A Method to Teach the Parameterization of All Stabilizing Controllers

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1 Preprints of the 8th FAC World Congress Milano (taly) August 8 - September, A Method to Teach the Parameterization of All Stabilizing Controllers Vladimír Kučera* *Czech Technical University in Prague, Faculty of Electrical Engineering, Czech Republic (Tel: ; kucera@fel.cvut.cz). Abstract: A simple insightful method to teach the oula-kučera parameterization of all controllers that stabilize a given plant is presented. The text is intended for first-year graduate students in engineering. The result is derived first using transfer functions. A state-space representation of all stabilizing controllers then evolves from the transfer function result. Thus, the transfer functions the state-space techniques are presented as connected approaches rather than isolated alternatives. Keywords: Linear systems, feedback stabilization, oula-kučera parameterization, control education.. TROUCTO The parameterization of all controllers that stabilize a given plant, often called the oula-kučera parameterization, is a fundamental result of control theory. t launched an entirely new area of research found application, among others, in optimal robust control. n its original form, the result was obtained for finitedimensional, linear time-invariant systems using transfer function methods, see Larin et al. (97), Kučera (975), oula et al. (976a, b), Kučera (979). t was then generalized to cover time-varying systems by e.g. ale Smith (993), infinite dimensional systems by e.g. esoer et al. (98), Vidyasagar (985), Quadrat (3), a class of non-linear systems by e.g. ammer (985), Paice Moore (99), Anderson (998). Quite naturally, a state-space representation of all stabilizing controllers was derived for finite dimensional, linear time-invariant systems; see ett et al. (984). Modern textbooks on the subject include Antsaklis Michel (6) Colaneri et al. (997). This paper presents a simple insightful method to teach the parameterization result. t is intended for first-year graduate students in engineering. The transfer function result is derived first. A state-space representation of all stabilizing controllers then evolves from the transfer function result. Thus, the transfer functions the state-space techniques are presented as connected approaches rather than isolated alternatives.. TRASFER FUCTO APPROAC Stability is achieved by feedback. Consider systems S S connected in the feedback configuration shown in Fig.. The systems are assumed to be continuous-time, linear timeinvariant. The case of discrete-time systems is analogous. Stability is taken to mean that the states of S S go to zero from any initial condition as time increases. Given S, the task is to determine all systems S so that the feedback system, S, is stable. Thus S is called the plant S the stabilizing controller. For any given plant, the set of stabilizing controllers is either empty or infinite. Thus it is convenient to describe the set in parametric form. Suppose that the plant is described by state-space equations of the form x Ax + Bu, y Cx + u () or by the transfer function A B ( C( s A) B + :. () C The transfer function () is a proper rational matrix such that y u holds, where y, u denotes the one-sided Laplace transforms of y, u. ere below, the arguments of time functions as well as of their Laplace transforms are dropped, for the sake of brevity, when no ambiguity occurs. The special notation for the transfer function matrix that is introduced in () will be found useful when studying the relations between the state-space the transfer-function descriptions. Fig.. Feedback system S. The feedback system S is required to be well posed, that is to say, described by state-space equations of the same form as S S are. Then, the first observation is that the feedback system S is controllable from inputs r r observable from outputs y y if only if the constituent systems S S are both controllable observable. This can be seen from the eigenvalue criterion. As a consequence, stability of the feedback system can be studied using transfer functions. Copyright by the nternational Federation of Automatic Control (FAC) 6355

2 Preprints of the 8th FAC World Congress Milano (taly) August 8 - September, The transfer function matrix,, of the feedback system S that relates inputs r, r outputs y, y is given by, (3) where are the transfer function matrices of S S, respectively. This can easily be obtained from the signal relationships in Fig.. Thus, suppose for the present that S S are jointly controllable observable. The feedback system S is seen to be well posed if only if ( is proper (that is, analytic at s ). Then the feedback system S is stable if only if ( is stable (that is, analytic in Re s ).. Fractional escriptions The set of stabilizing controllers for a given plant may contain controllers whose transfer function is not proper. n order to isolate only those with proper transfer functions, it is convenient to express the transfer function of the plant in the form of a proper stable matrix fractional description. Thus, let, (4) where, is a pair of right coprime, proper stable rational matrices, while, is a pair of left coprime, proper stable rational matrices. Similarly, for the controller, let, (5) where, is a pair of left coprime, proper stable rational matrices, while, is a pair of right coprime, proper stable rational matrices. Then the closed-loop system transfer function is proper stable if only if U (6) or, (7) U where U U are unimodular proper stable rational matrices, that is, proper stable rational matrices whose inverses exist are proper stable rational. To prove this result, consider (6) first. t follows that U Thus, in view of (3), U U. U [ ] +. Since, are right coprime, seen to be proper stable if only if stable. ow consider (7). t follows that U Thus, in view of (3), U left coprime, is U is proper U. U [ ] +. Since, are right coprime, left coprime, is seen to be proper stable if only if U is proper stable. This completes the proof of conditions (6) (7) for closed-loop stability in terms of fractional descriptions.. Bézout dentity The proper stable rational matrices, being right coprime, there exist two other proper stable rational matrices, such that +. (8) The proper stable rational matrices, being left coprime, there exist two other proper stable rational matrices, such that +. (9) t follows from (4) that. () Finally, the matrices, can be selected in such a way as to satisfy. () ndeed, if Q for some pair, that satisfies (9), then the pair + Q, Q will satisfy both (9) (). Collecting identities (8), (9), (), (), one obtains the matrix Bézout identity 6356

3 Preprints of the 8th FAC World Congress Milano (taly) August 8 - September,. () For the present, we assume that the plant S the controller S are jointly controllable observable. This proviso makes it possible to refer to the transfer function results..3 Parameterization of Stabilizing Controllers ote that () defines one stabilizing controller, namely. Any all controllers having proper transfer function, with the property that the feedback system S is well posed stable, are now provided in parametric form. Theorem. Let be coprime, proper stable matrix fractional descriptions. Let,, be proper stable rational matrices that satisfy the Bézout identity (). Then all proper that render the closed-loop feedback system well posed stable are given by ( + W) ( W ) (3) ( W )( + W ), where the parameter, W, is a proper stable rational matrix such that ( + W or ( + W exists is proper. ) ) Proof. All proper stabilizing are generated by all solutions of equation (6), written in the form ( U ) + ( U ). The solution set can be parameterized as U + W, U W, where W is an arbitrary proper stable rational matrix. This follows by direct substitution the use of (8), (). Thus, in view of (5), all proper stabilizing are given by ( U ) ( U ) ( + W) ( W ). Alternatively, all proper stabilizing are generated by all solutions of equation (7), written in the form ( U ) + ( U ). The solution set can be parameterized as U + W, U, W where W is an arbitrary proper stable rational matrix. This follows by direct substitution the use of (9), (). Thus, in view of (5), all proper stabilizing are given by. ( U )( U ) ( W )( + W ) W W (3) holds. This completes the ow () implies proof. 3. STATE SPACE APPROAC The best way of presenting the oula-kučera parameterization in state space is to follow the three steps of the transfer function approach translate each into the state-space parlance. This provides a connected result rather than two isolated alternatives. 3. Fractional escription Given the state-space description () of the plant to be stabilized, we seek appropriate stable systems, each giving rise to a pair of transfer functions that defines the desired fractional description. This can be achieved by applying a stabilizing state feedback a stabilizing output injection. To obtain a right fractional description for, consider any stabilizing state feedback u Fx + r around the plant. The resulting equations x ( A + BF) x + Br u Fx + r y ( C + F) x + r define a stable system with one input, r, two outputs, u y. enote A + BF B : F the transfer function between r u denote A + BF B : C + F the transfer function between r y. Then are proper stable rational matrices it holds y, r u r. (4) t follows that y u u so that, is a right matrix fractional description for. To obtain a left fractional description for, consider the dual situation, namely a state observer for the plant based on any stabilizing output injection Ke, where e y ( Cx + u) is the difference between the actual output the estimated output. The resulting equations x ( A KC) x + ( B K) u + Ky e y ( Cx + u) define a stable system with two inputs, u y, one output, e. enote A KC K : C the transfer function between y e denote A KC B K : C the transfer function between u e. Then are proper stable rational matrices it holds 6357

4 Preprints of the 8th FAC World Congress Milano (taly) August 8 - September, e y u ( ) r (5) because e is independent of r. t follows that so that, is a left matrix fractional description for. 3. Bézout dentity The matrix fractional descriptions for the controller can be constructed in a like fashion. To obtain a right fractional description for, consider a stabilizing state feedback u Fx around the observer with output y. The resulting equations x ( A + BF) x + Ke u Fx y ( C + F) x + e define a stable system with one input, e, two outputs, u y. enote A + BF K : (6) C + F the transfer function between e y denote A + BF K : (7) F the transfer function between e u. Then are proper stable rational matrices it holds û ê, ŷ ê. (8) t follows that û ŷ ŷ so that, is a right matrix fractional description for. To obtain a left fractional description for, consider a stabilizing state feedback u Fx + r around the observer with output r. The resulting equations x ( A KC) x + ( B K) u + Ky r Fx + u define a stable system with two inputs, u y, one output, r. enote A KC B K : (9) F the transfer function between u r denote A KC K : () F the transfer function between y r. Then are proper stable rational matrices it holds r ( ) y + u e () as r is independent of e. t follows that so that, is a left matrix fractional description for. Collecting equations (4), (5), (8), (), one obtains in matrix form r u () e y u r. (3) y e Thus, the Bézout identity () holds each of the four fractional descriptions involved is actually coprime. 3.3 Parameterization of Stabilizing Controllers Putting r in () (3), it follows that (4) is one stabilizing controller for. Thus it is plausible that involving a parameter in generating r one obtains all stabilizing controllers. Accordingly, we put r We. (5) Then () implies that [ r W ] [ W ] e ( + W ) u + ( W ) y u y while (3) implies that u W W e ( ) e. y ( + W ]) e ence the parameterization formula (3) follows ( + W) ( W ) ( W ) ( + W ). This shows that the parameter W introduced in (5) does generate any all stabilizing controllers. 3.4 State-Space Representation of Stabilizing Controllers A state-space representation of the stabilizing controller (4), which corresponds to r, is inferred from (6), (7) or (9), (). t consists of an asymptotic state observer a stabilizing state feedback from the estimated state. The entire 6358

5 Preprints of the 8th FAC World Congress Milano (taly) August 8 - September, set of stabilizing controllers is then generated by putting r We as follows, x Ax + Bu + K [ y ( Cx + u)] u Fx + W ( q)[ y ( Cx + u)], where the parameter W is an arbitrary proper stable rational matrix in the differential operator q : d / dt is such that the feedback system is well posed. A state-space representation of all stabilizing controllers is shown in Fig.. by s, ( (. s + s + The Bézout identity is completed by solving the equations + + to obtain, Thus,. s ( s ) W + W, s + s + where W is an arbitrary proper stable rational function. ote that the indicated inverse exists is proper for any W. For example, W implies. Following the state-space approach, we first construct an asymptotic state observer Fig.. A state-space representation of all stabilizing controllers. Thus, every stabilizing controller can be viewed as a combination of an asymptotic state observer a stabilizing feedback from the estimated state, plus an additional system that depends on the parameter W. The order of any stabilizing controller is the order of the plant plus the order of a statespace realization of W. Thus the set of stabilizing controllers contains controllers of arbitrarily high order. The initial assumption of controllability observability for the plant the controller can be relaxed. f the plant is uncontrollable /or unobservable, then the closed-loop system has an uncontrollable /or unobservable eigenvalue. Such an eigenvalue is invariant under state feedback /or output injection. Thus, for all such eigenvalues to be stable, the plant must be stabilizable detectable. This is a necessary sufficient condition for the stabilization of a plant given by () using the feedback configuration of Fig.. t is to be noted that a stabilizing controller can also turn uncontrollable /or unobservable provided it is jointly stabilizable detectable. x u + K( y x) (6) with an output injection gain K such that A KC K is stable, hence K >. We then apply a stabilizing state feedback u Fx + r with a gain F such that A + BF F is stable, hence F <. We complete the design of all stabilizing controllers that have a proper transfer function by adding a term that depends on the parameter, W, as follows u Fx + W ( q)( y x), (7) where W is an arbitrary proper stable rational function in the differential operator q. ote that the closed-loop system is well posed for any W. For example, W implies F K ( s ) F K. The stabilizing controllers described by (6) (7) are shown in Fig EAMPLE Simple examples are most illustrative. Consider an integrator plant described by the equations x u, y x, which correspond to () with A, B, C,. The transfer function of the plant is. s etermine all stabilizing controllers with proper transfer function using the transfer-function approach. Proper stable fractional descriptions of are provided, for example, Fig. 3. The set of all controllers that stabilize an integrator. Any stabilizing controller in the set has order at least one. The McMillan degree of its transfer function may be lower, however, in case the controller is uncontrollable /or unobservable. For example, the parameter 6359

6 Preprints of the 8th FAC World Congress Milano (taly) August 8 - September, q + K F + KF W ( q) q + yields the controller F + K K ( ) ( ) + K F + KF K F + KF F + that is uncontrollable unobservable, of McMillan degree zero. t corresponds to the controller effected byw. On the other h, if it is of interest to obtain the controller that corresponds to W in (5) by employing transfer function techniques, put s, ( s, (. s + K s + K Complete the Bézout identity while enforcing strictly proper. n this case, s + ( K F) KF, s + K s + K s + ( K F) KF, Then. s + ( K F) ( s + K KF. s + ( K F) 5. COCLUSOS KF KF s + ( K F) s + K The transfer function approach to determining all controllers with proper transfer function that stabilize a given plant consists of three steps: (i) determine coprime, proper stable fractional descriptions for, (ii) solve a Bézout equation to obtain one stabilizing controller, (iii) form the transfer functions of all stabilizing controllers using a parameter that is a proper stable rational matrix. The state space approach to determining all controllers S with proper transfer function that stabilize a given plant S consists of two steps: (i) obtain one stabilizing controller by constructing an asymptotic state observer for S applying a stabilizing state feedback from the estimated state, (ii) form all stabilizing controllers S by connecting to the initial controller a parameter that is a system with proper stable rational transfer function. Thus, in the state space approach, there is no need to construct proper stable fractional descriptions for the plant, or to solve the Bézout equation. The observer-based stabilizing controller is given directly by stabilizing gains F K. A particular selection of F K then corresponds to a choice of a particular proper stable fractional description used in the transfer-function approach. The transfer function result is the result of reference it is amenable to generalizations for time-varying, infinitedimensional, nonlinear systems. The state space result is simple useful in the design of control systems. ACKOWLEGMET Supported by the Ministry of Education of the Czech Republic, Project M567 Center for Applied Cybernetics. REFERECES Anderson, B..O. (998). From oula-kucera to identification, adaptive nonlinear control. Automatica, 34, Antsaklis, P.J. Michel, A.. (6). Linear Systems. Birkhäuser, Boston, MA. Colaneri, P., Geromel, J.C., Locatelli, A. (997). Control Theory esign: An R R Viewpoint. Academic Press, San iego, CA. ale, W.., Smith, M.C. (993). Stabilizability existence of system representations for discrete-time time-varying systems. SAM J. Control Optimization, 3, esoer, C.A., Liu, R.W., Murray, J., Saeks, R. (98). Feedback system design: The fractional representation approach to analysis synthesis. EEE Trans. Automatic Control, 5, ammer, J. (985). onlinear system stabilization coprimeness. nt. Journal of Control, 44, Kučera, V. (975). Stability of discrete linear control systems. n: Proc. 6th FAC World Congress, paper 44.. Boston, MA. Kučera, V. (979). iscrete Linear Control: The Polynomial Equation Approach. Wiley, Chichester, UK. Larin, V.B., aumenko, K.., Suntsev, V.. (97). Spectral Methods for Synthesis of Linear Systems with Feedback (in Russian). aukova umka, Kiev, Ukraine. ett, C.., Jacobson, C.A., Balas, M.J. (984). A connection between state-space doubly coprime fractional representations. EEE Trans. Automatic Control, 9, Paice, A..B., Moore, J.B. (99). On the oula-kucera parametrization of nonlinear systems. Systems & Control Letters, 4, -9. Quadrat, A. (3). On a generalization of the oula-kučera parametrization. Part : The fractional ideal approach to SSO systems. Systems & Control Letters, 5, Vidyasagar, M. (985). Control System Synthesis: A Factorization Approach. MT Press, Cambridge, MA. oula,.c., Bongiorno, J.J., Jabr,.A. (976a). Modern Wiener-opf design of optimal controllers, Part : The single-input case. EEE Trans. Auto. Control,, 3-4. oula,.c., Jabr,.A., Bongiorno, J.J. (976b). Modern Wiener-opf design of optimal controllers, Part : The multivariable case. EEE Trans. Automatic Control,,