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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 The test is conducted by applying a square wave reference [1] A. 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Kaufman et al. “Direct adaptive control algorithms theory and motion,” Proc. of the 37th IEEE Conference on Decision and Control, applications,” Springer-Verlag, 1993. 1998, Volume 1, 16-18 Dec. 1998, pp.186 – 191. [20] S. Ozcelik and H. Kaufman, “Design of MIMO robust direct model reference adaptive controllers,” Proceedings of the 36the Conference [6] Z. Gong et al., “A heuristic rule-based switching and adaptive PID on Decision & Control, December 1997, pp. 1890-1895. controller for a large autonomous tracked vehicle: from development [21] S. Ozcelik et al., “Implementation of direct adaptive control on 3-DOF to implementation,” Proc. of the 2004 IEEE International Conference spring-mass-damper system,” Proceedings of the American control on Control Applications, Volume 2, 2-4 Sept. 2004, pp.1272 – 1277. Conference, Jun 4-6, 2003, pp 3299-3304. [7] G. Zhenhai and Z. Bo; “Vehicle lane keeping of adaptive PID control [22] J. Y. Kim and J. Bentsman, “Robust model reference adaptive control with BP neural network self-tuning,” Proc. of the 2005 IEEE of parabolic and hyperbolic systems with spatially-varying Intelligent Vehicles Symposium, June 2005, pp. 84 – 87. parameters,” Proceedings of the 44th IEEE Conference on Decision [8] M. Zhang et al., “Adaptive PID control based on RBF neural network and Control, and the European control Conference 2005, December identification,” Proc. of the 17th IEEE International Conference on 12-15, 2005, pp. 1503-1508. Tools with Artificial Intelligence, 2005, Nov. 2005, pp. 3. [23] C. Rohrs, et al., “Robustness of adaptive control algorithms in the [9] P. Tsai et al., “The model reference control by adaptive PID-like presence of unmodeled dynamics,” Proc. of 21st IEEE Conference on fuzzy-neural controller,” Proc. of the 2005 IEEE International Decision and Control, 1982, pp.3-11. Conference on Systems, Man and Cybernetics, Vol. 1, Oct. 2005, pp. 239 - 244. Wei Su received the B.S degree in electrical [10] J. Wang et al., “Study of neuron adaptive PID controller in a single- engineering and the M.S degree in systems zone HVAC system,” Proc. of the First International Conference on engineering from Shanghai Jiao Tong University, Innovative Computing, Information and Control, 2006, Vol. 2, Aug. China, in 1983 and 1987, respectively. He received 2006, pp. 142 – 145. his Ph.D. degree in electrical engineering from The [11] X. Gao et al., “Simulation and research of fuzzy immune adaptive PID City University of New York, New York, in 1992. control in coke oven temperature control system,” Proc. of the Sixth He is a senior research engineer in U.S. Army World Congress on Intelligent Control and Automation, 2006, Vol. 1, Communication Electronics Research Development June 2006, pp. 3315 – 3319. and Engineering Center (CERDEC), at Fort [12] J. Xu et al., “Application of optimal fuzzy PID controller design: PI Monmouth, New Jersey since 1998. From 1991 to control for nonlinear induction motor,” Proc. of the Sixth World 1997, he was with US Army Research Laboratory at Congress on Intelligent Control and Automation, WCICA 2006, vol. Fort Monmouth, New Jersey. His research interests include wireless 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.

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