system tracks                                     r (t ) = ⎢ x m (t ) ⎥
                                                                                                            ⎢          ⎥
                                                                                                                                         (44)
   p

the output of the reference model asymptotically. That is                                                   ⎢ u m (t ) ⎥
                                                                                                            ⎣          ⎦
               y p (t ) → C p x * (t ) = C m x m (t ) = y m (t ) .
                                p
                                                                                        where K e (t ) = [ K pe (t )       K pf (t )]    (45)
                      ~ ~ ~                 ~
It is remarkable that K pe , K pf , K x and K u are virtual gain                                         ⎡ e yp (t ) ⎤
                                                                                         and e yv (t ) = ⎢           ⎥,                  (46)
matrices used in proving the existence of output tracking.                                               ⎣− y f (t )⎦
Neither their numerical values nor their implementation is
required.                                                                   we rewrite Eq. (18) into a vector form

    5. ADAPTIVE MECHANISM AND STABILITY                                                 u p (t ) = K (t )r (t )                          (47)
    Since the system is not explicitly known, the feedback
PID controller is needed to adaptively choose the values of                 The adaptive gain matrix K(t) is chosen to be a combination
Kp, KI, and KD (equivalently Kpe, Kpf, Kx, and Ku) in order to              of proportional and integral (PI) terms as follows [1, 2]:
stabilize the dynamics in (33) and eliminate the unwanted
terms in (35). The term asymptotic output reference model                                K (t ) = K P (t ) + K I (t )                    (48)
tracking means that the system output approaches the
reference model output, y p (t ) → y m (t ) , when the time is              where the proportional term is described by
sufficiently large. When output tracking occurs, the
corresponding state and control trajectories are defined to be                           K P (t ) = v(t )r T (t )T                       (49)
the ideal state x p (t ) = x * (t ) = C p C m x m (t ) and ideal
                             p
                                        +


control command u p (t ) = u * (t ) respectively. While                     and integral term is described by
                             p

           e yp (t ) = C a ( y m (t ) − y * (t ) ) → 0 ,
                                          p
                                                                                         K I (t ) = [v(t )r T (t ) − σK I (t )]T         (50)
the system is driven by the ideal input u * (t ) . If the tracking
                                          p

conditions discussed in the last section are satisfactory, we               with the initial gains given by
show that the output tracking can be implemented                                         K I (0) = [ K eI (0) K px (0) K pu (0)]
                                                                                                                I        I
                                                                                                                                         (51)
adaptively.
   To show the stability of the adaptive control mechanism,
we introduce an ideal dynamic system without input and                      The signal v(t) is chosen based upon the Lyapunov stability
output disturbances as shown below                                          analysis, which is in the form of

                                                                                        v(t ) = Qe yv (t ) + GK (t ) r (t )              (52)
                *              *
              x P (t ) = A p x P (t ) + B p u * (t )
                                              p
                                                                   (39)
                          y * (t ) = C p x * (t ) = y m (t )
                            p              p
                                                                   (40)     Where
                            *             *
                          y (t ) = A f y (t )                      (41)
                            f             f
                                                                                                   Q = [Q p     Qf ]                     (53)
                           ~               ~             ~
               *
              u p (t ) = − K pf y * (t ) + K x xm (t ) + K u um (t ) (42)
                                  f
                                                                                The matrices T, T , Q, and G are matrices selected by
Since the system is not known and the virtual matrices                      designers such that T and T are positive definite symmetric
 ~ ~ ~                ~                                                     and positive semi definite symmetric, respectively, and such
K pe , K pf , K x and K u cannot be computed. The adaptive
                                                                            that Q and G satisfy the sufficient conditions for stability,
structure is designed by computing the adaptive gain                        and positive scalar σ is introduced to guarantee robustness
matrices K pe (t ), K pf (t ), K x (t ) and K u (t ) to eliminate the       in the presence of disturbances.
                              SU: A Model Reference-Based Adaptive PID Controller for Robot Motion Control of Not Explicitly Known Systems 242




   The stability of the adaptive system must be studied to                                                 J + J T + G + GT < 0            (64)
insure all states and gains have bounded values. In order to
simplify the stability proof, a meta-state model is used by                      The sufficient conditions in (61)-(64) do not restrict to the
defining meta-vectors:                                                           assumption that the unknown system {Ap, Bp, Cp} is ASPR.
                                                                                 Comparing (59) to (60), the later is much less restrictive due
                       ⎡ x p (t ) ⎤                y (t )                        to the additional term LW.
               x(t ) = ⎢            and y (t ) = ⎡ p ⎤                    (54)
                       ⎣ x f (t )⎥⎦
                                                 ⎢ y (t )⎥
                                                 ⎣ f ⎦                                                       6. EXAMPLE
where       y f (t ) = x f (t ) . Combining the state-variable                      A classic example studied by many authors [14, 15, 19,
equations of the unknown system, Eqs. (1) and (2) with the                       23] is the so called Rohrs’ example described by
state-valuable equations of the adaptive PID controller in
(47), to form a metastate valuable system described by                                       y p (s)         2          229                (65)
                                                                                                       =
                                                                                             u p (s)       s + 1 s 2 + 30 s + 229
               x(t ) = Ax(t ) + BK (t )r (t ) + d i                       (55)
                         y (t ) = Cx(t ) + d o                            (56)   The open loop system in (65) is stable but has a pair of
                                                                                 complex unmodeled poles. The root locus of (65) shows that
                                                                                 the dominant second order term of the corresponding closed
where the system is described by meta-matrices:
                                                                                 loop system becomes unstable when the loop gain is larger
                                                                                 than the admissible limit as shown in Fig. 3. This example is
         ⎡ Ap              0 ⎤,   ⎡B p ⎤ , ⎡C p                0⎤         (57)
       A=⎢                    ⎥ B=⎢ ⎥ C=⎢                                        considered as a difficult case in adaptive control and is used
         ⎣− B f C p        Af ⎦   ⎣0⎦      ⎣0                  Ir ⎥
                                                                  ⎦              to test various adaptive controllers.

and the disturbances are described by metastates:

              ⎡        E p d ip (t )       ⎤                ⎡d op (t )⎤
   d i (t ) = ⎢                            ⎥ and d o (t ) = ⎢         ⎥
                                                                          (58)
              ⎣ B f ( y m (t ) − d op (t ))⎦                ⎣ 0 ⎦

    The adaptive control algorithm described in (55) and (56)
is stable [15] if there exist a real symmetric positive definite
                             ~
matrix P and real matrix K u and R, R + R T > 0 , such that

            ~                ~
    P( A − BK e C ) + ( A − BK e C ) T P = − LLT − R < 0 (59)
                           PB = C T Q T                                   (60)

where matrices T and T are positive definite symmetric and                                 Fig. 3 Root locus plot of Rohrs example
positive semi-definite symmetric, respectively. The
sufficient conditions in (59) and (60) only assume that the                         The output of the system in (65) is required to follow the
metasystem {A, B, C} is ASPR. In this case, the unknown                          output of the reference model, which is shown below
system {Ap, Bp, Cp} is not directly restricted by ASPR
conditions.                                                                                                 y m ( s)   1
                                                                                                                     =                     (66)
                                                                                                            u m (s) 1 + s / 3
    A less restrictive sufficient condition statement is
developed based on BIBO stability analysis [9], which                            To demonstrate the necessity of adopting an adaptive PID
states that all states and errors in the adaptive system in (59)                 controller, the system response of using a fixed PI controller
and (60) are bounded if there exist a real symmetric positive                    is tested by insert the function
                                                    ~
definite matrix P and real matrices L, W, K u , and R,                                                                 − (10 s + 35)       (67)
                                                                                                           H 0 (s) =
R + R > 0 , such that
        T                                                                                                                    s

           ~                ~                                                    into the feedback loop. As shown in Fig. 4, the dominant
   P( A − BK e C ) + ( A − BK e C ) T P = − LLT − R < 0                   (61)
                                                                                 second order term leads to a stable direction with the
                             ~                                            (62)
           PB = C T (Q T + K eT G T ) − LW                                       increasing of loop gain. The square wave response of the
                                                                                 closed loop systems in (65) with the PI controller in (67) is
                           W W =J+J
                              T               T
                                                                          (63)
 243 INTERNATIONAL JOURNAL OF INTELLIGENT CONTROL AND SYSTEMS, VOL. 12, NO. 3, SEPTEMBER 2007




stable but has a setting time greater than 500 seconds and a          parameters of the PI controller and the gains of the
damping ratio for the dominant poles of less than 0.001 as            feedforward in order to stabilize the system and eliminate
shown in Fig. 5. The performance is unsatisfactory.                   the output tracking errors. Qp=57.14, Qf=0, and G=0 are
                                                                      chosen in the simulation based on the design specifications
                                                                      [15]. As shown in Fig. 6, the stability of the closed loop
                                                                      system is significantly improved. The output of (65) tracks
                                                                      the output of the reference model in (66) asymptotically and
                                                                      achieves zero output tracking error in approximately two
                                                                      seconds.




    Fig. 4. Root locus plot of Rohrs example with PID

   Since the mathematic description of the robot is not
explicitly known and the parameters of the robot may vary
unexpectedly, the choice of a proper PID controller becomes               Fig. 6. Output tracking with Adaptive PID Controller
very difficult. Thus, an adaptive PID controller
                                                                                           8. CONCLUSION
                                               K pf
                        H ( s ) = K pe −                       (68)
                                                s                         Adaptive PID controller based on DMROT is developed
                                                                      by using both feedforward and feed back adaptive
is inserted into the feedback loop. The state-variable                mechanisms. The outputs of the not explicitly known
description of (68) is                                                MIMO system are forced to track the outputs of a known
                                                                      reference model asymptotically. This allows us to
                        x f (t ) = e yp (t )                   (69)   manipulate and control the multiple motions of a complex
                                                                      and not explicitly known robot using a single controller by
           z f (t ) =−K pf (t ) x f (t ) + Kpe(t ) e yp (t )   (70)
                                                                      simply operating on a known reference model. It also
                                                                      implies to manipulate an unstable and high-order robot by
                                                                      dealing with a stable and lower order reference model. The
                                                                      parameters of the PID controller are self-adjusted in time to
                                                                      achieve the best performance. The adaptive system tolerates
                                                                      the parameter change and input/output disturbances.
                                                                      Conditions for output matching and output tracking between
                                                                      the system and reference model are derived. A matching
                                                                      system outputs to reference model output (the ideal
                                                                      trajectory) exist if the rank condition in (21) is met. The
                                                                      system outputs track the reference model outputs if the rank
                                                                      condition in (34) or (36) is satisfied. The stability of the
                                                                      adaptive control is ensured for asymptotic or BIBO output
                                                                      tracking if the sufficient conditions in (61)-(64) are true.
                                                                      Simulation shows that the adaptive PID control eliminates
                                                                      output tracking error with a satisfactory performance.

       Fig. 5.. Output Tracking with PID Controller                                          REFERENCES
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                              SU: A Model Reference-Based Adaptive PID Controller for Robot Motion Control of Not Explicitly Known Systems 244




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[12] J. Xu et al., “Application of optimal fuzzy PID controller design: PI                             Monmouth, New Jersey since 1998. From 1991 to
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      1, June 2006 pp. 3953 – 3957.                                             communication, signal and image processing, and adaptive control.
[13] K. Sobel et al., “Implicit adaptive control for a class of MIMO                  He is the recipient of Superior Civilian Service Award and Medals,
      systems,” IEEE Trans. On Aerospace and electronic Systems, Vol.           2005 Army Research and Development Achievement Award, Army
      AES-18No.5, pp.576-590, September 1982.                                   Material Command Top 10 Employee Nomination, 2004 and 2007 AOC
[14] I. Bar-Kana and H. Kaufman, “Global stability and performance of a         International Research and Development Awards, and 2002 Thomas Alva
      simplified adaptive control algorithm,” International Journal Control,    Edison Patent Award. He is also recognized by Army CERDEC for
      Vol.42, No.6, pp.1941-1505, 1985.                                         Inventor’s Wall of Honor and Electronic Warfare & Information
[15] W. Su and K. Sobel, “A unified theory for CGT approach to adaptive         Operations Association for Electronic Warfare Technology Hall of Fame.
      control,” International Journal of Control, Vol. 56, No. 1, pp.143-171,   He is a senior member of IEEE.
      1992.